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User:IntegralPython

2024-11-05T18:48:18Z

IntegralPython: remove personal info

[[File:WikiProject Mathematics AD.gif|center]]
<table style="float: right; margin-left: 1em; margin-bottom: 0.5em; width: 250px; border: #99B3FF solid 1px">
<tr><td>{{Template:User WP Mathematics}}</td>
<td>{{Template:User interest mathematics}}</td></tr>
</table>

==My Articles==
*[[Quota rule]]
*[[Hand eye calibration problem]]
*[[Open set condition]]
===Other articles===
Articles I have spent a significant amount of effort on
*[[Genetic use restriction technology]]
*[[internet meme]]
*[[tetration]]

==Helpful links==
*[[User:IntegralPython/sandbox|My Sandbox]]
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===Math links===
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==Uploaded Pictures==
[[File:Koch Snowflake.svg|200px]]
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[[File:Pentation.jpg|200px]]
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Inscribed square problem

2024-07-16T01:09:26Z

IntegralPython: /* Lipschitz graphs */ Added recent improvement of the result

{{Short description|Unsolved problem about inscribing a square in a Jordan curve}}
{{unsolved|mathematics|Does every [[Jordan curve]] have an inscribed square?}}
[[Image:Inscribed square.svg|thumb|right|Example: The black dashed curve goes through all corners of several blue squares.]]
The '''inscribed square problem''', also known as the '''square peg problem''' or the '''Toeplitz' conjecture''', is an unsolved question in [[geometry]]: ''Does every [[Jordan curve|plane simple closed curve]] contain all four vertices of some [[Square (geometry)|square]]?'' This is true if the curve is [[convex set|convex]] or piecewise [[Smooth function|smooth]] and in other special cases. The problem was proposed by [[Otto Toeplitz]] in 1911.<ref>{{citation | last= Toeplitz | first = O. | authorlink = Otto Toeplitz | title = Über einige Aufgaben der Analysis situs | journal = Verhandlungen der Schweizerischen Naturforschenden Gesellschaft | volume = 94 | date = 1911 | page = 197 | language = de }}</ref> Some early positive results were obtained by [[Arnold Emch]]<ref name="Emch">{{citation |last=Emch |first=Arnold | authorlink=Arnold Emch |year=1916 |title=On some properties of the medians of closed continuous curves formed by analytic arcs |journal=American Journal of Mathematics |doi=10.2307/2370541 |mr=1506274 |volume=38 |issue=1 |pages=6–18|jstor=2370541 }}</ref> and [[Lev Schnirelmann]].<ref name="Schnirelmann 1944">{{citation |last=Šnirel'man |first=L. G. |author-link=Lev Schnirelmann |year=1944 |title=On certain geometrical properties of closed curves |journal=Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk |mr=0012531 |volume=10 |pages=34–44}}</ref> The general case remains open.<ref name=hartnett/>

== Problem statement ==
Let <math>C</math> be a [[Jordan curve]]. A [[polygon]] <math>P</math> is '''inscribed in <math>C</math> ''' if all vertices of <math>P</math> belong to <math>C</math>. The '''inscribed square problem''' asks:

: ''Does every Jordan curve admit an inscribed square?''

It is ''not'' required that the vertices of the square appear along the curve in any particular order.

==Examples==
Some figures, such as [[circle]]s and [[Square (geometry)|square]]s, admit infinitely many [[inscribed]] squares. If <math>C</math> is an [[obtuse triangle]] then it admits exactly one inscribed square; right triangles admit exactly two, and acute triangles admit exactly three.<ref>{{citation | last1 = Bailey | first1 = Herbert | last2 = DeTemple | first2 = Duane | title = Squares inscribed in angles and triangles | journal = [[Mathematics Magazine]] | volume = 71 | issue = 4 | date = 1998 | pages = 278–284 | doi=10.2307/2690699| jstor = 2690699 }}</ref>

== Resolved cases ==
It is tempting to attempt to solve the inscribed square problem by proving that a special class of well-behaved curves always contains an inscribed square, and then to approximate an arbitrary curve by a sequence of well-behaved curves and infer that there still exists an inscribed square as a [[limit (mathematics)|limit]] of squares inscribed in the curves of the sequence. One reason this argument has not been carried out to completion is that the limit of a sequence of squares may be a single point rather than itself being a square. Nevertheless, many special cases of curves are now known to have an inscribed square.<ref name="matschke">{{citation |last=Matschke |first=Benjamin |year=2014 |title=A survey on the square peg problem |journal=[[Notices of the American Mathematical Society]] |doi=10.1090/noti1100 |volume=61 |issue=4 |pages=346–352|doi-access=free }}</ref>

===Piecewise analytic curves===
{{harvs|first=Arnold|last=Emch|authorlink=Arnold Emch|year=1916|txt}} showed that [[piecewise]] [[analytic curve]]s always have inscribed squares. In particular this is true for [[polygon]]s. Emch's proof considers the curves traced out by the [[midpoint]]s of [[Secant line|secant]] [[line segments]] to the curve, parallel to a given line. He shows that, when these curves are intersected with the curves generated in the same way for a perpendicular family of secants, there are an odd number of crossings. Therefore, there always exists at least one crossing, which forms the center of a [[rhombus]] inscribed in the given curve. By rotating the two perpendicular lines continuously through a [[right angle]], and applying the [[intermediate value theorem]], he shows that at least one of these rhombi is a square.<ref name="matschke"/>

=== Locally monotone curves ===
[[Walter Stromquist|Stromquist]] has proved that every ''local monotone'' plane simple curve admits an inscribed square.<ref name="stromquist">{{citation |last=Stromquist |first=Walter |year=1989 |title=Inscribed squares and square-like quadrilaterals in closed curves |journal = [[Mathematika]]|mr=1045781|doi=10.1112/S0025579300013061|volume=36 |issue=2|pages= 187–197}}</ref> The condition for the admission to happen is that for any point {{mvar|p}}, the curve {{mvar|C}} should be locally represented as a graph of a function <math>y=f(x)</math>.

In more precise terms, for any given point <math>p</math> on <math>C</math>, there is a neighborhood <math>U(p)</math> and a fixed direction <math>n(p)</math> (the direction of the “<math>y</math>-axis”) such that no [[Chord (geometry)|chord]] of <math>C</math> -in this neighborhood- is parallel to <math>n(p)</math>.

Locally monotone curves include all types of [[polygon]]s, all closed [[convex set|convex]] curves, and all piecewise
[[Smooth function#Differentiability classes|<math>C^1</math>]] curves without any [[Cusp (singularity)|cusp]]s.

===Curves without special trapezoids===
An even weaker condition on the curve than local monotonicity is that, for some <math>\varepsilon>0</math>, the curve does not have any inscribed special trapezoids of size <math>\varepsilon</math>. A special trapezoid is an [[isosceles trapezoid]] with three equal sides, each longer than the fourth side, inscribed in the curve with a vertex ordering consistent with the clockwise ordering of the curve itself. Its size is the length of the part of the curve that extends around the three equal sides. Here, this length is measured in the domain of a fixed [[Parametrization_(geometry)|parametrization]] of <math>C</math>, as <math>C</math> may not be [[rectifiable curve|rectifiable]]. Instead of a limit argument, the proof is based on relative [[obstruction theory]]. This condition is open and dense in the space of all Jordan curves with respect to the [[compact-open topology]]. In this sense, the inscribed square problem is solved for [[Generic property#In topology|generic]] curves.<ref name="matschke"/>

===Curves in annuli===
If a Jordan curve is inscribed in an [[Annulus (mathematics)|annulus]] whose outer radius is at most <math>1+\sqrt{2}</math> times its inner radius, and it is drawn in such a way that it separates the inner circle of the annulus from the outer circle, then it contains an inscribed square. In this case, if the given curve is approximated by some well-behaved curve, then any large squares that contain the center of the annulus and are inscribed in the approximation are topologically separated from smaller inscribed squares that do not contain the center. The limit of a sequence of large squares must again be a large square, rather than a degenerate point, so the limiting argument may be used.<ref name="matschke"/>

===Symmetric curves===
The affirmative answer is also known for centrally symmetric curves, even [[Fractal|fractals]] such as the [[Koch snowflake]], and curves with reflective symmetry across a line.<ref name="nielsen-wright">{{citation |last1=Nielsen |first1=Mark J. |last2=Wright |first2=S. E. |year=1995 |title=Rectangles inscribed in symmetric continua |journal=[[Geometriae Dedicata]] |mr=1340790 |doi=10.1007/BF01263570 | doi-access=free|volume=56 |issue=3 |pages=285–297}}</ref>

===Lipschitz graphs===
In 2017, [[Terence Tao]] published a proof of the existence of a square in curves formed by the union of the [[graph of a function|graphs of two functions]], both of which have the same value at the endpoints of the curves and both of which obey a [[Lipschitz continuity]] condition with Lipschitz constant less than one. Tao also formulated several related conjectures.<ref>{{citation
| last = Tao | first = Terence
| doi = 10.1017/fms.2017.23
| journal = Forum of Mathematics
| mr = 3731730
| page = e30
| title = An integration approach to the Toeplitz square peg problem
| volume = 5
| year = 2017| doi-access = free
}}; see also [https://terrytao.wordpress.com/2016/11/22/an-integration-approach-to-the-toeplitz-square-peg-problem/ Tao's blog post on the same set of results]</ref>
In 2024, Joshua Greene and Andrew Lobb published a preprint improving this result to curves with Lipschitz constant less than <math>1 + \sqrt{2}</math>. <ref>{{cite arXiv
| last1 = Greene
| first1 = Joshua
| last2 = Lobb
| first2 = Andrew
| date = 2024
| title = Square pegs between two graphs
| eprint = 2407.07798
}}
</ref>

=== Jordan curves close to a <math>C^2</math> Jordan curve ===
In March 2022, Gregory R. Chambers showed that if <math>\gamma</math> is a Jordan curve which is close to a <math>C^2</math> Jordan curve <math>\beta</math> in <math>\mathbb{R}^2</math>, then <math>\gamma</math> contains an inscribed square. He showed that, if <math>\kappa>0</math> is the maximum unsigned curvature of <math>\beta</math> and there is a map <math>f</math> from the image of <math>\gamma</math> to the image of <math>\beta</math> with <math>\|f(x)-x\|<1/10\kappa</math> and <math>f\circ\gamma</math> having [[winding number]] <math>1</math>, then <math>\gamma</math> has an inscribed square of positive sidelength.<ref>{{Cite arXiv |last=Chambers |first=Gregory |date=March 2022 |title=On the square peg problem |class=math.GT |eprint=2203.02613 }}</ref>

== Variants and generalizations ==
One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle <math>T</math> and Jordan curve <math>C</math>, there is a triangle similar to <math>T</math> and inscribed in <math>C</math>.<ref name="meyerson">{{citation |last=Meyerson |first=Mark D. |year=1980 |title=Equilateral triangles and continuous curves |journal=Fundamenta Mathematicae |mr=600575 |volume=110 |issue=1 |pages=1–9 |doi=10.4064/fm-110-1-1-9|doi-access=free }}</ref><ref>{{citation |last1=Kronheimer |first1=E. H. |last2=Kronheimer |first2=P. B. |author2-link=Peter B. Kronheimer |year=1981 |title=The tripos problem |journal=Journal of the London Mathematical Society |series=Second Series |mr=623685 |doi=10.1112/jlms/s2-24.1.182 |volume=24 |issue=1 |pages=182–192}}</ref> Moreover, the set of the vertices of such triangles is [[dense set|dense]] in <math>C</math>.<ref>{{citation |last=Nielsen |first=Mark J. |year=1992 |title=Triangles inscribed in simple closed curves |journal=[[Geometriae Dedicata]] |mr=1181760 |doi=10.1007/BF00151519 | doi-access=free |volume=43 |issue=3 |pages=291–297}}</ref> In particular, there is always an inscribed [[equilateral triangle]].

It is also known that any Jordan curve admits an inscribed [[rectangle]]. This was proved by Vaughan by reducing the problem to the non-embeddability of the [[projective plane]] in <math>\mathbb{R}^3</math>; his proof from around 1977 is published in Meyerson.<ref>{{citation |last=Meyerson |first=Mark D. |year=1981 |title=Balancing acts |journal=Topology Proceedings |volume=6 |issue=1 |pages=71 |url=http://topology.nipissingu.ca/tp/reprints/v06/tp06107.pdf |access-date=2023-10-06 }}</ref>
In 2020, Morales and Villanueva characterized locally connected plane continua that admit at least one inscribed rectangle.<ref name="morales-villanueva">{{citation |last1=Morales-Fuentes |first1=Ulises |last2=Villanueva-Segovia |first2=Cristina |title=Rectangles Inscribed in Locally Connected Plane Continua |journal=Topology Proceedings |date=2021 |volume=58 |pages=37–43}}</ref> In 2020, Joshua Evan Greene and Andrew Lobb proved that for every smooth Jordan curve <math>C</math> and rectangle <math>R</math> in the Euclidean plane there exists a rectangle similar to <math>R</math> whose vertices lie on <math>C</math>.<ref name=hartnett>{{citation|last=Hartnett|first=Kevin|title=New geometric perspective cracks old problem about rectangles|url=https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/|access-date=2020-06-26|magazine=Quanta Magazine|date=June 25, 2020}}</ref><ref>{{citation |last1=Greene |first1=Joshua Evan |last2=Lobb |first2=Andrew |title=The rectangular peg problem |journal=Annals of Mathematics |date=September 2021 |volume=194 |issue=2 |pages=509–517 |doi=10.4007/annals.2021.194.2.4|arxiv=2005.09193|s2cid=218684701 }}</ref><ref>{{Cite journal|last=Schwartz|first=Richard Evan|date=2021-09-13|title=Rectangles, curves, and Klein bottles|url=https://www.ams.org/bull/2022-59-01/S0273-0979-2021-01755-1/|journal=Bulletin of the American Mathematical Society|language=en|volume=59|issue=1|pages=1–17|doi=10.1090/bull/1755|issn=0273-0979|doi-access=free}}</ref> This generalizes both the existence of rectangles (of arbitrary shape) and the existence of squares on smooth curves, which has been known since the work of {{harvtxt|Šnirel'man|1944}}.<ref name="Schnirelmann 1944"/> In 2021, Green and Lobb extended their 2020 result and proved that every smooth Jordan curve inscribes every cyclic quadrilateral (modulo an orientation-preserving similarity).<ref>{{Cite journal |last=Greene |first=Joshua Evan |last2=Lobb |first2=Andrew |date=2023 |title=Cyclic quadrilaterals and smooth Jordan curves |url=https://link.springer.com/10.1007/s00222-023-01212-6 |journal=Inventiones mathematicae |language=en |volume=234 |issue=3 |pages=931–935 |doi=10.1007/s00222-023-01212-6 |issn=0020-9910}}</ref>

Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general [[continuum (topology)|continua]] in higher dimensional [[Euclidean space]]s. For example, Stromquist proved that every continuous closed curve <math>C</math> in <math>\mathbb{R}^n</math> satisfying "Condition A" that no two chords of <math>C</math> in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals.<ref name="stromquist"/> This class of curves includes all <math>C^2</math> curves. Nielsen and Wright proved that any symmetric continuum <math>K</math> in <math>\mathbb{R}^n</math> contains many inscribed rectangles.<ref name="nielsen-wright"/>

== References ==
{{reflist|colwidth=30em}}

== Further reading ==
*{{citation|last1=Klee|first1=Victor|author1-link=Victor Klee|last2=Wagon|first2=Stan|author2-link=Stan Wagon|contribution=Inscribed squares|isbn=978-0-88385-315-3|pages=58–65, 137–144|publisher=Cambridge University Press|series=The Dolciani Mathematical Expositions|title=Old and New Unsolved Problems in Plane Geometry and Number Theory|volume=11|year=1991}}

== External links ==
* Mark J. Nielsen, [http://www.webpages.uidaho.edu/~markn/squares/ Figures Inscribed in Curves. A short tour of an old problem]
* [http://quomodocumque.wordpress.com/2007/08/31/inscribed-squares-denne-speaks/ Inscribed squares: Denne speaks] at Jordan Ellenberg's blog
* Grant Sanderson, [https://www.youtube.com/watch?v=AmgkSdhK4K8 Who cares about topology? (Inscribed rectangle problem)], [[3Blue1Brown]], YouTube a – video showing a topological solution to a simplified version of the problem.

[[Category:Curves]]
[[Category:Unsolved problems in geometry]]

Function of several complex variables

2024-07-04T04:15:15Z

IntegralPython: /* Sheaf */ fixed several issues with language (perhaps translated?) and also some formatting. I also removed some unnecessary subsectioning into non-sheaf and sheaf definitions since the sections were short as is and it was easy to make it clear via prose when the sheaves were used.

{{Short description|Type of mathematical functions}}
{{Use American English|date = March 2019}}

The theory of '''functions of several complex variables''' is the branch of [[mathematics]] dealing with functions defined on [[#The complex coordinate space|the complex coordinate space]] <math>\mathbb C^n</math>, that is, {{mvar|n}}-tuples of [[complex number]]s. The name of the field dealing with the properties of these functions is called '''several complex variables''' (and [[analytic space]]), which the [[Mathematics Subject Classification]] has as a top-level heading.

As in [[complex analysis|complex analysis of functions of one variable]], which is the case {{math|1=''n'' = 1}}, the functions studied are ''[[holomorphic function|holomorphic]]'' or ''complex analytic'' so that, locally, they are [[power series]] in the variables {{mvar|zi}}. Equivalently, they are locally [[uniform convergence|uniform limits]] of [[polynomial]]s; or locally [[Lp space|square-integrable]] solutions to the {{mvar|n}}-dimensional [[#Holomorphic functions|Cauchy–Riemann equations]].<ref name="Hörmander1965">{{cite journal |doi=10.1007/BF02391775|title=L2 estimates and existence theorems for the <math>\bar \partial</math> operator |year=1965 |last1=Hörmander |first1=Lars |journal=Acta Mathematica |volume=113 |pages=89–152 |s2cid=120051843 |doi-access=free }}</ref><ref>{{cite book |isbn=978-1-4704-4636-9|title=Analysis of Several Complex Variables|last1=Ohsawa|first1=Takeo|year=2002|url={{Google books|IXhoWbo1oCkC|title=Analysis of Several Complex Variables|page=18-IA8|plainurl=yes}}}}</ref><ref name =Błocki2014>{{cite journal |doi=10.1007/s13373-014-0058-2|title=Cauchy–Riemann meet Monge–Ampère|year=2014|last1=Błocki|first1=Zbigniew|journal=Bulletin of Mathematical Sciences|volume=4|issue=3|pages=433–480|s2cid=53582451|doi-access=free}}</ref> For one complex variable, every [[Domain (mathematical analysis)|domain]]<ref name="domain" group="note">That is an [[open set|open]] [[connected space|connected]] [[subset]].
</ref>(<math>D \subset \mathbb C</math>), is the [[domain of holomorphy]] of some function, in other words every domain has a function for which it is the domain of holomorphy.<ref name=Siu1978>{{cite journal |doi=10.1090/S0002-9904-1978-14483-8|title=Pseudoconvexity and the problem of Levi|year=1978|last1=Siu|first1=Yum-Tong|journal=Bulletin of the American Mathematical Society|volume=84|issue=4|pages=481–513|mr=0477104|doi-access=free}}</ref><ref name="Chen2000">{{cite journal |last1=Chen |first1=So-Chin |title=Complex analysis in one and several variables |journal=Taiwanese Journal of Mathematics |year=2000 |volume=4 |issue=4 |pages=531–568 |doi=10.11650/twjm/1500407292|doi-access=free|zbl=0974.32001|jstor=43833225|mr=1799753}}</ref> For several complex variables, this is not the case; there exist domains (<math>D \subset \mathbb C^n,\ n \geq 2</math>) that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field.{{R|Siu1978}} Patching the local data of [[meromorphic function]]s, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and [[Projective variety#Complex projective varieties|complex projective varieties]] (<math>\mathbb{CP}^n</math>)<ref name="IWS">{{cite journal |last1=Chong |first1=C.T. |last2=Leong |first2=Y.K. |title=An interview with Jean-Pierre Serre |journal=The Mathematical Intelligencer |year=1986 |volume=8 |issue=4 |pages=8–13|doi=10.1007/BF03026112|s2cid=121138963 }}</ref> and has a different flavour to complex analytic geometry in <math>\mathbb{C}^n</math> or on [[Stein manifold]]s, these are much similar to study of algebraic varieties that is study of the [[algebraic geometry]] than complex analytic geometry.

== Historical perspective ==

Many examples of such functions were familiar in nineteenth-century mathematics; [[abelian variety|abelian functions]], [[theta function]]s, and some [[hypergeometric series]], and also, as an example of an inverse problem; the [[Jacobi inversion problem]].<ref name=":0">{{cite book |doi=10.1007/978-3-642-20554-5_5|chapter=Analytic Functions of Several Complex Variables |title=Complex Analysis 2 |series=Universitext |year=2011 |last1=Freitag |first1=Eberhard |pages=300–346 |isbn=978-3-642-20553-8 }}</ref> Naturally also same function of one variable that depends on some complex [[parameter]] is a candidate. The theory, however, for many years didn't become a full-fledged field in [[mathematical analysis]], since its characteristic phenomena weren't uncovered. The [[Weierstrass preparation theorem]] would now be classed as [[commutative algebra]]; it did justify the local picture, [[Ramification (mathematics)|ramification]], that addresses the generalization of the [[branch point]]s of [[Riemann surface]] theory.

With work of [[Friedrich Hartogs]], {{ill|Pierre Cousin (mathematician)|lt=Pierre Cousin|fr|Pierre Cousin (mathématicien)}}, [[E. E. Levi]], and of [[Kiyoshi Oka]] in the 1930s, a general theory began to emerge; others working in the area at the time were [[Heinrich Behnke]], [[Peter Thullen]], [[Karl Stein (mathematician)|Karl Stein]], [[Wilhelm Wirtinger]] and [[Francesco Severi]]. Hartogs proved some basic results, such as every [[isolated singularity]] is [[removable singularity|removable]], for every analytic function
<math display=block>f : \mathbb C^n \to \Complex</math>
whenever {{math|1=''n'' > 1}}. Naturally the analogues of [[Line integral|contour integrals]] will be harder to handle; when {{math|1=''n'' = 2}} an integral surrounding a point should be over a three-dimensional [[manifold]] (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a [[double integral]] over a two-dimensional surface. This means that the [[residue calculus]] will have to take a very different character.

After 1945 important work in France, in the seminar of [[Henri Cartan]], and Germany with [[Hans Grauert]] and [[Reinhold Remmert]], quickly changed the picture of the theory. A number of issues were clarified, in particular that of [[analytic continuation]]. Here a major difference is evident from the one-variable theory; while for every open connected set ''D'' in <math>\Complex</math> we can find a function that will nowhere continue analytically over the boundary, that cannot be said for {{math|''n'' > 1}}. In fact the ''D'' of that kind are rather special in nature (especially in complex coordinate spaces <math>\mathbb C^n</math> and Stein manifolds, satisfying a condition called ''[[#Pseudoconvexity|pseudoconvexity]]''). The natural domains of definition of functions, continued to the limit, are called ''[[Stein manifold]]s'' and their nature was to make [[sheaf cohomology]] groups vanish, on the other hand, the [[Grauert–Riemenschneider vanishing theorem]] is known as a similar result for compact complex manifolds, and the Grauert–Riemenschneider conjecture is a special case of the conjecture of Narasimhan.{{R|Siu1978}} In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for [[algebraic geometry]], in particular from Grauert's work).

From this point onwards there was a foundational theory, which could be applied to [[Algebraic geometry#Analytic geometry|analytic geometry]], {{refn|group=note|A name adopted, confusingly, for the geometry of [[Complex analytic variety|zeroes of analytic functions]]; this is not the [[analytic geometry]] learned at school. (In other words, in the sense of GAGA on Serre.)<ref name="GAGA">{{cite journal | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Géométrie algébrique et géométrie analytique | url=http://www.numdam.org/numdam-bin/item?id=AIF_1956__6__1_0 | mr=0082175 | year=1956 | journal=[[Annales de l'Institut Fourier]] | issn=0373-0956 | volume=6 | pages=1–42 | doi=10.5802/aif.59| doi-access=free |language=fr|zbl=0075.30401}}</ref>}} [[automorphic form]]s of several variables, and [[partial differential equation]]s. The [[complex structure deformation|deformation theory of complex structures]] and [[complex manifold]]s was described in general terms by [[Kunihiko Kodaira]] and [[D. C. Spencer]]. The celebrated paper ''[[GAGA]]'' of [[Jean-Pierre Serre|Serre]]{{R|GAGA|}} pinned down the crossover point from ''géometrie analytique'' to ''géometrie algébrique''.

[[C. L. Siegel]] was heard to complain that the new ''theory of functions of several complex variables'' had few ''functions'' in it, meaning that the [[special function]] side of the theory was subordinated to sheaves. The interest for [[number theory]], certainly, is in specific generalizations of [[modular form]]s. The classical candidates are the [[Hilbert modular form]]s and [[Siegel modular form]]s. These days these are associated to [[algebraic group]]s (respectively the [[Weil restriction]] from a [[totally real number field]] of {{math|[[general linear group|GL]](2)}}, and the [[symplectic group]]), for which it happens that [[automorphic representation]]s can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.

Subsequent developments included the [[hyperfunction]] theory, and the [[edge-of-the-wedge theorem]], both of which had some inspiration from [[quantum field theory]]. There are a number of other fields, such as [[Banach algebra]] theory, that draw on several complex variables.

== The complex coordinate space ==
The [[complex coordinate space]] <math>\mathbb C^n</math> is the [[Cartesian product]] of {{mvar|''n''}} copies of <math>\mathbb C</math>, and when <math>\mathbb C^n</math> is a domain of holomorphy, <math>\mathbb C^n</math> can be regarded as a [[Stein manifold]], and more generalized Stein space. <math>\mathbb C^n</math> is also considered to be a [[complex projective variety]], a [[Kähler manifold]],<ref>{{cite journal |doi=10.2977/prims/1195181825|title=Vanishing theorems on complete Kähler manifolds|year=1984|last1=Ohsawa|first1=Takeo|journal=Publications of the Research Institute for Mathematical Sciences|volume=20|pages=21–38|doi-access=free}}</ref> etc. It is also an [[dimension (vector space)|{{mvar|n}}-dimensional vector space]] over the [[complex number]]s, which gives its dimension {{math|2''n''}} over <math>\mathbb R</math>.<ref group=note>The field of complex numbers is a 2-dimensional vector space over real numbers.</ref> Hence, as a set and as a [[topological space]], <math>\mathbb C^n</math> may be identified to the [[real coordinate space]] <math>\mathbb R^{2n}</math> and its [[topological dimension]] is thus {{math|2''n''}}.

In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where [[almost complex manifold|a complex structure]] is specified by a [[linear operator]] {{mvar|J}} (such that {{math|1=''J'' 2 = [[reflection through the origin|−''I'']]}}) which defines [[scalar multiplication|multiplication]] by the [[imaginary unit]] {{mvar|i}}.

Any such space, as a real space, is [[orientation (vector space)|oriented]]. On the [[complex plane]] thought of as a [[Cartesian plane]], [[multiplication]] by a complex number {{math|1=''w'' = ''u'' + ''iv''}} may be represented by the real [[square matrix|matrix]]
:<math>\begin{pmatrix}
u & -v \\
v & u
\end{pmatrix},</math>
with [[determinant]]
:<math>u^2 + v^2 = |w|^2.</math>

Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be [[block matrix|composed from 2 × 2 blocks]] of the aforementioned form), then its determinant equals to the [[square (algebra)#In complex analysis|square of absolute value]] of the corresponding complex determinant. It is a non-negative number, which implies that [[real coordinate space#Orientation|the (real) orientation of the space is never reversed]] by a complex operator. The same applies to [[Jacobian matrix and determinant|Jacobians]] of [[holomorphic function]]s from <math>\mathbb C^n</math> to <math>\mathbb C^n</math>.

== Holomorphic functions ==
=== Definition ===

A function ''f'' defined on a domain <math>D\subset \mathbb{C}^n</math> and with values in <math>\mathbb{C}</math> is said to be holomorphic at a point <math>z\in D</math> if it is complex-differentiable at this point, in the sense that there exists a complex linear map <math>L:\mathbb{C}^n \to \mathbb{C}</math> such that

<math>
f(z+h) = f(z) + L(h) + o(\lVert h\rVert)
</math>

The function ''f'' is said to be holomorphic if it is holomorphic at all points of its domain of definition ''D''.

If ''f'' is holomorphic, then all the partial maps :

<math>z \mapsto f(z_1,\dots,z_{i-1},z,z_{i+1},\dots,z_n)
</math>

are holomorphic as functions of one complex variable : we say that ''f'' is holomorphic in each variable separately. Conversely, if ''f'' is holomorphic in each variable separately, then ''f'' is in fact holomorphic : this is known as [[Hartogs's theorem on separate holomorphicity|Hartog's theorem]], or as [[Osgood's lemma]] under the additional hypothesis that ''f'' is [[continuous function|continuous]].

=== Cauchy–Riemann equations ===

In one complex variable, a function <math>f:\mathbb{C}\to \mathbb{C}</math> defined on the plane is holomorphic at a point <math>p\in \mathbb{C}</math> if and only if its real part <math>u</math> and its imaginary part <math>v</math> satisfy the so-called [[Cauchy–Riemann equations|Cauchy-Riemann equations]] at <math>p</math> :
<math display="block">\frac{\partial u}{\partial x}(p) = \frac{\partial v}{\partial y}(p) \quad \text{ and } \quad\frac{\partial u}{\partial y} (p)=-\frac{\partial v}{\partial x}(p)</math>

In several variables, a function <math>f:\mathbb{C}^n\to \mathbb{C}</math> is holomorphic if and only if it is holomorphic in each variable separately, and hence if and only if the real part <math>u</math> and the imaginary part <math>v</math> of <math>f</math> satisfiy the Cauchy Riemann equations :
<math display="block">\forall i\in \{1,\dots,n\},\quad\frac{\partial u}{\partial x_i} = \frac{\partial v}{\partial y_i} \quad \text{ and } \quad\frac{\partial u}{\partial y_i} = -\frac{\partial v}{\partial x_i}</math>

Using the formalism of [[Wirtinger derivatives]], this can be reformulated as :
<math display="block">\forall i\in \{1,\dots,n\},\quad \frac{\partial f}{\partial \overline{z_i}} = 0,</math>
or even more compactly using the formalism of [[complex differential form]]s, as :
<math display="block">\bar\partial f=0.</math>

=== Cauchy's integral formula I (Polydisc version) ===
Prove the sufficiency of two conditions (A) and (B). Let ''f'' meets the conditions of being continuous and separately homorphic on domain ''D''. Each disk has a [[rectifiable curve]] <math>\gamma</math>, <math>\gamma_\nu</math> is piecewise [[smoothness]], class <math>\mathcal{C}^1</math> Jordan closed curve. (<math>\nu=1,2,\ldots,n</math>) Let <math>D_\nu</math> be the domain surrounded by each <math>\gamma_\nu</math>. Cartesian product closure <math>\overline{D_1\times D_2\times\cdots\times D_n}</math> is <math>\overline{D_1\times D_2\times\cdots\times D_n} \in D </math>. Also, take the closed [[polydisc]] <math>\overline{\Delta}</math> so that it becomes <math>\overline{\Delta}\subset{D_1 \times D_2 \times \cdots \times D_n}</math>. (<math>\overline{\Delta}(z,r) = \left\{\zeta=(\zeta_1, \zeta_2, \dots, \zeta_n)\in \Complex^n ; \left|\zeta_\nu - z_\nu\right| \leq r_\nu \text{ for all } \nu = 1,\dots,n\right\}</math> and let <math> \{z\}^n_{\nu=1} </math> be the center of each disk.) Using the [[Cauchy's integral formula]] of one variable repeatedly, <ref group=note>Note that this formula only holds for polydisc. See [[#Bochner–Martinelli formula (Cauchy's integral formula II)|§Bochner–Martinelli formula]] for the Cauchy's integral formula on the more general domain.</ref>

: <math>
\begin{align}
f(z_1,\ldots,z_n) & =\frac{1}{2 \pi i}\int_{\partial D_1}\frac{f(\zeta_1,z_2,\ldots,z_n)}{\zeta_{1}-z_1} \, d\zeta_1 \\[6pt]
& = \frac{1}{(2 \pi i)^{2}} \int_{\partial D_2} \, d\zeta_2\int_{\partial D_1}\frac{f(\zeta_1,\zeta_2,z_3,\ldots,z_n)}{(\zeta_1 - z_1)(\zeta_2 - z_2)} \, d\zeta_1 \\[6pt]
& = \frac{1}{(2 \pi i)^n} \int_{\partial D_n} \, d\zeta_n \cdots \int_{\partial D_2} \, d\zeta_2 \int_{\partial D_1} \frac{f(\zeta_1,\zeta_2,\ldots,\zeta_n)}{(\zeta_1-z_1)(\zeta_2-z_2)\cdots(\zeta_n - z_n)} \, d\zeta_1
\end{align}
</math>

Because <math>\partial D</math> is a rectifiable Jordanian closed curve<ref group=note>According to the Jordan curve theorem, domain ''D'' is bounded closed set, that is, each domain <math>D_\nu</math> is compact.</ref> and ''f'' is continuous, so the order of products and sums can be exchanged so the [[iterated integral]] can be calculated as a [[multiple integral]]. Therefore,

{{NumBlk|::|<math>f(z_1,\dots,z_n)=\frac{1}{(2\pi i)^n}\int_{\partial D_1}\cdots\int_{\partial D_n}\frac{f(\zeta_1,\dots,\zeta_n)}{(\zeta_1 - z_1) \cdots (\zeta_n - z_n)} \, d\zeta_1\cdots d\zeta_n</math>|{{EquationRef|1}}}}

==== Cauchy's evaluation formula ====
Because the order of products and sums is interchangeable, from ({{EquationNote|1}}) we get

{{NumBlk|::|<math>\frac{\partial^{k_1 + \cdots + k_n}f(\zeta_1,\zeta_2,\ldots,\zeta_n)}{\partial{z_1}^{k_1} \cdots \partial{z_n}^{k_n} } = \frac{k_1 \cdots k_n!}{(2\pi i)^n} \int_{\partial D_1} \cdots \int_{\partial D_n} \frac{f(\zeta_1,\dots,\zeta_n)}{(\zeta_1 - z_1)^{k_1+1} \cdots (\zeta_n - z_n)^{k_n + 1} } \, d\zeta_1\cdots d\zeta_n.</math>|{{EquationRef|2}}}}

''f'' is class <math>\mathcal{C}^{\infty}</math>-function.

From (2), if ''f'' is holomorphic, on polydisc <math>\left\{ \zeta=(\zeta_1, \zeta_2, \dots, \zeta_n) \in \Complex^n ; | \zeta_\nu - z_\nu | \leq r_\nu, \text{ for all } \nu = 1,\dots,n \right\}</math> and <math>|f| \leq {M}</math>, the following evaluation equation is obtained.

: <math>\left|\frac{\partial^{k_1 + \cdots + k_n} f(\zeta_1,\zeta_2,\ldots,\zeta_n)}{{\partial z_1}^{k_1} \cdots \partial {z_n}^{k_n}} \right| \leq \frac{Mk_1 \cdots k_n!}{{r_1}^{k_1} \cdots {r_n}^{k_n}}</math>

Therefore, [[Liouville's theorem (complex analysis)|Liouville's theorem]] hold.

==== Power series expansion of holomorphic functions on polydisc ====
If function ''f'' is holomorphic, on polydisc <math>\{ z=(z_1, z_2, \dots, z_n) \in \mathbb C^n ; | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \}</math>, from the Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series.

: <math>\begin{align}
& f(z)=\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n} (z_1 - a_1)^{k_1} \cdots (z_n - a_n)^{k_n}\ , \\
& c_{k_1 \cdots k_n}=\frac{1}{(2\pi i)^n}\int_{\partial D_1}\cdots\int_{\partial D_n}\frac{f(\zeta_1,\dots,\zeta_n)}{(\zeta_1 - a_1)^{k_1 + 1} \cdots(\zeta_n - a_n)^{k_n + 1} } \, d\zeta_1\cdots d\zeta_n
\end{align} </math>

In addition, ''f'' that satisfies the following conditions is called an analytic function.

For each point <math>a=(a_1,\dots,a_n)\in D \subset \mathbb C^n</math>, <math>f(z)</math> is expressed as a power series expansion that is convergent on ''D'' :

: <math>f(z)=\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1 - a_1)^{k_1}\cdots(z_n - a_n)^{k_n}\ ,</math>

We have already explained that holomorphic functions on polydisc are analytic. Also, from the theorem derived by Weierstrass, we can see that the analytic function on polydisc (convergent power series) is holomorphic.

:If a sequence of functions <math>f_1,\ldots,f_n</math> which converges uniformly on compacta inside a domain ''D'', the limit function ''f'' of <math>f_v</math> also uniformly on compacta inside a domain ''D''. Also, respective partial derivative of <math>f_v</math> also compactly converges on domain ''D'' to the corresponding derivative of ''f''.

:<math>\frac{\partial^{k_1 + \cdots + k_n}f}{\partial{z_1}^{k_1} \cdots \partial{z_n}^{k_n}} = \sum_{v=1}^\infty \frac{\partial^{k_1 + \cdots + k_n} f_v}{\partial{z_1}^{k_1} \cdots \partial{z_n}^{k_n}} </math><ref>{{Eom| title = Weierstrass theorem | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 49192}}</ref>

==== Radius of convergence of power series ====
It is possible to define a combination of positive real numbers <math>\{r_\nu \ (\nu = 1,\dots,n) \}</math> such that the power series <math display="inline">\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1-a_1)^{k_1}\cdots(z_n-a_n)^{k_n}\ </math> converges uniformly at <math>\left\{ z=(z_1, z_2, \dots, z_n) \in \Complex^n ; | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \right\}</math> and does not converge uniformly at <math>\left\{ z=(z_1, z_2, \dots, z_n) \in \Complex^n ; | z_\nu - a_\nu | > r_\nu, \text{ for all } \nu = 1,\dots,n \right\}</math>.

In this way it is possible to have a similar, combination of radius of convergence<ref group=note>But there is a point where it converges outside the circle of convergence. For example if one of the variables is 0, then some terms, represented by the product of this variable, will be 0 regardless of the values taken by the other variables. Therefore, even if you take a variable that diverges when a variable is other than 0, it may converge.</ref> for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

==== Laurent series expansion ====
Let <math>\omega(z)</math> be holomorphic in the [[Annulus (mathematics)|annulus]] <math>\left\{ z=(z_1, z_2, \dots, z_n) \in \Complex^n ; r_\nu < |z| <R_\nu, \text{ for all } \nu + 1,\dots, n\right\}</math> and continuous on their circumference, then there exists the following expansion ;

: <math>
\begin{align}\omega(z) & = \sum_{k=0}^{\infty}\frac{1}{k!}\frac{1}{(2\pi i)^n} \int_{|\zeta_\nu|=R_\nu}\cdots\int\omega(\zeta)\times\left[\frac{d^k}{dz^k}\frac{1}{\zeta-z}\right]_{z=0}df_{\zeta}\cdot z^k \\[6pt]
&+\sum_{k=1}^\infty \frac{1}{k!}\frac{1}{2\pi i}\int_{|\zeta_\nu| = r_\nu}\cdots\int\omega(\zeta) \times \left(0,\cdots,\sqrt{\frac{k!}{\alpha_{1}!\cdots\alpha_{n}!}}\cdot\zeta_{n}^{\alpha_1-1}\cdots\zeta_{n}^{\alpha_n-1},\cdots 0\right)df_{\zeta}\cdot\frac{1}{z^k}\ (\alpha_1 + \cdots + \alpha_n = k)
\end{align}
</math>

The integral in the second term, of the right-hand side is performed so as to see the zero on the left in every plane, also this integrated series is uniformly convergent in the annulus <math>r'_\nu < |z| < R'_\nu</math>, where <math>r'_\nu > r_\nu</math> and <math>R'_\nu < R_\nu</math>, and so it is possible to integrate term.<ref>{{cite journal | last1=Ozaki | first1=Shigeo | last2=Onô | first2=Isao | title=Analytic Functions of Several Complex Variables | journal=Science Reports of the Tokyo Bunrika Daigaku, Section A | volume=4 | issue=98/103 | date=February 1, 1953 | pages=262–270|jstor=43700400}}</ref>

=== Bochner–Martinelli formula (Cauchy's integral formula II) ===

The Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many possible domains, so we introduce the [[Bochner–Martinelli formula]].

Suppose that ''f'' is a continuously differentiable function on the closure of a domain ''D'' on <math>\mathbb C^n</math> with piecewise smooth boundary <math>\partial D</math>, and let the symbol <math>\land</math> denotes the exterior or [[wedge product]] of differential forms. Then the Bochner–Martinelli formula states that if ''z'' is in the domain ''D'' then, for <math>\zeta</math>, ''z'' in <math>\mathbb C^n</math> the Bochner–Martinelli kernel <math>\omega(\zeta,z)</math> is a [[differential form]] in <math>\zeta</math> of bidegree <math>(n,n-1)</math>, defined by
:<math>\omega(\zeta,z) = \frac{(n-1)!}{(2\pi i)^n}\frac{1}{|z-\zeta|^{2n}}
\sum_{1\le j\le n}(\overline\zeta_j-\overline z_j) \, d\overline\zeta_1 \land d\zeta_1 \land \cdots \land d\zeta_j \land \cdots \land d\overline\zeta_n \land d\zeta_n</math>
:<math>\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z) - \int_D\overline\partial f(\zeta)\land\omega(\zeta,z).</math>

In particular if ''f'' is holomorphic the second term vanishes, so
:<math>\displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z). </math>

=== Identity theorem ===
Holomorphic functions of several complex variables satisfy an [[identity theorem]], as in one variable : two holomorphic functions defined on the same connected open set <math>D\subset \mathbb{C}^n</math> and which coincide on an open subset ''N'' of ''D'', are equal on the whole open set ''D''. This result can be proven from the fact that holomorphics functions have power series extensions, and it can also be deduced from the one variable case. Contrary to the one variable case, it is possible that two different holomorphic functions coincide on a set which has an accumulation point, for instance the maps <math>f(z_1,z_2)=0</math> and <math>g(z_1,z_2)=z_1 </math>coincide on the whole complex line of <math>\mathbb{C}^2</math> defined by the equation <math>z_1=0</math>.

The [[maximal principle]], [[inverse function theorem]], and implicit function theorems also hold. For a generalized version of the implicit function theorem to complex variables, see the [[Weierstrass preparation theorem]].

=== Biholomorphism ===
From the establishment of the inverse function theorem, the following mapping can be defined.

For the domain ''U'', ''V'' of the ''n''-dimensional complex space <math>\Complex^n</math>, the bijective holomorphic function <math>\phi:U\to V</math> and the inverse mapping <math>\phi^{-1}:V\to U</math> is also holomorphic. At this time, <math>\phi</math> is called a ''U'', ''V'' biholomorphism also, we say that ''U'' and ''V'' are biholomorphically equivalent or that they are biholomorphic.

==== The Riemann mapping theorem does not hold ====
When <math>n > 1</math>, open balls and open polydiscs are ''not'' biholomorphically equivalent, that is, there is no [[biholomorphic mapping]] between the two.<ref name=FieldCM1982>{{cite book |doi=10.1017/CBO9781107325562.005|chapter=Complex Manifolds |title=Several Complex Variables and Complex Manifolds I |year=1982 |pages=134–186 |isbn=9780521283014|last1=Field|first1=M}}</ref> This was proven by [[Henri Poincaré|Poincaré]] in 1907 by showing that their [[automorphism group]]s have different dimensions as [[Lie group]]s.{{R|Chen2000}}<ref>{{cite journal |last1=Poincare |first1=M. Henri |title=Les fonctions analytiques de deux variables et la représentation conforme |journal=Rendiconti del Circolo Matematico di Palermo |date=1907 |volume=23 |pages=185–220 |doi=10.1007/BF03013518|doi-access=free|s2cid=123480258 |url=https://zenodo.org/record/2037590 }}</ref> However, even in the case of several complex variables, there are some results similar to the results of the theory of uniformization in one complex variable.<ref>{{cite book |last1=Siu |first1=Yum-Tong |editor1-last=Wu |editor1-first=Hung-Hsi |title=Contemporary Geometry |isbn=978-1-4684-7950-8 |url={{Google books|title=Contemporary Geometry|u53VBwAAQBAJ|page=95|plainurl=yes}}|page=494|doi=10.1007/978-1-4684-7950-8|chapter=Uniformization in Several Complex Variables|year=1991 |chapter-url=https://doi.org/10.1007/978-1-4684-7950-8_5}}</ref>

=== Analytic continuation ===
Let ''U, V'' be domain on <math>\mathbb{C}^n</math>, such that <math>f \in \mathcal{O}(U)</math> and <math>g \in \mathcal{O}(V)</math>, (<math>\mathcal{O}(U)</math> is the set/ring of holomorphic functions on ''U''.) assume that <math>U,\ V,\ U \cap V \ne \varnothing</math> and <math>W</math> is a [[connected component (topology)|connected component]] of <math>U \cap V</math>. If <math>f|_W =g|_W</math> then ''f'' is said to be connected to ''V'', and ''g'' is said to be analytic continuation of ''f''. From the identity theorem, if ''g'' exists, for each way of choosing ''W'' it is unique. When n > 2, the following phenomenon occurs depending on the shape of the boundary <math>\partial U</math>: there exists domain ''U'', ''V'', such that all holomorphic functions <math>f</math> over the domain ''U'', have an analytic continuation <math>g \in \mathcal{O}(V)</math>. In other words, there may be not exist a function <math>f \in \mathcal{O}(U)</math> such that <math>\partial U</math> as the natural boundary. There is called the Hartogs's phenomenon. Therefore, researching when domain boundaries become natural boundaries has become one of the main research themes of several complex variables. In addition, when <math>n \geq 2</math>, it would be that the above ''V'' has an intersection part with ''U'' other than ''W''. This contributed to advancement of the notion of sheaf cohomology.

== Reinhardt domain ==
In polydisks, the Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but polydisks and open unit balls are not biholomorphic mapping because the Riemann mapping theorem does not hold, and also, polydisks was possible to separation of variables, but it doesn't always hold for any domain. Therefore, in order to study of the domain of convergence of the power series, it was necessary to make additional restriction on the domain, this was the Reinhardt domain. Early knowledge into the properties of field of study of several complex variables, such as Logarithmically-convex, Hartogs's extension theorem, etc. , were given in the Reinhardt domain.

Let <math>D \subset \Complex^n</math> (<math> n \geq 1</math>) to be a domain, with centre at a point <math>a = (a_1,\dots,a_n) \in \Complex^n</math>, such that, together with each point <math>z^0 = (z_1^0,\dots,z_n^0)\in D</math>, the domain also contains the set

: <math> \left\{ z = (z_1, \dots, z_n) ; \left|z_\nu - a_\nu \right| = \left|z_\nu^0 - a_\nu\right|,\ \nu = 1, \dots, n \right\} .
</math>

A domain ''D'' is called a Reinhardt domain if it satisfies the following conditions:<ref>{{cite book |doi=10.4171/049 |title=First Steps in Several Complex Variables: Reinhardt Domains |date=2008 |last1=Jarnicki |first1=Marek |last2=Pflug |first2=Peter |isbn=978-3-03719-049-4|url={{Google books|TZuc66RB-rQC|keywords=reinhardt domains|plainurl=yes}}}}</ref><ref>{{cite journal |doi=10.1017/S0027763000013465|title=Meromorphic or Holomorphic Completion of a Reinhardt Domain|year=1970|last1=Sakai|first1=Eiichi|journal=Nagoya Mathematical Journal|volume=38|pages=1–12|s2cid=118248529 |doi-access=free}}</ref>

Let <math>\theta_\nu \;(\nu = 1,\dots,n)</math> is a arbitrary real numbers, a domain ''D'' is invariant under the rotation: <math>\left\{z^0 - a_\nu \right\} \to \left\{e^{i\theta_\nu} (z_\nu^0 - a_\nu) \right\}</math>.

The Reinhardt domains (subclass of the Hartogs domains<ref>{{Eom| title = Hartogs domain | author-last1 = Chirka| author-first1 = E.M.| oldid = 43472}}</ref>) which are defined by the following condition; Together with all points of <math>z^0 \in D</math>, the domain contains the set

: <math> \left\{ z = ( z_1, \dots, z_n ) ; z = a + \left(z^0 - a\right) e^{i \theta} ,\ 0 \leq \theta < 2 \pi \right\}.</math>

A Reinhardt domain ''D'' is called a complete Reinhardt domain with centre at a point ''a'' if together with all point <math>z^0\in D</math> it also contains the polydisc

: <math>
\left\{ z = ( z_1, \dots, z_n) ; \left|z_\nu - a_\nu \right| \leq \left|z_\nu^0 - a_\nu \right| , \ \nu = 1, \dots, n \right\}.
</math>

A complete Reinhardt domain ''D'' is [[Star-like domain|star-like]] with regard to its centre ''a''. Therefore, the complete Reinhardt domain is [[simply connected]], also when the complete Reinhardt domain is the boundary line, there is a way to prove the [[Cauchy's integral theorem]] without using the [[Jordan curve theorem]].

=== Logarithmically-convex ===
A Reinhardt domain ''D'' is called [[logarithmically convex]] if the image <math>\lambda(D^{*})</math> of the set

: <math>
D ^{*} = \{ z = (z_1, \dots, z_n) \in D ; z_1, \dots, z_n \neq 0 \}
</math>

under the mapping

: <math>
\lambda ; z \rightarrow \lambda(z) = (\ln|z_1|, \dots, \ln |z_n|)
</math>

is a [[convex set]] in the real coordinate space <math>\R^n</math>.

Every such domain in <math>\Complex^n</math> is the interior of the set of points of absolute convergence of some power series in <math display="inline">\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1 - a_1)^{k_1}\cdots(z_n - a_n)^{k_n}\ </math>, and conversely; The domain of convergence of every power series in <math>z_1,\dots,z_n</math> is a logarithmically-convex Reinhardt domain with centre <math>a = 0</math>.
<ref group=note>When described using the [[#Domain of holomorphy|domain of holomorphy]], which is a generalization of the convergence domain, a Reinhardt domain is a domain of holomorphy if and only if logarithmically convex.</ref> But, there is an example of a complete Reinhardt domain D which is not logarithmically convex.<ref>{{cite book |page=10.1007/978-1-4757-1918-5_2|doi=10.1007/978-1-4757-1918-5_2 |chapter=Domains of Holomorphy and Pseudoconvexity |title=Holomorphic Functions and Integral Representations in Several Complex Variables |series=Graduate Texts in Mathematics |date=1986 |last1=Range |first1=R. Michael |volume=108 |isbn=978-1-4419-3078-1 |url={{Google books|mv_pBwAAQBAJ|page=80|plainurl=yes}}}}</ref>

=== Some results ===

==== Hartogs's extension theorem and Hartogs's phenomenon ====
When examining the domain of convergence on the Reinhardt domain, Hartogs found the Hartogs's phenomenon in which holomorphic functions in some domain on the <math>\mathbb{C}^n</math> were all connected to larger domain.<ref>{{cite journal |doi=10.1080/17476930701747716 |title=The Hartogs extension phenomenon redux |year=2008 |last1=Krantz |first1=Steven G. |journal=Complex Variables and Elliptic Equations |volume=53 |issue=4 |pages=343–353 |s2cid=121700550 }}</ref>

:On the polydisk consisting of two disks <math>\Delta^2=\{z\in\Complex^2;|z_1|<1,|z_2|<1\}</math> when <math>0 <\varepsilon < 1</math>.

:Internal domain of <math>H_\varepsilon = \{z=(z_1,z_2)\in\Delta^2;|z_1|<\varepsilon\ \cup \ 1-\varepsilon< |z_2|\}\ (0 <\varepsilon < 1) </math>

::Hartogs's extension theorem (1906);<ref name=Hartogs1906>{{Citation|last = Hartogs|first = Fritz|author-link = Friedrich Hartogs|title = Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.| journal = Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse| language = de| volume = 36| pages = 223–242| year = 1906| url = https://archive.org/stream/bub_gb_N-sAAAAAYAAJ#page/n229/mode/1up| jfm = 37.0443.01
}}</ref> Let ''f'' be a [[holomorphic function]] on a [[Set (mathematics)|set]] {{math|''G'' \ ''K''}}, where {{mvar|G}} is a bounded (surrounded by a rectifiable closed Jordan curve) domain{{refn|group=note|1=This theorem holds even if the condition is not restricted to the bounded. i.e. The theorem holds even if this condition is replaced with an open set.<ref name=Simonič2016>{{cite arXiv|eprint=1608.00950|last1=Simonič|first1=Aleksander|title=Elementary approach to the Hartogs extension theorem|year=2016|class=math.CV }}</ref>}} on <math>\Complex^n</math> ({{math|''n'' ≥ 2}}) and ''K'' is a compact subset of ''G''. If the [[Complement (set theory)|complement]] {{math|''G'' \ ''K''}} is connected, then every holomorphic function ''f'' regardless of how it is chosen can be each extended to a unique holomorphic function on ''G''.<ref>{{cite journal |last1=Laufer |first1=Henry B. |title=Some remarks about a theorem of Hartogs |journal=[[Proceedings of the American Mathematical Society]] |date=1 June 1966 |volume=17 |issue=6 |pages=1244–1249 |doi=10.1090/S0002-9939-1966-0201675-2 |doi-access=free |jstor=2035718}}</ref>{{R|Simonič2016}}
:It is also called Osgood–Brown theorem is that for holomorphic functions of several complex variables, the singularity is a accumulation point, not an isolated point. This means that the various properties that hold for holomorphic functions of one-variable complex variables do not hold for holomorphic functions of several complex variables. The nature of these singularities is also derived from [[Weierstrass preparation theorem]]. A generalization of this theorem using the same method as Hartogs was proved in 2007.<ref>{{cite journal |doi=10.1007/BF02922095|doi-access=free|title=A Morse-theoretical proof of the Hartogs extension theorem|year=2007|last1=Merker|first1=Joël|last2=Porten|first2=Egmont|journal=Journal of Geometric Analysis|volume=17|issue=3|pages=513–546|s2cid=449210|arxiv=math/0610985}}</ref><ref>{{cite journal |doi=10.1016/j.jmaa.2013.01.049|title=Hartogs extension for generalized tubes in Cn|year=2013|last1=Boggess|first1=A.|last2=Dwilewicz|first2=R.J.|last3=Slodkowski|first3=Z.|journal=Journal of Mathematical Analysis and Applications|volume=402|issue=2|pages=574–578|doi-access=free}}</ref>

From Hartogs's extension theorem the domain of convergence extends from <math>H_\varepsilon</math> to <math>\Delta^2</math>. Looking at this from the perspective of the Reinhardt domain, <math>H_\varepsilon</math> is the Reinhardt domain containing the center z = 0, and the domain of convergence of <math>H_\varepsilon</math> has been extended to the smallest complete Reinhardt domain <math>\Delta^2</math> containing <math>H_\varepsilon</math>.<ref>{{Cite journal|first=Henri|last=Cartan|title= Les fonctions de deux variables complexes et le problème de la représentation analytique|journal=[[Journal de Mathématiques Pures et Appliquées]]|volume=10|year=1931|pages=1–116|zbl =0001.28501}}</ref>

==== Thullen's classic results ====

[[Peter Thullen|Thullen]]'s<ref>{{cite journal |doi=10.1007/bf01457933|doi-access=free|title=Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Veränderlichen die Invarianz des Mittelpunktes von Kreiskörpern|year=1931|last1=Thullen|first1=Peter|journal=Mathematische Annalen|volume=104|pages=244–259|s2cid=121072397}}</ref> classical result says that a 2-dimensional bounded Reinhard domain containing the origin is [[biholomorphic]] to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:

# <math>\{(z,w)\in \Complex^2;~|z| < 1,~|w| < 1\}</math> (polydisc);
# <math>\{(z,w)\in \Complex^2;~|z|^2 + |w|^2 < 1\}</math> (unit ball);
# <math>\{(z,w)\in \Complex^2;~|z|^2 + |w|^{\frac{2}{p}} < 1\}\, (p > 0,\neq 1)</math> (Thullen domain).

==== Sunada's results ====
[[Toshikazu Sunada]] (1978)<ref>{{cite journal |doi=10.1007/BF01405009|doi-access=free|title=Holomorphic equivalence problem for bounded Reinhardt domains|year=1978|last1=Sunada|first1=Toshikazu|journal=Mathematische Annalen|volume=235|issue=2|pages=111–128|s2cid=124324696}}</ref> established a generalization of Thullen's result:
:Two ''n''-dimensional bounded Reinhardt domains <math>G_1</math> and <math>G_2</math> are mutually biholomorphic if and only if there exists a transformation <math>\varphi:\Complex^n\to \Complex^n</math> given by <math>z_i\mapsto r_iz_{\sigma(i)} (r_i>0)</math>, <math>\sigma</math> being a permutation of the indices), such that <math>\varphi(G_1)=G_2</math>.

== Natural domain of the holomorphic function (domain of holomorphy) ==
When moving from the theory of one complex variable to the theory of several complex variables, depending on the range of the domain, it may not be possible to define a holomorphic function such that the boundary of the domain becomes a natural boundary. Considering the domain where the boundaries of the domain are natural boundaries (In the complex coordinate space <math>\Complex^n</math> call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of ''H''. Cartan and Thullen.<ref>{{cite journal | last1=Cartan | first1=Henri | last2=Thullen | first2=Peter | date = 1932 | title =Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen Regularitäts-und Konvergenzbereiche| journal = Mathematische Annalen | volume = 106 | pages = 617–647 | doi =10.1007/BF01455905 | doi-access=free}}</ref> Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for <math>\Complex^2</math>,<ref name="Oka'sVI">{{Citation | last1=Oka | first1=Kiyoshi | title=Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes | year=1943 | journal=[[Tohoku Mathematical Journal]] | series=First Series | issn=0040-8735 | volume=49 | pages=15–52 |zbl = 0060.24006 | url=https://www.jstage.jst.go.jp/article/tmj1911/49/0/49_0_15/_article/-char/en}}</ref> later extended to <math>\Complex^n</math>.<ref name="Oka'sIX">{{Citation | last1=Oka | first1=Kiyoshi | title=Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur | year=1953 | journal= Japanese Journal of Mathematics: Transactions and Abstracts| issn=0075-3432 | volume=23 | pages=97–155|doi=10.4099/jjm1924.23.0_97| doi-access=free }}</ref><ref>{{Citation | author = Hans J. Bremermann | year = 1954 | title =Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum vonn komplexen Veränderlichen| journal = Mathematische Annalen | volume = 106 | pages = 63–91 | doi =10.1007/BF01360125 | doi-access = free | s2cid = 119837287 }}</ref>)<ref name="Huckleberry2013" >{{cite journal |last1=Huckleberry |first1=Alan |title=Hans Grauert (1930–2011) |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |year=2013 |volume=115 |pages=21–45 |doi=10.1365/s13291-013-0061-7|arxiv=1303.6933|s2cid=119685542 }}</ref> [[Kiyoshi Oka]]'s{{refn|name=Oka'sVII|1={{cite journal |doi=10.24033/bsmf.1408|title=Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques|year=1950|last1=Oka|first1=Kiyoshi|journal=Bulletin de la Société Mathématique de France |volume=2|pages=1–27|doi-access=free}}, {{cite journal |title=Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques|year=1961|last1=Oka|first1=Kiyoshi|journal=Iwanami Shoten, Tokyo (Oka's Original Version)|url=http://www.nara-wu.ac.jp/aic/gdb/nwugdb/oka/ko_ron/pdf/ko-f77.pdf}}{{refn|group=note|1=Oka says that<ref>{{cite web |last1=Oka |first1=Kiyoshi |title=Sur les formes objectives et les contenus subjectifs dans les sciences math'ematiques; Propos post'erieur |url=https://www.ms.u-tokyo.ac.jp/~noguchi/oka/oka-propos-posterieurs-v6.pdf|editor-last1=Merker|editor-first1=j.|editor-last2=Noguchi|editor-first2=j.|year=1953}}</ref> the contents of these two papers are different.<ref>{{cite web |last1=Noguchi |first1=J.|url=https://www.ms.u-tokyo.ac.jp/~noguchi/oka/|title=Related to Works of Dr. Kiyoshi OKA}}</ref>}}}}<ref name="Oka'sVIII">{{Citation|last1=Oka | first1=Kiyoshi |title = Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental|journal=Journal of the Mathematical Society of Japan |volume=3|issue=1| year=1951|pages=204–214|doi = 10.2969/jmsj/00310204|doi-access=free}}, {{Citation| last1=Oka | first1=Kiyoshi |title = Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental (Suite)|journal=Journal of the Mathematical Society of Japan|issue=2|year=1951| volume=3 |pages=259–278|doi = 10.2969/jmsj/00320259|doi-access=free}}</ref> notion of [[#Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf))|''idéal de domaines indéterminés'']] is interpreted theory of [[sheaf cohomology]] by
''H''. Cartan and more development Serre.<ref group="note">The idea of the [[Sheaf (mathematics)|sheaf]] itself is by [[Jean Leray]].</ref><ref name=Cartan1950>{{cite journal |doi=10.24033/bsmf.1409|title=Idéaux et modules de fonctions analytiques de variables complexes|year=1950|last1=Cartan|first1=Henri|journal=Bulletin de la Société Mathématique de France|volume=2|pages=29–64|doi-access=free}}</ref><ref name = Cartan1953>{{cite journal |last1=Cartan |first1=Henri |title=Variétés analytiques complexes et cohomologie |journal=Colloque sur les fonctions de plusieurs variables, Bruxelles |year=1953 |pages=41–55|zbl=0053.05301|mr=64154}}</ref><ref name="CT3">{{cite journal|website=numdam.org |last1=Cartan |first1=H. |last2=Eilenberg |first2=Samuel |last3=Serre |first3=J-P. |title=Séminaire Henri Cartan, Tome 3 (1950-1951) |url=http://www.numdam.org/volume/SHC_1950-1951__3/}}</ref><ref name=Chorlay2010>{{cite journal |last1=Chorlay |first1=Renaud |title=From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept |journal=Archive for History of Exact Sciences |date=January 2010 |volume=64 |issue=1 |pages=1–73 |doi=10.1007/s00407-009-0052-3|jstor=41342411|s2cid=73633995 }}</ref><ref>{{cite book |doi=10.1007/978-3-662-02661-8|title=Sheaves on Manifolds|series=Grundlehren der mathematischen Wissenschaften|year=1990|volume=136|isbn=978-3-642-08082-1|url={{Google books|EyzqCAAAQBAJ|Sheaves on Manifolds|page=12|plainurl=yes|chapterA Short History: Les débuts de la théorie des faisceaux|chapter-url=https://doi.org/10.1007/978-3-662-02661-8_2}}}}</ref><ref>{{cite journal |last1=Serre |first1=Jean-Pierre |title=Quelques problèmes globaux rélatifs aux variétés de Stein |journal=Centre Belge Rech. Math., Colloque Fonctions Plusieurs Variables, Bruxelles du 11 Au 14 Mars |date=1953|pages=67–58|zbl=0053.05302|url={{Google books|eaUoAKOAbUsC|Oeuvres - Collected Papers I: 1949 - 1959|page=259|plainurl=yes}}}}</ref>{{R|IWS}} In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds.<ref name="CT4">{{cite web |last1=Cartan |first1=H. |last2=Bruhat |first2=F. |last3=Cerf |first3=Jean. |last4=Dolbeault |first4=P. |last5=Frenkel |first5=Jean. |last6=Hervé |first6=Michel |last7=Malatian. |last8=Serre |first8=J-P. |title=Séminaire Henri Cartan, Tome 4 (1951-1952) |url=http://www.numdam.org/volume/SHC_1951-1952__4/ |archive-url=https://web.archive.org/web/20201020181016/http://www.numdam.org/volume/SHC_1951-1952__4/ |url-status=dead |archive-date=October 20, 2020 }}</ref> The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization.{{R|Siu1978}}

=== Domain of holomorphy ===
[[Image:Domain of holomorphy illustration.svg|thumb|right|The sets in the definition. Note: On this section, replace <math>\Omega</math> in the figure with ''D'']]
When a function ''f'' is holomorpic on the domain <math>D\subset \Complex^n</math> and cannot directly connect to the domain outside ''D'', including the point of the domain boundary <math>\partial D</math>, the domain ''D'' is called the domain of holomorphy of ''f'' and the boundary is called the natural boundary of ''f''. In other words, the domain of holomorphy ''D'' is the supremum of the domain where the holomorphic function ''f'' is holomorphic, and the domain ''D'', which is holomorphic, cannot be extended any more. For several complex variables, i.e. domain <math>D\subset \Complex^n\ (n\geq 2)</math>, the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries.<ref name="Forstnerič2011§SteinManifolds">{{cite book |doi=10.1007/978-3-642-22250-4|title=Stein Manifolds and Holomorphic Mappings |year=2011 |last1=Forstnerič |first1=Franc |isbn=978-3-642-22249-8 |chapter=Stein Manifolds|series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |volume=56 |chapter-url={{Google books|5ibv2wzZ2IQC|An Introduction to Complex Analysis in Several Variables|page=43|plainurl=yes}}}}</ref>

Formally, a domain ''D'' in the ''n''-dimensional complex coordinate space <math>\Complex^n</math> is called a ''domain of holomorphy'' if there do not exist non-empty domain <math>U \subset D</math> and <math>V \subset \Complex^n</math>, <math>V \not\subset D</math> and <math>U \subset D \cap V</math> such that for every holomorphic function ''f'' on ''D'' there exists a holomorphic function ''g'' on ''V'' with <math>f = g</math> on ''U''.

For the <math>n=1</math> case, the every domain (<math>D\subset\mathbb{C}</math>) was the domain of holomorphy; we can define a holomorphic function with zeros [[accumulation point|accumulating]] everywhere on the [[boundary (topology)|boundary]] of the domain, which must then be a [[analytic continuation#Natural boundary|natural boundary]] for a domain of definition of its reciprocal.

==== Properties of the domain of holomorphy ====
* If <math>D_1, \dots, D_n</math> are domains of holomorphy, then their intersection <math display="inline">D = \bigcap_{\nu=1}^n D_\nu</math> is also a domain of holomorphy.
* If <math>D_1 \subseteq D_2 \subseteq \cdots</math> is an increasing sequence of domains of holomorphy, then their union <math display="inline">D = \bigcup_{n=1}^\infty D_n</math> is also a domain of holomorphy (see [[Behnke–Stein theorem]]).<ref>{{cite journal |last1=Behnke |first1=H. |last2=Stein |first2=K. |title=Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität |journal=Mathematische Annalen |year=1939 |volume=116 |pages=204–216 |doi=10.1007/BF01597355|s2cid=123982856 }}</ref>
* If <math>D_1</math> and <math>D_2</math> are domains of holomorphy, then <math> D_1 \times D_2</math> is a domain of holomorphy.
* The first [[Cousin problems|Cousin problem]] is always solvable in a domain of holomorphy, also Cartan showed that the converse of this result was incorrect for <math>n\geq 3</math>.<ref>{{cite journal |last1=Kajiwara |first1=Joji |title=Relations between domains of holomorphy and multiple Cousin's problems |journal=Kodai Mathematical Journal |date=1 January 1965 |volume=17 |issue=4 |doi=10.2996/kmj/1138845123|doi-access=free }}</ref> this is also true, with additional topological assumptions, for the second Cousin problem.

=== Holomorphically convex hull ===
Let <math>G \subset \Complex^n</math> be a domain , or alternatively for a more general definition, let <math>G</math> be an <math>n</math> dimensional [[complex analytic manifold]]. Further let <math>{\mathcal{O}}(G)</math> stand for the set of holomorphic functions on ''G''. For a compact set <math>K \subset G</math>, the '''holomorphically convex hull''' of ''K'' is

:<math> \hat{K}_G := \left \{ z \in G ; |f(z)| \leq \sup_{w \in K} |f(w)| \text{ for all } f \in \mathcal{O}(G) . \right \} .</math>

One obtains a narrower concept of '''polynomially convex hull''' by taking <math>\mathcal O(G)</math> instead to be the set of complex-valued polynomial functions on ''G''. The polynomially convex hull contains the holomorphically convex hull.

The domain <math>G</math> is called '''holomorphically convex''' if for every compact subset <math>K, \hat{K}_G</math> is also compact in ''G''. Sometimes this is just abbreviated as ''holomorph-convex''.

When <math>n=1</math>, every domain <math>G</math> is holomorphically convex since then <math>\hat{K}_G</math> is the union of ''K'' with the relatively compact components of <math>G \setminus K \subset G</math>.

When <math>n\geq 1</math>, if ''f'' satisfies the above holomorphic convexity on ''D'' it has the following properties. <math>\text{dist} (K, D^c) = \text{dist} (\hat{K}_D, D^c
)</math> for every compact subset ''K'' in ''D'', where
<math>\text{dist} (K, D^c)</math> denotes the distance between K and <math>D^c = \mathbb{C}^n \setminus D</math>. Also, at this time, D is a domain of holomorphy. Therefore, every convex domain <math>(D\subset\Complex^n)</math> is domain of holomorphy.{{R|Chen2000}}

=== Pseudoconvexity ===
Hartogs showed that

{{blockquote|Hartogs (1906):{{R|Hartogs1906}} Let ''D'' be a Hartogs's domain on <math>\mathbb{C}</math> and ''R'' be a positive function on ''D'' such that the set <math>\Omega</math> in <math>\mathbb{C}^2</math> defined by <math>z_1 \in D</math> and <math>|z_2| < R (z_1)</math> is a domain of holomorphy. Then <math>-\log {R} (z_1)</math> is a subharmonic function on ''D''.{{R|Siu1978}}}}

If such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex.{{refn|group=note|1=In fact, this was proved by Kiyoshi Oka{{R|Oka'sVI|}} with respect to <math>\Complex^n</math> domain.See [[Oka's lemma]]. }} The [[subharmonic function]] looks like a kind of [[convex function]], so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity). Pseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy. A domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain.<ref>{{cite journal |doi=10.1090/noti798|title=WHAT IS...a Pseudoconvex Domain?|year=2012|last1=Range|first1=R. Michael|journal=Notices of the American Mathematical Society|volume=59|issue=2|page=1|doi-access=free}}</ref>

==== Definition of plurisubharmonic function ====
:A function
:<math>f \colon D \to {\mathbb{R}}\cup\{-\infty\},</math>
:with ''domain'' <math>D \subset {\mathbb{C}}^n</math>
is called '''plurisubharmonic''' if it is [[semi-continuous function|upper semi-continuous]], and for every complex line

:<math>\{ a + b z ; z \in \mathbb{C} \}\subset \mathbb{C}^n</math> with <math>a, b \in \mathbb{C}^n</math>

:the function <math>z \mapsto f(a + bz)</math> is a subharmonic function on the set

:<math>\{ z \in \mathbb{C} ; a + b z \in D \}.</math>

:In ''full generality'', the notion can be defined on an arbitrary complex manifold or even a Complex analytic space <math>X</math> as follows. An [[semi-continuity|upper semi-continuous function]]
:<math>f \colon X \to \mathbb{R} \cup \{ - \infty \}</math>
:is said to be plurisubharmonic if and only if for any [[holomorphic map]]
<math>\varphi\colon\Delta \to X</math> the function
:<math>f\circ\varphi \colon \Delta \to \mathbb{R} \cup \{ -\infty \}</math>
is subharmonic, where <math>\Delta \subset \mathbb{C}</math> denotes the unit disk.

In one-complex variable, necessary and sufficient condition that the real-valued function <math>u=u(z)</math>, that can be second-order differentiable with respect to ''z'' of one-variable complex function is subharmonic is <math>\Delta=4\left(\frac{\partial^2 u}{\partial z\,\partial\overline{z}}\right)\geq0</math>. Therefore, if <math>u</math> is of class <math>\mathcal{C}^2</math>, then <math>u</math> is plurisubharmonic if and only if the [[hermitian matrix]] <math>H_u=(\lambda_{ij}),\lambda_{ij}=\frac{\partial^2u}{\partial z_i\,\partial\bar z_j}</math> is positive semidefinite.

Equivalently, a <math>\mathcal{C}^2</math>-function ''u'' is plurisubharmonic if and only if <math>\sqrt{-1}\partial\bar\partial f</math> is a [[positive form|positive (1,1)-form]].<ref name="CAaDG" >[https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf Complex Analytic and Differential Geometry]</ref>{{rp|pages=39–40}}

===== Strictly plurisubharmonic function =====
When the hermitian matrix of ''u'' is positive-definite and class <math>\mathcal{C}^2</math>, we call ''u'' a strict plurisubharmonic function.

==== (Weakly) pseudoconvex (p-pseudoconvex) ====
Weak pseudoconvex is defined as : Let <math>X\subset {\mathbb{C}}^n</math> be a domain. One says that ''X'' is ''pseudoconvex'' if there exists a [[continuous function|continuous]] [[plurisuperharmonic function|plurisubharmonic function]] <math>\varphi</math> on ''X'' such that the set <math>\{ z \in X ; \varphi(z) \leq \sup x \}</math> is a [[relatively compact]] subset of ''X'' for all real numbers ''x''. <ref group="note" >This is a hullomorphically convex hull condition expressed by a plurisubharmonic function. For this reason, it is also called p-pseudoconvex or simply p-convex.</ref> i.e. there exists a smooth plurisubharmonic exhaustion function <math>\psi \in \text{Psh}(X)\cap\mathcal{C}^{\infty}(X)</math>. Often, the definition of pseudoconvex is used here and is written as; Let ''X'' be a complex ''n''-dimensional manifold. Then is said to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function <math>\psi \in \text{Psh}(X)\cap\mathcal{C}^{\infty}(X)</math>.{{R|CAaDG|}}{{rp|page=49}}

==== Strongly (Strictly) pseudoconvex ====
Let ''X'' be a complex ''n''-dimensional manifold. '''Strongly (or Strictly) pseudoconvex''' if there exists a smooth '''strictly''' plurisubharmonic exhaustion function <math>\psi \in \text{Psh}(X)\cap\mathcal{C}^{\infty}(X)</math>,i.e., <math>H\psi</math> is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain.{{R|CAaDG|}}{{rp|page=49}} Strongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete<ref>{{cite book |url={{Google books|I3rSBwAAQBAJ|pg=267|plainurl=yes}} | title=From Holomorphic Functions to Complex Manifolds | isbn=9781468492736 | last1=Fritzsche | first1=Klaus | last2=Grauert | first2=Hans | date=6 December 2012 | publisher=Springer }}</ref>) are often used interchangeably,<ref>{{cite book |url={{Google books|bL8jDwAAQBAJ|pg=136|plainurl=yes}}| title=Function Theory of Several Complex Variables | isbn=9780821827246 | last1=Krantz | first1=Steven George | year=2001 | publisher=American Mathematical Soc. }}</ref> see Lempert<ref>{{cite journal |url=https://eudml.org/doc/87405 |title=La métrique de Kobayashi et la représentation des domaines sur la boule |journal=Bulletin de la Société Mathématique de France |year=1981 |volume=109 |pages=427–474 |last1=Lempert |first1=Laszlo |doi=10.24033/bsmf.1948 |doi-access=free }}</ref> for the technical difference.

==== Levi form ====

===== (Weakly) Levi(–Krzoska) pseudoconvexity =====
If <math>\mathcal{C}^2</math> boundary , it can be shown that ''D'' has a defining function; i.e., that there exists <math>\rho: \mathbb{C}^n \to \mathbb{R} </math> which is <math>\mathcal{C}^2</math> so that <math>D = \{\rho < 0 \}</math>, and <math>\partial D = \{\rho =0\}</math>. Now, ''D'' is pseudoconvex iff for every <math>p \in \partial D</math> and <math>w</math> in the complex tangent space at p, that is,

:<math> \nabla \rho(p) w = \sum_{i=1}^n \frac{\partial \rho (p)}{ \partial z_j }w_j =0 </math>, we have
:<math>H(\rho) = \sum_{i,j=1}^n \frac{\partial^2 \rho(p)}{\partial z_i \, \partial \bar{z_j} } w_i \bar{w_j} \geq 0.</math>{{R|Chen2000}}<ref>{{cite journal |doi=10.2206/kyushumfs.41.45 |title=Stein Neighborhood Bases for Product Sets of Polydiscs and Open Intervals |year=1987 |last1=Shon |first1=Kwang Ho |journal=Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics |volume=41 |pages=45–80 |doi-access=free }}</ref>

If ''D'' does not have a <math>\mathcal{C}^2</math> boundary, the following approximation result can be useful.

'''Proposition 1''' ''If ''D'' is pseudoconvex, then there exist [[bounded set|bounded]], strongly Levi pseudoconvex domains <math>D_k \subset D</math> with class <math>\mathcal{C}^\infty</math>-boundary which are relatively compact in ''D'', such that''

:<math>D = \bigcup_{k=1}^\infty D_k.</math>

This is because once we have a <math>\varphi</math> as in the definition we can actually find a <math>\mathcal{C}^\infty</math> exhaustion function.

===== Strongly (or Strictly) Levi (–Krzoska) pseudoconvex (a.k.a. Strongly (Strictly) pseudoconvex) =====
When the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly (or strictly) pseudoconvex.{{R|Chen2000}}

==== Levi total pseudoconvex ====
If for every boundary point <math>\rho</math> of ''D'', there exists an [[analytic variety]] <math>\mathcal{B}</math> passing <math>\rho</math> which lies entirely outside ''D'' in some neighborhood around <math>\rho</math>, except the point <math>\rho</math> itself. Domain ''D'' that satisfies these conditions is called Levi total pseudoconvex.<ref name="Hitomatsu1958" >{{Citation | author = Sin Hitomatsu | title=On some conjectures concerning pseudo-convex domains | year=1958 | journal=Journal of the Mathematical Society of Japan| volume=6 | issue=2 | pages=177–195| zbl=0057.31503|doi=10.2969/jmsj/00620177 | doi-access=free}}</ref>

==== Oka pseudoconvex ====
===== Family of Oka's disk =====
Let ''n''-functions <math>\varphi:z_j = \varphi_j(u, t)</math> be continuous on <math>\Delta:|U|\leq1, 0\leq t\leq1</math>, holomorphic in <math>|u|< 1</math> when the parameter ''t'' is fixed in [0, 1], and assume that <math>\frac{\partial\varphi_j}{\partial u}</math> are not all zero at any point on <math>\Delta</math>. Then the set <math>Q(t):= \{Z_j= \varphi_j(u, t);|u|\leq 1\}</math> is called an analytic disc de-pending on a parameter ''t'', and <math>B(t):= \{Z_j= \varphi_j(u, t);|u|= 1\}</math> is called its shell. If <math>Q(t)\subset D \ (0 <t)</math> and <math>B(0)\subset D</math>, ''Q(t)'' is called Family of Oka's disk.{{R|Hitomatsu1958|}}<ref name="Kajiwara1959">{{cite journal |doi=10.2206/kyushumfs.13.37|title=Some Results on the Equivalence of Complex-Analytic Fibre Bundles|year=1959|last1=Kajiwara|first1=Joji|journal=Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics|volume=13|pages=37–48|doi-access=free}}</ref>

===== Definition =====
When <math>Q(0)\subset D</math> holds on any family of Oka's disk, ''D'' is called Oka pseudoconvex.{{R|Hitomatsu1958|}} Oka's proof of Levi's problem was that when the [[unramified]] Riemann domain over <math>\mathbb{C}^n</math><ref>{{Eom| title = Riemannian domain | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 44356}}</ref> was a domain of holomorphy (holomorphically convex), it was proved that it was necessary and sufficient that each boundary point of the domain of holomorphy is an Oka pseudoconvex.{{R|Oka'sIX|}}{{R|Kajiwara1959|}}

==== Locally pseudoconvex (a.k.a. locally Stein, Cartan pseudoconvex, local Levi property) ====
For every point <math>x \in \partial D</math> there exist a neighbourhood ''U'' of ''x'' and ''f'' holomorphic. ( i.e. <math>U \cap D</math> be holomorphically convex.) such that ''f'' cannot be extended to any neighbourhood of ''x''. i.e., let <math>\psi : X \to Y</math> be a holomorphic map, if every point <math>y\in Y</math> has a neighborhood U such that <math>\psi^{-1}(U)</math> admits a <math> \mathcal{C}^{\infty}</math>-plurisubharmonic exhaustion function (weakly 1-complete<ref>{{cite journal |doi=10.5802/aif.3226|title=On the local pseudoconvexity of certain analytic families of <math>\mathbb{C}</math>|year=2018|last1=Ohsawa|first1=Takeo|journal=Annales de l'Institut Fourier|volume=68|issue=7|pages=2811–2818|doi-access=free}}</ref>), in this situation, we call that ''X'' is locally pseudoconvex (or locally Stein) over ''Y''. As an old name, it is also called Cartan pseudoconvex. In <math>\Complex^n</math> the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy.<ref name=Ohsawa2021>{{cite journal |hdl=2433/263965|title=NISHIno's Rigidity, Locally pseudoconvex maps, and holomorphic motions (Topology of pseudoconvex domains and analysis of reproducing kernels)|journal=RIMS Kôkyûroku|date=February 2021|volume=2175|pages=27–46|last1=Ohsawa|first1=Takeo}}</ref>{{R|Hitomatsu1958|}} For example, Diederich–Fornæss<ref>{{cite journal |doi=10.1007/BF01312449|title=A smooth pseudoconvex domain without pseudoconvex exhaustion|year=1982|last1=Diederich|first1=Klas|last2=Fornæss|first2=John Erik|journal=Manuscripta Mathematica|volume=39|pages=119–123|s2cid=121224216 |url=http://eudml.org/doc/154876}}</ref> found local pseudoconvex bounded domains <math>\Omega</math> with smooth boundary on non-Kähler manifolds such that <math>\Omega</math> is not weakly 1-complete.<ref>{{cite journal |doi=10.4064/ap106-0-19| url=http://eudml.org/doc/281083 | title=Hartogs type extension theorems on some domains in Kähler manifolds | year=2012 | last1=Ohsawa | first1=Takeo | journal=[[Annales Polonici Mathematici]] | volume=106 | pages=243–254 | s2cid=123827662 | doi-access=free }}</ref>{{refn|group=note|1=Definition of weakly 1-complete.<ref>{{cite journal |doi=10.2977/prims/1195186709 |title=Weakly 1-Complete Manifold and Levi Problem|year=1981 |last1=Ohsawa |first1=Takeo |journal=Publications of the Research Institute for Mathematical Sciences |volume=17 |pages=153–164 |doi-access=free }}</ref>}}

=== Conditions equivalent to domain of holomorphy ===
For a domain <math>D \subset \mathbb C^n</math> the following conditions are equivalent:{{refn|group=note|1=In algebraic geometry, there is a problem whether it is possible to remove the singular point of the complex analytic space by performing an operation called modification<ref>{{Citation | author = Heinrich Behnke & Karl Stein | date = 1951 | title =Modifikationen komplexer Mannigfaltigkeiten und Riernannscher Gebiete| journal = Mathematische Annalen | volume = 124 | pages = 1–16 | doi =10.1007/BF01343548|zbl=0043.30301| s2cid = 120455177|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00228264X }}</ref><ref>{{Eom| title = Modification | author-last1 = Onishchik| author-first1 = A.L.| oldid = 47868}}</ref> on the complex analytic space (when n = 2, the result by Hirzebruch,<ref>{{Citation | author = Friedrich Hirzebruch | date = 1953 | title =Über vierdimensionaleRIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen| journal = Mathematische Annalen | volume = 126 | pages = 1–22 | doi =10.1007/BF01343146| hdl = 21.11116/0000-0004-3A47-C | s2cid = 122862268 | hdl-access = free }}</ref> when n = 3 the result by Zariski<ref>{{Citation | author = Oscar Zariski | date = 1944 | title =Reduction of the Singularities of Algebraic Three Dimensional Varieties| journal = Annals of Mathematics |series=Second Series| volume = 45 | issue = 3 | pages = 472–542 | doi =10.2307/1969189| jstor = 1969189 }}</ref> for algebraic varietie.), but, Grauert and Remmert has reported an example of a domain that is neither pseudoconvex nor holomorphic convex, even though it is a domain of holomorphy:
<ref>{{Citation | author = Hans Grauert & Reinhold Remmert | date = 1956 | title = Konvexität in der komplexen Analysis. Nicht-holomorph-konvexe Holomorphiegebiete und Anwendungen auf die Abbildungstheorie | journal = Commentarii Mathematici Helvetici| volume = 31 | pages = 152–183 | doi =10.1007/BF02564357|zbl =0073.30301| s2cid = 117913713 }}</ref>}}
<ol type="1">
<li> ''D'' is a domain of holomorphy.</li>
<li> ''D'' is holomorphically convex.</li>
<li> ''D'' is the union of an increasing sequence of [[analytic polyhedron]]s in ''D''.</li>
<li> ''D'' is pseudoconvex.</li>
<li> ''D'' is Locally pseudoconvex.</li></ol>

The implications <math>1 \Leftrightarrow 2 \Leftrightarrow 3 </math>,{{refn|group=note|1=This relation is called the Cartan–Thullen theorem.<ref>{{cite journal |jstor=43698735|title=Some properties of holomorphic convexity in general function algebras|last1=Tsurumi|first1=Kazuyuki|last2=Jimbo|first2=Toshiya|journal=Science Reports of the Tokyo Kyoiku Daigaku, Section A|year=1969|volume=10|issue=249/262|pages=178–183}}</ref>}} <math>1 \Rightarrow 4</math>,<ref group=note>See [[Oka's lemma]]</ref> and <math>4\Rightarrow 5</math> are standard results. Proving <math>5 \Rightarrow 1</math>, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the [[Levi problem]] (after [[Eugenio Elia Levi|E. E. Levi]]) and was solved for unramified Riemann domains over <math>\mathbb{C}^n</math> by Kiyoshi Oka,<ref group=note>Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.</ref> but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity,<ref>{{cite journal |doi=10.1007/BF01420649|title=A counterexample for the Levi problem for branched Riemann domains over <math>\mathbb{C}^n</math> |year=1978 |last1=Fornæss |first1=John Erik |journal=Mathematische Annalen |volume=234 |issue=3 |pages=275–277 |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002315858}}</ref> and then by [[Lars Hörmander]] using methods from functional analysis and partial differential equations (a consequence of <math>\bar{\partial}</math>-problem(equation) with a [[Ohsawa–Takegoshi L2 extension theorem|L2 methods]]).{{R|Hörmander1965|}}{{R|Forstnerič2011§SteinManifolds|}}{{R|Błocki2014|}}{{R|Noguchi2019}}

== Sheaves ==
The introduction of [[Sheaf (mathematics)|sheaves]] into several complex variables allowed the reformulation of and solution to several important problems in the field.

=== Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf)) ===
Oka introduced the notion which he termed "idéal de domaines indéterminés" or "ideal of indeterminate domains".{{R|Oka'sVII}}{{R|Oka'sVIII}} Specifically, it is a set <math>(I)</math> of pairs <math>(f, \delta)</math>, <math>f</math> holomorphic on a non-empty open set <math>\delta</math>, such that
<ol type="1">
<li> If <math>(f, \delta) \in (I)</math> and <math>(a, \delta')</math> is arbitrary, then <math>(af, \delta \cap \delta') \in (I)</math>.</li>
<li> For each <math>(f, \delta), (f', \delta') \in (I)</math>, then <math>(f + f', \delta \cap \delta') \in (I).</math></li>
</ol>

The origin of indeterminate domains comes from the fact that domains change depending on the pair <math>(f, \delta)</math>. Cartan{{R|Cartan1950}}{{R|Cartan1953}} translated this notion into the notion of the '''[[coherent sheaf|coherent]]''' ('''[[Sheaf (mathematics)|sheaf]]''') (Especially, coherent analytic sheaf) in sheaf cohomology.<ref name=Noguchi2019>{{cite journal |doi=10.4310/ICCM.2019.V7.N2.A2|title=A brief chronicle of the Levi (Hartog's inverse) problem, coherence and open problem |year=2019 |last1=Noguchi |first1=Junjiro |journal=Notices of the International Congress of Chinese Mathematicians |volume=7 |issue=2 |pages=19–24 |arxiv=1807.08246 |s2cid=119619733 }}</ref><ref>{{cite book|last1=Noguchi |first1=Junjiro |title=Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.x) |date=2016 |isbn=978-981-10-0289-2 |page=XVIII, 397|doi=10.1007/978-981-10-0291-5|s2cid=125752012 |url={{Google books|4gHdDAAAQBAJ|Analytic Function Theory of Several Variables: Elements of Oka's Coherence|page=PR10|plainurl=yes}}}}</ref> This name comes from
H. Cartan.<ref>{{cite book|last1=Noguchi |first1=Junjiro |title=Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.33) |date=2016 |isbn=978-981-10-0289-2 |page=XVIII, 397|doi=10.1007/978-981-10-0291-5|s2cid=125752012 |url={{Google books|4gHdDAAAQBAJ|Analytic Function Theory of Several Variables: Elements of Oka's Coherence|page=33|plainurl=yes}}}}</ref> Also, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf.<ref name="Serre1955">{{Citation|author1-first=Jean-Pierre|author1-last=Serre|author1-link=Jean-Pierre Serre|title=Faisceaux algébriques cohérents|journal=Annals of Mathematics|volume=61|pages=197–278|year=1955|issue=2|doi=10.2307/1969915|jstor=1969915|mr=0068874|url=https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf}}</ref> The notion of coherent ([[coherent sheaf cohomology]]) helped solve the problems in several complex variables.{{R|Chorlay2010|}}

=== Coherent sheaf ===

==== Definition ====
The definition of the coherent sheaf is as follows.{{R|Serre1955|}}<ref>{{cite journal| last1 = Grothendiec | first1 = Alexander | last2 =Dieudonn | first2 = Jean | year = 1960| title = Éléments de géométrie algébrique: I. Le langage des schémas (ch.0 § 5. FAISCEAUX QUASI-COHÉRENTS ET FAISCEAUX COHÉRENTS (0.5.1–0.5.3))| journal = Publications Mathématiques de l'IHÉS | volume = 4 | mr = 0217083 | url = http://www.numdam.org/item/PMIHES_1960__4__5_0| doi = 10.1007/bf02684778 | s2cid = 121855488 }}</ref><ref>{{cite book |doi=10.1007/978-3-662-09873-8_2|chapter=Local Theory of Complex Spaces |title=Several Complex Variables VII §6. Calculs of Coherent sheaves |series=Encyclopaedia of Mathematical Sciences |year=1994 |last1=Remmert |first1=R. |volume=74 |pages=7–96 |isbn=978-3-642-08150-7 | url={{Google books|WqnvCAAAQBAJ|page=40|plainurl=yes}}}}</ref><ref>{{cite book |isbn=9784431568513|doi=10.1007/978-4-431-55747-0|title=L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds|last1=Ohsawa|first1=Takeo|series=Springer Monographs in Mathematics|date=10 December 2018|url={{Google books|DIJ8DwAAQBAJ|page=25|plainurl=yes}}}}</ref>
{{R|CAaDG}}{{rp|pages=83–89}}
A '''quasi-coherent sheaf''' on a [[ringed space]] <math>(X, \mathcal O_X)</math> is a sheaf <math>\mathcal F</math> of <math>\mathcal O_X</math>-[[sheaf of modules|modules]] which has a local presentation, that is, every point in <math>X</math> has an open neighborhood <math>U</math> in which there is an [[exact sequence]]
:<math>\mathcal{O}_X^{\oplus I}|_{U} \to \mathcal{O}_X^{\oplus J}|_{U} \to \mathcal{F}|_{U} \to 0</math>
for some (possibly infinite) sets <math>I</math> and <math>J</math>.

A '''coherent sheaf''' on a ringed space <math>(X, \mathcal O_X)</math> is a sheaf <math>\mathcal F</math> satisfying the following two properties:
<ol type="1">
<li> <math>\mathcal F</math> is of ''finite type'' over <math>\mathcal O_X</math>, that is, every point in <math>X</math> has an [[open neighborhood]] <math>U</math> in <math>X</math> such that there is a surjective morphism <math>\mathcal{O}_X^{\oplus n}|_{U} \to \mathcal{F}|_{U} </math> for some natural number <math>n</math>;</li>
<li> for each open set <math>U\subseteq X</math>, integer <math>n > 0</math>, and arbitrary morphism <math>\varphi: \mathcal{O}_X^{\oplus n}|_{U} \to \mathcal{F}|_{U} </math> of <math>\mathcal O_X</math>-modules, the kernel of <math>\varphi</math> is of finite type.</li>
</ol>

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of <math>\mathcal O_X</math>-modules.

Also, [[Jean-Pierre Serre]] (1955){{R|Serre1955|}} proves that

:If in an exact sequence <math>0\to \mathcal{F}_1|_U\to\mathcal{F}_2|_U\to\mathcal{F}_3|_U\to 0</math> of sheaves of <math>\mathcal{O}</math>-modules two of the three sheaves <math>\mathcal{F}_{j}</math> are coherent, then the third is coherent as well.

==== (Oka–Cartan) coherent theorem ====
(Oka–Cartan) coherent theorem{{R|Oka'sVII|}} says that each sheaf that meets the following conditions is a coherent.<ref name="Noguchi">{{Citation | last1=Noguchi | first1=Junjiro | title=A Weak Coherence Theorem and Remarks to the Oka Theory |journal=Kodai Math. J.|volume=42|url=https://www.ms.u-tokyo.ac.jp/~noguchi/WeakcohOka_3.pdf | year=2019 | issue=3 | arxiv = 1704.07726|doi =10.2996/kmj/1572487232|pages=566–586| s2cid=119697608 }}</ref>

<ol type="i">
<li> the sheaf <math>\mathcal{O} := \mathcal{O}_{\mathbb{C}_n}</math> of [[germ (mathematics)|germs]] of holomorphic functions on <math>\mathbb{C}_n</math>, or the structure sheaf <math>\mathcal{O}_X</math> of complex submanifold or every complex analytic space <math>(X, \mathcal{O}_X)</math><ref>{{cite book |isbn=978-3-642-69582-7|title=Coherent Analytic Sheaves|last1=Grauert|first1=H.|last2=Remmert|first2=R.|date=6 December 2012|page=60|publisher=Springer |url={{Google books|title=Coherent Analytic Sheaves|blPxCAAAQBAJ|page=60|plainurl=yes}}}}</ref></li>
<li> the ideal sheaf <math>\mathcal{I} \langle A \rangle</math> of an analytic subset A of an open subset of <math>\mathbb{C}_n</math>. (Cartan 1950{{R|Cartan1950}})<ref>{{cite book |isbn=978-3-642-69582-7|title=Coherent Analytic Sheaves|last1=Grauert|first1=H.|last2=Remmert|first2=R.|date=6 December 2012|page=84|publisher=Springer |url={{Google books|title=Coherent Analytic Sheaves|blPxCAAAQBAJ|page=84|plainurl=yes}}}}</ref><ref>{{cite web |last1=Demailly |first1=Jean-Pierre |title=Basic results on Sheaves and Analytic Sets |url=https://www-fourier.ujf-grenoble.fr/~demailly/analytic_geometry_2019/sheaves_and_analytic_sets.pdf |publisher=Institut Fourier}}</ref></li>
<li> the normalization of the structure sheaf of a complex analytic space<ref>{{cite book |doi=10.1007/978-3-642-69582-7_8|chapter=Normalization of Complex Spaces |title=Coherent Analytic Sheaves |series=Grundlehren der mathematischen Wissenschaften |year=1984 |last1=Grauert |first1=Hans |last2=Remmert |first2=Reinhold |volume=265 |pages=152–166 |isbn=978-3-642-69584-1 }}</ref></li>
</ol>
From the above Serre(1955) theorem, <math>\mathcal{O}^p</math> is a coherent sheaf, also, (i) is used to prove [[Cartan's theorems A and B]].

=== Cousin problem ===
In the case of one variable complex functions, [[Mittag-Leffler's theorem]] was able to create a global meromorphic function from a given and principal parts (Cousin I problem), and [[Weierstrass factorization theorem]] was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem). However, these theorems do not hold in several complex variables because the singularities of [[Analytic function#Analytic functions of several variables|analytic function in several complex variables]] are not isolated points; these problems are called the Cousin problems and are formulated in terms of sheaf cohomology. They were first introduced in special cases by Pierre Cousin in 1895.<ref>{{cite journal |last1=Cousin |first1=Pierre |title=Sur les fonctions de ''n'' variables complexes |journal=Acta Mathematica |volume=19 |year=1895|pages=1–61 |doi=10.1007/BF02402869|doi-access=free }}</ref> It was Oka who showed the conditions for solving first Cousin problem for the domain of holomorphy{{refn|group=note|There are some counterexamples in the domain of holomorphicity regarding second Cousin problem.{{R|OkaIII|}}<ref>{{cite book |last1=Serre |first1=Jean-Pierre |title=Oeuvres - Collected Papers I |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-39815-5|page=XXIII, 598|chapter=Quelques problèmes globaux rélatifs aux variétés de Stein|chapter-url={{Google books|eaUoAKOAbUsC|Quelques problèmes globaux rélatifs aux variétés de Stein|page=265|plainurl=yes}}|language=fr}}</ref>
}} on the complex coordinate space,<ref>{{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. I. Domaines convexes par rapport aux fonctions rationnelles|journal=Journal of Science of the Hiroshima University|volume=6|year=1936|pages=245–255|doi=10.32917/hmj/1558749869|doi-access=free}}</ref><ref>{{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie|journal=Journal of Science of the Hiroshima University|volume=7|year=1937|pages=115–130|doi=10.32917/hmj/1558576819|doi-access=free}}</ref><ref name=OkaIII>{{Cite journal|first=Kiyoshi|last=Oka|title= Sur les fonctions analytiques de plusieurs variables. III–Deuxième problème de Cousin|journal=Journal of Science of the Hiroshima University|volume=9|year=1939|pages=7–19|doi=10.32917/hmj/1558490525|doi-access=free}}</ref>{{refn|group=note|This is called the classic Cousin problem.{{R|Chorlay2010|}}}} also solving the second Cousin problem with additional topological assumptions. The Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic property are pure topological;{{R|OkaIII|}}{{R|Chorlay2010|}}{{R|Huckleberry2013|}} Serre called this the [[#Definition using sheaf cohomology words|Oka principle]].<ref>{{cite journal |url=http://www.numdam.org/item/SHC_1951-1952__4__A20_0/|title=Applications de la théorie générale à divers problèmes globaux|journal=Séminaire Henri Cartan|volume=4|pages=1–26|last1=Serre|first1=J. -P}}</ref> They are now posed, and solved, for arbitrary complex manifold ''M'', in terms of conditions on ''M''. ''M'', which satisfies these conditions, is one way to define a Stein manifold. The study of the cousin's problem made us realize that in the study of several complex variables, it is possible to study of global properties from the patching of local data,{{R|Cartan1950|}} that is it has developed the theory of sheaf cohomology. (e.g.Cartan seminar.{{R|CT4|}}){{R|Chorlay2010|}}

==== First Cousin problem ====
Without the language of sheaves, the problem can be formulated as follows. On a complex manifold ''M'', one is given several meromorphic functions <math>f_i</math> along with domains <math>U_i</math> where they are defined, and where each difference <math>f_i-f_j</math> is holomorphic (wherever the difference is defined). The first Cousin problem then asks for a meromorphic function <math>f</math> on ''M'' such that <math>f-f_i</math> is ''holomorphic'' on <math>U_i</math>; in other words, that <math>f</math> shares the [[mathematical singularity|singular]] behaviour of the given local function.

Now, let '''K''' be the sheaf of meromorphic functions and '''O''' the sheaf of holomorphic functions on ''M''. The first Cousin problem can always be solved if the following map is surjective:

:<math>H^0(M,\mathbf{K}) \xrightarrow{\phi} H^0(M,\mathbf{K}/\mathbf{O}).</math>

By the [[long exact sequence in homology|long exact cohomology sequence]],

:<math>H^0(M,\mathbf{K}) \xrightarrow{\phi} H^0(M,\mathbf{K}/\mathbf{O})\to H^1(M,\mathbf{O})</math>

is exact, and so the first Cousin problem is always solvable provided that the first cohomology group ''H''1(''M'','''O''') vanishes. In particular, by [[Cartan's theorems A and B|Cartan's theorem B]], the Cousin problem is always solvable if ''M'' is a Stein manifold.

==== Second Cousin problem ====
The second Cousin problem starts with a similar set-up to the first, specifying instead that each ratio <math>f_i/f_j</math> is a non-vanishing holomorphic function (where said difference is defined). It asks for a meromorphic function <math>f</math> on ''M'' such that <math>f/f_i</math> is holomorphic and non-vanishing.

Let <math>\mathbf{O}^*</math> be the sheaf of holomorphic functions that vanish nowhere, and <math>\mathbf{K}^*</math> the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of [[abelian group]]s, and the quotient sheaf <math>\mathbf{K}^*/\mathbf{O}^*</math> is well-defined. If the following map <math>\phi</math> is surjective, then Second Cousin problem can be solved:

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*).</math>

The long exact sheaf cohomology sequence associated to the quotient is

:<math>H^0(M,\mathbf{K}^*)\xrightarrow{\phi} H^0(M,\mathbf{K}^*/\mathbf{O}^*)\to H^1(M,\mathbf{O}^*)</math>

so the second Cousin problem is solvable in all cases provided that <math>H^1(M,\mathbf{O}^*)=0.</math>

The cohomology group <math>H^1(M,\mathbf{O}^*)</math> for the multiplicative structure on <math>\mathbf{O}^*</math> can be compared with the cohomology group <math>H^1(M,\mathbf{O})</math> with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

:<math>0\to 2\pi i\Z\to \mathbf{O} \xrightarrow{\exp} \mathbf{O}^* \to 0</math>

where the leftmost sheaf is the locally constant sheaf with fiber <math>2\pi i\Z</math>. The obstruction to defining a logarithm at the level of ''H''1 is in <math>H^2(M,\Z)</math>, from the long exact cohomology sequence

:<math>H^1(M,\mathbf{O})\to H^1(M,\mathbf{O}^*)\to 2\pi i H^2(M,\Z) \to H^2(M, \mathbf{O}).</math>

When ''M'' is a Stein manifold, the middle arrow is an isomorphism because <math>H^q(M,\mathbf{O}) = 0</math> for <math>q > 0</math> so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that <math>H^2(M,\Z)=0.</math> (This condition called Oka principle.)

== Manifolds and analytic varieties with several complex variables ==

=== Stein manifold (non-compact complex manifold) ===
Since a non-compact (open) Riemann surface<ref name =Weyl1913>{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The concept of a Riemann surface | orig-year=1913 | url=https://archive.org/details/dieideederrieman00weyluoft | publisher=[[Dover Publications]] | location=New York | edition=3rd | isbn=978-0-486-47004-7 | year=2009 | mr=0069903}}</ref> always has a non-constant single-valued holomorphic function,<ref name ="Behnke–Stein1948">{{Citation | author = Heinrich Behnke & Karl Stein | title=Entwicklung analytischer Funktionen auf Riemannschen Flächen | year=1948 | journal=Mathematische Annalen| volume=120 | pages=430–461|doi=10.1007/BF01447838|zbl =0038.23502 | s2cid=122535410 }}</ref> and satisfies the [[second axiom of countability]], the open Riemann surface is in fact a ''1''-dimensional complex manifold possessing a holomorphic mapping into the complex plane <math>\mathbb C</math>. (In fact, Gunning and Narasimhan have shown (1967)<ref>{{cite journal |doi=10.1007/BF01360812|title=Immersion of open Riemann surfaces |year=1967 |last1=Gunning |first1=R. C. |last2=Narasimhan |first2=Raghavan |journal=Mathematische Annalen |volume=174 |issue=2 |pages=103–108 |s2cid=122162708 }}</ref> that every non-compact Riemann surface actually has a holomorphic ''immersion'' into the complex plane. In other words, there is a holomorphic mapping into the complex plane whose derivative never vanishes.)<ref>{{cite book |first1=J.E.|last1 =Fornaess |last2=Forstneric |first2=F |last3=Wold |first3=E.F |editor1-first=Daniel |editor1-last=Breaz |editor2-first=Michael Th. |editor2-last=Rassias |title=Advancements in Complex Analysis – Holomorphic Approximation |chapter=The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan |date=2020 |publisher=[[Springer Nature]] |pages=133–192|doi=10.1007/978-3-030-40120-7|arxiv=1802.03924 |isbn =978-3-030-40119-1 |s2cid =220266044 }}</ref> The [[Whitney embedding theorem]] tells us that every smooth ''n''-dimensional manifold can be [[Embedding|embedded]] as a smooth submanifold of <math>\mathbb{R}^{2n}</math>, whereas it is "rare" for a complex manifold to have a holomorphic embedding into <math>\mathbb C^n</math>. For example, for an arbitrary compact connected complex manifold ''X'', every holomorphic function on it is constant by Liouville's theorem, and so it cannot have any embedding into complex n-space. That is, for several complex variables, arbitrary complex manifolds do not always have holomorphic functions that are not constants. So, consider the conditions under which a complex manifold has a holomorphic function that is not a constant. Now if we had a holomorphic embedding of ''X'' into <math>\mathbb C^n</math>, then the coordinate functions of <math>\mathbb C^n</math> would restrict to nonconstant holomorphic functions on ''X'', contradicting compactness, except in the case that ''X'' is just a point. Complex manifolds that can be holomorphic embedded into <math>\mathbb C^n</math> are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.<ref>{{cite journal |doi=10.1016/j.crma.2010.11.020|title=On complex Banach manifolds similar to Stein manifolds|year=2011|last1=Patyi|first1=Imre|journal=Comptes Rendus Mathematique|volume=349|issue=1–2|pages=43–45|arxiv=1010.3738|s2cid=119631664}}</ref>

A '''Stein manifold''' is a complex [[submanifold]] of the [[vector space]] of ''n'' complex dimensions. They were introduced by and named after Karl Stein (1951).<ref>{{citation|mr=0043219|last=Stein|first= Karl|title=Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem|language=German|journal=Math. Ann. |volume=123|year=1951|pages=201–222|doi=10.1007/bf02054949|s2cid=122647212}}</ref> A '''Stein space''' is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of [[affine variety|affine varieties]] or [[affine scheme]]s in algebraic geometry. If the univalent domain on <math>\mathbb C^n</math> is connection to a manifold, can be regarded as a [[complex manifold]] and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity. Therefore, the Stein manifold is the properties of the domain of definition of the (maximal) [[analytic continuation]] of an analytic function.

==== Definition ====
Suppose ''X'' is a [[paracompact]] [[complex manifold]]s of complex dimension <math>n</math> and let <math>\mathcal O(X)</math> denote the ring of holomorphic functions on ''X''. We call ''X'' a '''Stein manifold''' if the following conditions hold:<ref>{{cite journal |arxiv=1108.2078|last1=Noguchi|first1=Junjiro|title=Another Direct Proof of Oka's Theorem (Oka IX)|year=2011|mr=3086750|journal=J. Math. Sci. Univ. Tokyo|url=https://www.ms.u-tokyo.ac.jp/journal/pdf/jms190407.pdf|volume=19|issue=4}}</ref>
<ol type="1">
<li>''X'' is holomorphically convex, i.e. for every compact subset <math>K \subset X</math>, the so-called ''holomorphically convex hull'',
:<math>\bar K = \left \{z \in X ; |f(z)| \leq \sup_{w \in K} |f(w)|, \ \forall f \in \mathcal O(X) \right \},</math>
is also a ''compact'' subset of ''X''.</li>
<li>''X'' is [[holomorphically separable]],<ref group = note>From this condition, we can see that the Stein manifold is not compact.</ref> i.e. if <math>x \neq y</math> are two points in ''X'', then there exists <math>f \in \mathcal O(X)</math> such that <math>f(x) \neq f(y).</math></li>
<li> The open neighborhood of every point on the manifold has a holomorphic [[Atlas (topology)#Charts|chart]] to the <math>\mathcal O(X)</math>.</li></ol>

Note that condition (3) can be derived from conditions (1) and (2).<ref>{{cite journal |doi=10.1007/BF01362369|title=Charakterisierung der holomorph vollständigen komplexen Räume |year=1955 |last1=Grauert |first1=Hans |journal=Mathematische Annalen |volume=129 |pages=233–259 |s2cid=122840967|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002284383}}</ref>

==== Every non-compact (open) Riemann surface is a Stein manifold ====

Let ''X'' be a connected, non-compact (open) [[Riemann surface]]. A [[Behnke–Stein theorem on Stein manifolds#Method of proof|deep theorem]] of Behnke and Stein (1948){{R|Behnke–Stein1948|}} asserts that ''X'' is a Stein manifold.

Another result, attributed to [[Hans Grauert]] and [[Helmut Röhrl]] (1956), states moreover that every [[holomorphic vector bundle]] on ''X'' is trivial. In particular, every line bundle is trivial, so <math>H^1(X, \mathcal O_X^*) =0 </math>. The [[exponential sheaf sequence]] leads to the following exact sequence:

:<math>H^1(X, \mathcal O_X) \longrightarrow H^1(X, \mathcal O_X^*) \longrightarrow H^2(X, \Z) \longrightarrow H^2(X, \mathcal O_X) </math>

Now [[Cartan's theorems A and B|Cartan's theorem B]] shows that <math>H^1(X,\mathcal{O}_X) = H^2(X,\mathcal{O}_X)=0 </math>, therefore <math>H^2(X,\Z) = 0</math>.

This is related to the solution of the [[Cousin problems|second (multiplicative) Cousin problem]].

==== Levi problems ====
Cartan extended Levi's problem to Stein manifolds.<ref>{{cite journal |last1=Cartan |first1=Henri |title=Variétés analytiques réelles et variétés analytiques complexes |journal=Bulletin de la Société Mathématique de France |year=1957 |volume=85 |pages=77–99 |doi=10.24033/bsmf.1481|doi-access=free }}</ref>
:If the [[Relatively compact subspace|relative compact open subset]] <math>D\subset X</math> of the Stein manifold X is a Locally pseudoconvex, then ''D'' is a Stein manifold, and conversely, if ''D'' is a Locally pseudoconvex, then ''X'' is a Stein manifold. i.e. Then ''X'' is a Stein manifold if and only if ''D'' is locally the Stein manifold.<ref>{{cite journal |last1=Barth |first1=Theodore J. |title=Families of nonnegative divisors |journal=Trans. Amer. Math. Soc. |year=1968 |volume=131 |pages=223–245 |doi=10.1090/S0002-9947-1968-0219751-3|doi-access=free }}</ref>

This was proved by Bremermann<ref>{{cite journal |last1=Bremermann |first1=Hans J. |title=On Oka's theorem for Stein manifolds.|journal=Seminars on Analytic Functions. Institute for Advanced Study (Princeton, N.J.) |year=1957 |volume=1 |pages=29–35|zbl=0192.18304}}</ref> by embedding it in a sufficiently high dimensional <math>\mathbb{C}^n</math>, and reducing it to the result of Oka.{{R|Oka'sIX|}}

Also, Grauert proved for arbitrary '''complex''' manifolds ''M''.{{refn|group=note|1=Levi problem is not true for domains in arbitrary manifolds.{{R|Huckleberry2013|}}<ref name="Sibony2018">{{cite journal |doi=10.1007/s00208-017-1539-x|title=Levi problem in complex manifolds|year=2018|last1=Sibony|first1=Nessim|journal=Mathematische Annalen|volume=371|issue=3–4|pages=1047–1067|arxiv=1610.07768|s2cid=119670805}}</ref><ref name="Grauert1963">{{cite journal |last1=Grauert |first1=Hans |title=Bemerkenswerte pseudokonvexe Mannigfaltigkeiten |journal=Mathematische Zeitschrift |year=1963 |volume=81 |issue=5 |pages=377–391 |doi=10.1007/BF01111528|s2cid=122214512 }}</ref>}}<ref name="Grauert1958" >{{Citation | author = Hans Grauert | title=On Levi's Problem and the Imbedding of Real-Analytic Manifolds | year=1958 | journal=Annals of Mathematics |series=Second Series| volume=68 | issue=2 | pages=460–472| zbl=0108.07804|doi=10.2307/1970257 | jstor=1970257 }}</ref>{{R|Huckleberry2013|}}{{R|Sibony2018|}}
:If the relative compact subset <math>D\subset M</math> of a arbitrary complex manifold ''M'' is a '''strongly pseudoconvex''' on ''M'', then ''M'' is a holomorphically convex (i.e. Stein manifold). Also, ''D'' is itself a Stein manifold.

And Narasimhan<ref name=NarasimhanI>{{cite journal |last1=Narasimhan |first1=Raghavan |title=The Levi problem for complex spaces |journal=Mathematische Annalen |year=1961 |volume=142 |issue=4 |pages=355–365 |doi=10.1007/BF01451029|s2cid=120565581 }}</ref><ref name=NarasimhanII>{{cite journal |last1=Narasimhan |first1=Raghavan |title=The Levi problem for complex spaces II|journal=Mathematische Annalen |year=1962 |volume=146 |issue=3 |pages=195–216 |doi=10.1007/BF01470950|s2cid=179177434 }}</ref> extended Levi's problem to [[complex analytic space]], a generalized in the singular case of complex manifolds.
:A Complex analytic space which admits a continuous strictly plurisubharmonic exhaustion function (i.e.strongly pseudoconvex) is Stein space.{{R|Siu1978}}

Levi's problem remains unresolved in the following cases;

:Suppose that ''X'' is a singular Stein space,{{refn|group=note|1=In the case of Stein space with isolated singularities, it has already been positively solved by Narasimhan.{{R|Siu1978}}{{R|Coltoiu2009}}}} <math>D \subset\subset X</math> . Suppose that for all <math>p\in \partial D</math> there is an open neighborhood <math>U (p)</math> so that <math>U\cap D</math> is Stein space. Is ''D'' itself Stein?{{R|Siu1978}}<ref>{{cite journal |last1=Fornæss |first1=John Erik |last2=Sibony |first2=Nessim |title=Some open problems in higher dimensional complex analysis and complex dynamics |year=2001 |journal = Publicacions Matemàtiques |volume=45 |issue=2 |pages=529–547 |doi=10.5565/PUBLMAT_45201_11|jstor =43736735 |url=http://ddd.uab.cat/record/1973 }}</ref><ref name = Coltoiu2009>{{cite arXiv |eprint=0905.2343|last1=Coltoiu|first1=Mihnea|title=The Levi problem on Stein spaces with singularities. A survey|year=2009|class=math.CV}}</ref>

more generalized

:Suppose that ''N'' be a Stein space and ''f'' an injective, and also <math>f :M \to N</math> a Riemann unbranched domain, such that map ''f'' is a locally pseudoconvex map (i.e. Stein morphism). Then ''M'' is itself Stein ?{{R|Coltoiu2009}}<ref>{{cite book |isbn=9784431568513|doi=10.1007/978-4-431-55747-0|url=https://books.google.com/books?id=DIJ8DwAAQBAJ&pg=PA109|title=L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds|last1=Ohsawa|first1=Takeo|series=Springer Monographs in Mathematics|date=10 December 2018}}</ref>{{rp|page=109}}

and also,

:Suppose that ''X'' be a Stein space and <math>D = \bigcup_{n\in\mathbb{N}} D_n</math> an increasing union of Stein open sets. Then ''D'' is itself Stein ?

This means that Behnke–Stein theorem, which holds for Stein manifolds, has not found a conditions to be established in Stein space. {{R|Coltoiu2009}}

===== K-complete =====
Grauert introduced the concept of K-complete in the proof of Levi's problem.

Let ''X'' is complex manifold, ''X'' is K-complete if, to each point <math>x_0\in X</math>, there exist finitely many holomorphic map <math>f_1,\dots,f_k</math> of ''X'' into <math>\Complex^p</math>, <math>p = p(x_0)</math>, such that <math>x_0</math> is an isolated point of the set <math>A = \{x\in X;f^{-1}f(x_0)\ (v=1,\dots,k)\}</math>.{{R|Grauert1958|}} This concept also applies to complex analytic space.<ref name=Andreotti&Narasimhan1964>{{cite journal |doi=10.1090/S0002-9947-1964-0159961-3|jstor=1994247|title=Oka's Heftungslemma and the Levi Problem for Complex Spaces|last1=Andreotti|first1=Aldo|last2=Narasimhan|first2=Raghavan|journal=Transactions of the American Mathematical Society|year=1964|volume=111|issue=2|pages=345–366|doi-access=free}}</ref>

==== Properties and examples of Stein manifolds ====
* The standard<ref group="note"><math>\Complex^n \times \mathbb{P}_m</math> (<math>\mathbb{P}_m</math> is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.</ref> complex space <math>\Complex^n</math> is a Stein manifold.
* Every domain of holomorphy in <math>\Complex^n</math> is a Stein manifold.{{R|FieldCM1982}}
* It can be shown quite easily that every closed complex submanifold '''of a Stein manifold''' is a Stein manifold, too.
* The embedding theorem for Stein manifolds states the following: Every Stein manifold ''X'' of complex dimension ''n'' can be embedded into <math>\Complex^{2 n+1}</math> by a [[biholomorphic]] [[proper map]].<ref>{{cite journal |last1=Raghavan |first1=Narasimhan |title=Imbedding of Holomorphically Complete Complex Spaces |journal=American Journal of Mathematics |year=1960 |volume=82 |issue=4 |pages=917–934 |doi=10.2307/2372949|jstor=2372949 }}</ref><ref>{{cite journal |last1=Eliashberg |first1=Yakov |last2=Gromov |first2=Mikhael |title=Embeddings of Stein Manifolds of Dimension n into the Affine Space of Dimension 3n/2 +1 |journal=Annals of Mathematics |series=Second Series |year=1992 |volume=136 |issue=1 |pages=123–135|doi =10.2307/2946547 |jstor=2946547 }}</ref><ref>{{cite journal |title =Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes |journal= Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris| pages=118–121| last1=Remmert|first1=Reinhold|year=1956|volume=243|url=https://gallica.bnf.fr/ark:/12148/bpt6k3195v/f118.item|language=fr|zbl=0070.30401}}</ref>

These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the [[ambient space]] (because the embedding is biholomorphic).
* Every Stein manifold of (complex) dimension ''n'' has the homotopy type of an ''n''-dimensional CW-Complex.<ref>{{cite journal |doi=10.1090/S0002-9904-1967-11839-1|title=Some remarks on parallelizable Stein manifolds |year=1967 |last1=Forster |first1=Otto |journal=Bulletin of the American Mathematical Society |volume=73 |issue=5 |pages=712–716 |doi-access=free }}</ref>
* In one complex dimension the Stein condition can be simplified: a connected [[Riemann surface]] is a Stein manifold [[if and only if]] it is not compact. This can be proved using a version of the [[Runge theorem]]<ref>{{cite journal |first=R. R. |last=Simha |title= The Behnke-Stein Theorem for Open Riemann Surfaces |journal=[[Proceedings of the American Mathematical Society]] |volume=105 |issue=4 |year=1989 |pages=876–880 |doi=10.1090/S0002-9939-1989-0953748-X |jstor=2047046|doi-access=free }}</ref> for Riemann surfaces,<ref group = note>The proof method uses an approximation by the [[Analytic polyhedron|polyhedral domain]], as in [[Oka-Weil theorem]].</ref> due to Behnke and Stein.{{R|Behnke–Stein1948|}}
* Every Stein manifold ''X'' is holomorphically spreadable, i.e. for every point <math>x \in X</math>, there are ''n'' holomorphic functions defined on all of ''X'' which form a local coordinate system when restricted to some open neighborhood of ''x''.
* The first Cousin problem can always be solved on a Stein manifold.
* Being a Stein manifold is equivalent to being a (complex) ''strongly pseudoconvex manifold''. The latter means that it has a strongly pseudoconvex (or [[plurisubharmonic function|plurisubharmonic]]) exhaustive function,{{R|Grauert1958|}} i.e. a smooth real function <math>\psi</math> on ''X'' (which can be assumed to be a [[Morse theory|Morse function]]) with <math>i \partial \bar \partial \psi >0</math>,{{R|Grauert1958|}} such that the subsets <math>\{z \in X \mid \psi (z)\leq c \}</math> are compact in ''X'' for every real number ''c''. This is a solution to the so-called '''Levi problem''',<ref>{{Eom| title = Levi problem | author-last1 = Onishchik| author-first1 = A.L.| oldid = 47620}}</ref> named after [[Eugenio Elia Levi|E. E. Levi]] (1911). The function <math>\psi</math> invites a generalization of ''Stein manifold'' to the idea of a corresponding class of compact complex manifolds with boundary called '''Stein domain'''.<ref>{{cite journal |doi=10.2977/prims/1195183303|title=A Stein domain with smooth boundary which has a product structure|year=1982|last1=Ohsawa|first1=Takeo|journal=Publications of the Research Institute for Mathematical Sciences|volume=18|issue=3|pages=1185–1186|doi-access=free}}</ref> A Stein domain is the preimage <math>\{z \mid -\infty\leq\psi(z)\leq c\}</math>. Some authors call such manifolds therefore strictly pseudoconvex manifolds.
*Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface ''X'' with a real-valued Morse function ''f'' on ''X'' such that, away from the critical points of ''f'', the field of complex tangencies to the preimage <math>X_c=f^{-1}(c)</math> is a [[Contact geometry|contact structure]] that induces an orientation on ''Xc'' agreeing with the usual orientation as the boundary of <math>f^{-1}(-\infty, c).</math> That is, <math>f^{-1}(-\infty, c)</math> is a Stein [[Symplectic filling|filling]] of ''Xc''.

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" [[holomorphic function]]s taking values in the complex numbers. See for example [[Cartan's theorems A and B]], relating to [[sheaf cohomology]].

In the [[GAGA]] set of analogies, Stein manifolds correspond to [[affine variety|affine varieties]].<ref>{{cite journal |doi=10.2307/2007052|jstor=2007052 |last1=Neeman |first1=Amnon |title=Steins, Affines and Hilbert's Fourteenth Problem |journal=Annals of Mathematics |year=1988 |volume=127 |issue=2 |pages=229–244 }}</ref>

Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is [[fibrant object|fibrant]] in the sense of so-called "holomorphic homotopy theory".

=== Complex projective varieties (compact complex manifold) ===
Meromorphic function in one-variable complex function were studied in a
compact (closed) Riemann surface, because since the [[Riemann-Roch theorem]] ([[Riemann's inequality]]) holds for compact Riemann surfaces (Therefore the theory of compact Riemann surface can be regarded as the theory of (smooth (non-singular) projective) [[algebraic curve]] over <math>\mathbb{C}</math><ref>{{cite book |doi=10.1090/gsm/005|title=Algebraic Curves and Riemann Surfaces |series=Graduate Studies in Mathematics |year=1995 |volume=5 |isbn=9780821802687|first=Rick|last= Miranda|url={{Google books|aN4bfzgHvvkC|page=195|plainurl=yes}}}}</ref><ref>{{cite book |url={{Google books|FQslb7pH8EgC|page=130|plainurl=yes}} | title=Algebraic Geometry over the Complex Numbers | isbn=9781461418092 | last1=Arapura | first1=Donu | date=15 February 2012 | publisher=Springer }}</ref>). In fact, compact Riemann surface had a non-constant single-valued meromorphic function{{R|Weyl1913|}}, and also a compact Riemann surface had enough meromorphic functions. A compact one-dimensional complex manifold was a Riemann sphere <math>\widehat\mathbb{C} \cong \mathbb{CP}^1</math>. However, the abstract notion of a compact Riemann surface is always algebraizable (The [[Riemann's existence theorem]], [[Kodaira embedding theorem]].),<ref group=note>Note that the Riemann extension theorem and its references explained in the linked article includes a generalized version of the Riemann extension theorem by Grothendieck that was proved using the GAGA principle, also every one-dimensional compact complex manifold is a Hodge manifold.</ref> but it is not easy to verify which compact complex analytic spaces are algebraizable.<ref>{{cite book |doi=10.1007/978-3-642-60925-1_1|chapter=Cohomology of Algebraic Varieties |title=Algebraic Geometry II |series=Encyclopaedia of Mathematical Sciences |year=1996 |last1=Danilov |first1=V. I. |volume=35 |pages=1–125 |isbn=978-3-642-64607-2| url={{Google books|nDAiCQAAQBAJ|page=70|plainurl=yes}}}}</ref> In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions.{{R|Ohsawa2021|}} However, there is a Siegel result that gives the necessary conditions for compact complex manifolds to be algebraic. <ref>{{Cite book| last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | zbl=0367.14001 | year=1977 | volume=52 | url={{Google books|7z4mBQAAQBAJ|Algebraic Geometry|page=442|plainurl=yes}}|doi=10.1007/978-1-4757-3849-0| s2cid=197660097 }}</ref> The generalization of the Riemann-Roch theorem to several complex variables was first extended to compact analytic surfaces by Kodaira,<ref>{{cite journal |doi=10.2307/2372120|jstor=2372120 |title=The Theorem of Riemann-Roch on Compact Analytic Surfaces |last1=Kodaira |first1=Kunihiko |journal=American Journal of Mathematics |year=1951 |volume=73 |issue=4 |pages=813–875 }}</ref> Kodaira also extended the theorem to three-dimensional,<ref>{{cite journal |doi=10.2307/1969802|jstor=1969802 |last1=Kodaira |first1=Kunihiko |title=The Theorem of Riemann-Roch for Adjoint Systems on 3-Dimensional Algebraic Varieties |journal=Annals of Mathematics |year=1952 |volume=56 |issue=2 |pages=298–342 }}</ref> and n-dimensional Kähler varieties.<ref>{{cite journal |jstor=88542 |last1=Kodaira |first1=Kunihiko |title=On the Theorem of Riemann-Roch for Adjoint Systems on Kahlerian Varieties |journal=Proceedings of the National Academy of Sciences of the United States of America |year=1952 |volume=38 |issue=6 |pages=522–527 |doi=10.1073/pnas.38.6.522 |pmid=16589138 |pmc=1063603 |bibcode=1952PNAS...38..522K |doi-access=free }}</ref> Serre formulated the Riemann-Roch theorem as a problem of dimension of [[coherent sheaf cohomology]],{{R|IWS}} and also Serre proved [[Serre duality]].<ref>{{Citation | author1-last=Serre | author1-first=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Un théorème de dualité | journal=Commentarii Mathematici Helvetici | volume=29 | year=1955 | pages=9–26 | mr=0067489 | doi=10.1007/BF02564268| s2cid=123643759 |url=https://doi.org/10.5169/seals-23275}}</ref> Cartan–Serre proved the following property:<ref>{{cite journal |url=https://gallica.bnf.fr/ark:/12148/bpt6k3189t/f128.item| zbl=0050.17701 | title=Un théorème de finitude concernant les variétés analytiques compactes | journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris | year=1953 | volume=237 | pages=128–130 | last1=Cartan | first1=Henri | last2=Serre | first2=Jean-Pierre }}</ref> the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M.<ref>{{cite book |doi=10.1007/BFb0093697|chapter=Vector bundles over complex manifolds |title=Holomorphic Vector Bundles over Compact Complex Surfaces |series=Lecture Notes in Mathematics |year=1996 |last1=Brînzănescu |first1=Vasile |volume=1624 |pages=1–27 |isbn=978-3-540-61018-2 }}</ref> Riemann–Roch on a Riemann surface for a vector bundle was proved by [[Vector bundles on algebraic curves|Weil]] in 1938.<ref>{{cite journal |doi=10.1515/crll.1938.179.129|url=http://eudml.org/doc/150043 |title=Zur algebraischen Theorie der algebraischen Funktionen. (Aus einem Brief an H. Hasse.) |year=1938 |last1=Weil |first1=A. |journal=Journal für die reine und angewandte Mathematik |volume=179 |pages=129–133|s2cid=116472982 }}</ref>
[[Hirzebruch–Riemann–Roch theorem|Hirzebruch]] generalized the theorem to compact complex manifolds in 1994<ref>{{cite book |doi=10.1007/978-3-642-62018-8|title=Topological Methods in Algebraic Geometry |year=1966 |last1=Hirzebruch |first1=Friedrich |isbn=978-3-540-58663-0 }}</ref> and [[Grothendieck–Hirzebruch–Riemann–Roch theorem|Grothendieck]] generalized it to a relative version (relative statements about [[morphism]]s.).<ref>{{cite book | last = Berthelot | first = Pierre |editor=Alexandre Grothendieck |editor2=Luc Illusie | title = Théorie des Intersections et Théorème de Riemann-Roch | series = Lecture Notes in Mathematics | year = 1971 | volume = 225 | publisher = Springer Science+Business Media| pages = xii+700 |doi=10.1007/BFb0066283 |isbn= 978-3-540-05647-8}}</ref><ref>
{{Cite journal | last1=Borel | first1=Armand | last2=Serre | first2=Jean-Pierre | title=Le théorème de Riemann–Roch | mr=0116022 | year=1958 | journal=Bulletin de la Société Mathématique de France | volume=86 | pages=97–136 | doi=10.24033/bsmf.1500 | doi-access=free }}</ref> Next, we generalize the result that the compact Riemann surfaces are projective, to the high-dimensional case, specifically, consider the conditions that when embedding of compact complex submanifold ''X'' into the complex projective space <math>\mathbb{CP}^n</math>. <ref group=note>This is the standard method for compactification of <math>\mathbb{C}^n</math>, but not the only method like the Riemann sphere that was compactification of <math>\mathbb{C}</math>.</ref> i.e., gives the conditions when a compact complex manifold is projective. The [[Kodaira vanishing theorem]] (1954) and its generalization [[Nakano vanishing theorem]] etc. gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of [[Positive form|positivity]]. As an example given by this theorem, [[Kodaira embedding theorem]]<ref>{{cite journal |last1=Kodaira |first1=K. |title=On Kahler Varieties of Restricted Type (An Intrinsic Characterization of Algebraic Varieties)|journal=Annals of Mathematics |series=Second Series |year=1954 |volume=60 |issue=1 |pages=28–48 |doi=10.2307/1969701|jstor=1969701 }}</ref> says that a compact [[Kähler manifold]] ''M'', with a Hodge metric, there is a complex-analytic embedding of ''M'' into [[complex projective space]] of enough high-dimension ''N''. [[GAGA#Chow's theorem|Chow's theorem]]<ref>{{cite journal |last1=Chow |first1=Wei-Liang |title=On Compact Complex Analytic Varieties |journal=American Journal of Mathematics |year=1949 |volume=71 |issue=2 |pages=893–914 |doi=10.2307/2372375|jstor=2372375 }}</ref> shows that the complex analytic subspace (subvariety) of a closed complex projective space to be an algebraic that is, so it is the common zero of some homogeneous polynomials, such a relationship is one example of what is called Serre's [[Algebraic geometry and analytic geometry|GAGA principle]].{{R|GAGA}} The complex analytic sub-space(variety) of the complex projective space has both algebraic and analytic properties. Then combined with Kodaira's result, a compact Kähler manifold ''M'' embeds as an algebraic variety. This gives an example of a complex manifold with enough meromorphic functions. Similarities in the Levi problems on the complex projective space <math>\mathbb{CP}^n</math>, have been proved in some patterns, for example by Takeuchi.{{R|Siu1978}}<ref>{{cite journal |doi=10.1215/21562261-1625181|title=On the complement of effective divisors with semipositive normal bundle|year=2012|last1=Ohsawa|first1=Takeo|journal=Kyoto Journal of Mathematics|volume=52|issue=3|s2cid=121799985 |doi-access=free}}</ref><ref>{{cite journal |last1=Matsumoto |first1=Kazuko |title=Takeuchi's equality for the levi form of the Fubini–Study distance to complex submanifolds in complex projective spaces |journal=Kyushu Journal of Mathematics |date=2018 |volume=72 |issue=1 |pages=107–121 |doi=10.2206/kyushujm.72.107|doi-access=free }}</ref><ref>{{cite journal |doi=10.2969/jmsj/01620159|title=Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projecti|year=1964|last1=Takeuchi|first1=Akira|journal=Journal of the Mathematical Society of Japan|volume=16|issue=2|s2cid=122894640 |doi-access=free}}</ref> Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as [[Hodge theory]]. Also, the [[deformation theory]] of compact complex manifolds has developed as Kodaira–Spencer theory. However, despite being a compact complex manifold, there are counterexample of that cannot be embedded in projective space and are not algebraic.<ref>{{cite journal |doi=10.2307/1969750|jstor=1969750|last1=Calabi|first1=Eugenio|last2=Eckmann|first2=Beno|title=A Class of Compact, Complex Manifolds Which are not Algebraic|journal=Annals of Mathematics|year=1953|volume=58|issue=3|pages=494–500}}</ref>

==See also==
*[[Bicomplex number]]
*[[Complex geometry]]
*[[CR manifold]]
*[[Dolbeault cohomology]]
*[[Harmonic map]]s
*[[Harmonic morphism]]s
*[[Infinite-dimensional holomorphy]]
*[[Oka–Weil theorem]]

== Annotation ==
{{Reflist|group=note}}

==References==

=== Inline citations ===
{{Reflist}}

=== Textbooks ===
{{Refbegin|2}}
*{{cite book |doi=10.1007/978-3-642-99659-7|title=Theorie der Funktionen mehrerer komplexer Veränderlichen|year=1934|last1=Behnke|first1=H.|last2=Thullen|first2=P.|isbn=978-3-642-98844-8}}
*{{cite book | last1=Bochner | first1=S. | last2=Martin | first2=W.T.| title=Several Complex Variables | publisher=Princeton University Press | series=Princeton mathematical series | year=1948 | isbn=978-0-598-34865-4 }}
*{{cite book|last1=Forster | first1=Otto | author1-link= Otto Forster | title=Lectures on Riemann surfaces | publisher=Springer Verlag | location=New-York | series=Graduate Text in Mathematics | isbn=0-387-90617-7 | year=1981 | volume=81}}
*{{cite book|title=Введение в теорию аналитических функции многих комплексных переменных|last1=Фукс|first1=Б.А.|oclc=896179082|year=1962|language=ru}}
**{{cite book |isbn=978-1-4704-4428-0|title=Theory of Analytic Functions of Several Complex Variables|last1=Fuks|first1=Boris Abramovich|year=1963|publisher=American Mathematical Society }}
*{{cite book |last1=Cartan |first1=Henri |last2=Takahashi |first2=Reiji |title=Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes |date=1992 |publisher=Paris : Hermann |isbn=9782705652159 |page=231 |edition=6é. ed., nouv. tir |language=French}}
** {{cite book |last1=Cartan |first1=Henri |title=Elementary Theory of Analytic Functions of One or Several Complex Variables |date=1992 |publisher=Courier Corporation |isbn=9780486318677 |page=228 |url={{Google books|xUHDAgAAQBAJ|Elementary Theory of Analytic Functions of One or Several Complex Variables|plainurl=yes}}}}
*{{cite book |last1=Freitag |first1=Eberhard |author1-link= Eberhard Freitag |title=Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions |series=Universitext |date=2011 |publisher=Springer |doi=10.1007/978-3-642-20554-5 |isbn=978-3-642-20554-5 |edition=2 |url=https://link.springer.com/book/10.1007/978-3-642-20554-5}}
*{{cite book |doi=10.1007/978-3-642-22250-4|title=Stein Manifolds and Holomorphic Mappings |year=2011 |last1=Forstnerič |first1=Franc |series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |volume=56 |isbn=978-3-642-22249-8 }}
*{{citation|mr=0580152|last1=Grauert|first1= Hans|last2= Remmert|first2= Reinhold|title=Theory of Stein spaces|series=Grundlehren der Mathematischen Wissenschaften |volume=236|publisher= Springer-Verlag|place= Berlin-New York|year= 1979|isbn= 3-540-90388-7 }}
*{{cite book |last1=Шабат |first1=Б.В. |title=Введение в комплексный анализ / Vvedenie v kompleksnyĭ analiz |publisher=Nauka, Glav. red. fiziko-matematicheskoĭ lit-ry, Moskva|oclc=14003250|year=1985|language=ru}}
**{{cite book|doi=10.1090/mmono/110|title=Introduction to Complex Analysis Part II. Functions of Several Variables|series=Translations of Mathematical Monographs|year=1992|volume=110|isbn=9780821819753}}
*{{Citation |author =[[Lars Hörmander]] |date=1990 |orig-year=1966 |title=An Introduction to Complex Analysis in Several Variables |edition=3rd |publisher=North Holland |isbn=978-1-493-30273-4 |url={{Google books|MaM7AAAAQBAJ|An Introduction to Complex Analysis in Several Variables|plainurl=yes}}}}
*{{cite book |isbn=978-0-8218-2165-7|title=Analytic Functions of Several Complex Variables|last1=Gunning|first1=Robert Clifford|last2=Rossi|first2=Hugo|year=2009|publisher=American Mathematical Soc. |url={{Google books|title=Analytic Functions of Several Complex Variables|wsqFAwAAQBAJ|plainurl=yes}}}}
*{{cite book |isbn=9783110838350|title=Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|last1=Kaup|first1=Ludger|last2=Kaup|first2=Burchard|date=9 May 2011|publisher=Walter de Gruyter |url={{Google books|4YVXCgewhTIC|Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|plainurl=yes}}}}
*{{Cite book|last=Kodaira |first=Kunihiko |title=Complex Manifolds and Deformation of Complex Structures|doi=10.1007/b138372 |series=Classics in Mathematics|date=17 November 2004 |publisher=Springer |isbn=3-540-22614-1|url={{Google books|AmvdBwAAQBAJ|An Introduction to Complex Analysis in Several Variables|plainurl=yes}}}}
*{{cite book|last1=Krantz|first1=Steven G.|title=Function Theory of Several Complex Variables |date=1992 |publisher=AMS Chelsea Publishing |isbn=978-0-8218-2724-6 |page=340 |edition=Second|doi =10.1090/chel/340}}
*{{cite book |doi=10.1007/978-1-4757-1918-5|title=Holomorphic Functions and Integral Representations in Several Complex Variables|series=Graduate Texts in Mathematics|year=1986|volume=108|isbn=978-1-4419-3078-1}}
*{{cite book |doi=10.1007/3-7643-7491-8|title=Introduction to Complex Analysis in Several Variables|year=2005|isbn=3-7643-7490-X}}
*{{cite book|last1=Noguchi |first1=Junjiro |title=Analytic Function Theory of Several Variables Elements of Oka's Coherence |date=2016 |isbn=978-981-10-0289-2 |page=XVIII, 397|doi=10.1007/978-981-10-0291-5|s2cid=125752012 }}
*{{cite book |isbn=9780486458120|title=Methods of the Theory of Functions of Many Complex Variables|last1=Vladimirov|first1=Vasiliy Sergeyevich|last2=Technica|first2=Scripta|date=January 2007|publisher=Courier Corporation |url={{Google books|JO4dlEr79GYC|Methods of the Theory of Functions of Many Complex Variables|plainurl=yes}}}}
{{Refend}}

=== Encyclopedia of Mathematics ===
{{Refbegin|2}}
*{{Eom| title = Analytic function | author-last1 = Gonchar| author-first1 = A.A.| author-last2 = Shabat| author-first2 =B.V.| oldid = 51340}}
*{{Eom| title = Power series | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 44404}}
*{{Eom| title = Biholomorphic mapping | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 33098}}
*{{Eom| title = Reinhardt domain | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 48495}}
*{{Eom| title = Hartogs theorem | author-last1 = Chirka| author-first1 = E.M.| oldid = 40754}}
*{{Eom| title =Domain of holomorphy| author-last1 = Gonchar| author-first1 = A.A.| author-last2 = Vladimirov| author-first2 =V.S.| oldid = 46763}}
*{{Eom| title = Pseudo-convex and pseudo-concave | author-last1 = Onishchik| author-first1 = A.L.| oldid = 48344}}
*{{Eom| title = Plurisubharmonic function | author-last1 = Solomentsev| author-first1 = E.D.| oldid = 48192}}
*{{Eom| title = Quasi-coherent sheaf | author-last1 = Danilov| author-first1 = V.I.| oldid = 48377}}
*{{Eom| title = Coherent sheaf | author-last1 = Onishchik| author-first1 = A.L.| oldid = 30768}}
*{{Eom| title = Coherent analytic sheaf | author-last1 = Onishchik| author-first1 = A.L.| oldid = 33071}}
*{{Eom| title = Coherent algebraic sheaf | author-last1 = Danilov| author-first1 = V.I.| oldid = 41096}}
*{{Eom| title = Oka theorems | author-last1 = Chirka| author-first1 = E.M.| oldid = 44640}}
*{{Eom| title = Cousin problems | author-last1 = Chirka| author-first1 = E.M.| oldid = 46538}}
*{{Eom| title = Stein manifold | author-last1 = Onishchik| author-first1 = A.L.| oldid = 48831}}
*{{Eom| title = Finiteness theorems | author-last1 = Parshin| author-first1 = A.N.| oldid = 44303}}
{{Refend}}

== Further reading ==
{{Refbegin|2}}
*{{Cite book| last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | zbl=0367.14001 | year=1977 | volume=52|doi=10.1007/978-1-4757-3849-0| s2cid=197660097 }}
*{{Citation|last1=Krantz |first1=Steven G. |title=What is Several Complex Variables? |journal=The American Mathematical Monthly |year=1987 |volume=94 |issue=3 |pages=236–256 |doi=10.2307/2323391|jstor=2323391 }}
*{{cite journal |doi=10.2307/2316199|jstor=2316199 |last1=Seebach |first1=J. Arthur |last2=Seebach |first2=Linda A. |last3=Steen |first3=Lynn A. |title=What is a Sheaf? |journal=The American Mathematical Monthly |year=1970 |volume=77 |issue=7 |pages=681–703 }}
*{{Citation|first1=Kiyoshi|last1=Oka | editor1-last=Remmert|editor1-first=R.|title=Collected Papers |date=1984 |publisher=Springer-Verlag Berlin Heidelberg |isbn=978-3-662-43412-3 |page=XIV, 226}}
*{{cite journal |doi=10.1090/S0002-9904-1956-10013-X|title=Scientific report on the second summer institute, several complex variables. Part I. Report on the analysis seminar|year=1956|last1=Martin|first1=W. T.|journal=Bulletin of the American Mathematical Society|volume=62|issue=2|pages=79–102|doi-access=free}}
*{{cite journal |doi=10.1090/S0002-9904-1956-10015-3|title=Scientific report on the second summer institute, several complex variables. Part II. Complex manifolds|year=1956|last1=Chern|first1=Shiing-Shen|journal=Bulletin of the American Mathematical Society|volume=62|issue=2|pages=101–118|doi-access=free}}
*{{cite journal |doi=10.1090/S0002-9904-1956-10018-9|title=Scientific report on the second summer institute, several complex variables. Part III. Algebraic sheaf theory|year=1956|last1=Zariski|first1=Oscar|journal=Bulletin of the American Mathematical Society|volume=62|issue=2|pages=117–142|doi-access=free}}
*{{cite journal |last1=Remmert |first1=Reinhold |title=From Riemann Surfaces to Complex Spaces |journal=Séminaires et Congrès |date=1998|zbl=1044.01520|url=https://www.emis.de/journals/SC/1998/3/pdf/smf_sem-cong_3_203-241.pdf}}
{{Refend}}

==External links==
* [https://www.jirka.org/scv/ Tasty Bits of Several Complex Variables] open source book by Jiří Lebl
* [https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf Complex Analytic and Differential Geometry] ([https://www-fourier.ujf-grenoble.fr/~demailly/documents.html OpenContent book See B2])
*{{cite book
| last = Demailly | first = Jean-Pierre
| editor1-last = Harinck | editor1-first = Pascale
| editor2-last = Plagne | editor2-first = Alain
| editor3-last = Sabbah | editor3-first = Claude
| contribution = Henri Cartan et les fonctions holomorphes de plusieurs variables
| contribution-url = https://www.math.polytechnique.fr/xups/xups12-03.pdf
| isbn = 978-2-7302-1610-4
| location = Palaiseau
| pages = 99–168
| publisher = Les Éditions de l'École Polytechnique
| title = Henri Cartan et André Weil. Mathématiciens du XXesiècle. Journées mathématiques X-UPS, Palaiseau, France, May 3–4, 2012
| year = 2012}}
* Victor Guillemin. 18.117 [https://ocw.mit.edu/courses/mathematics/18-117-topics-in-several-complex-variables-spring-2005/lecture-notes Topics in Several Complex Variables]. Spring 2005. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons [[BY-NC-SA]].
*{{PlanetMath attribution
|urlname=ReinhardtDomain |title=Reinhardt domain
|urlname2=Holomorphicallyconvex |title2=Holomorphically convex
|urlname3=Domainofholomorphy |title3=Domain of holomorphy
|urlname4=polydisc |title4=polydisc
|urlname5=biholomorphicallyequivalent |title5=biholomorphically equivalent
|urlname6=Levipseudoconvex |title6=Levi pseudoconvex
|urlname7=pseudoconvex |title7=Pseudoconvex
|urlname8=ExhaustionFunction |title8=exhaustion function
}}
{{Authority control}}

[[Category:Several complex variables| ]]
[[Category:Multivariable calculus]]

Bar Kokhba revolt

2023-12-15T04:52:50Z

IntegralPython: /* Rebel Judean statehood */ deleted accidental repetition of the word Bar Kokhba?

{{pp|small=yes}}
{{Expand German|topic=hist}}
{{Short description|Jewish rebellion against Roman rule (132–136 CE)}}
{{Infobox military conflict
| conflict = Bar Kokhba revolt {{nobold|{{lang|he|{{Script/Hebrew|מֶרֶד בַּר כּוֹכְבָא}}|rtl=yes}}}}
| width = 315px
| partof = the [[Jewish–Roman wars]]
| image = Knesset Menorah P5200010 Bar Kochvah.JPG
| image_size = 300
| alt = Close-up view of the rebellion's leader on a large menorah sculpture in Jerusalem
| caption = Detail of [[Simon bar Kokhba]] from Israel's [[Knesset Menorah]]
| date = 132–136 CE (main phase: autumn 132 – summer 135)
| place = [[Judaea (Roman province)|Judea]], Roman Empire
| territory = Destruction of the rebels' [[#Rebel Judean statehood|Jewish state]] by the Roman army
| result = Roman victory{{bulletedlist
| Restructuring of Judea as [[Syria Palaestina]]
| Massacre of the Judean populace
| Suppression of Jewish religious/political autonomy by [[Hadrian]]
| Expulsion of the Jews from [[Jerusalem]]}}
| combatant1 = [[Roman Empire]]
| combatant2 = [[Judea|Judeans]]
| commander1 = {{unbulletedlist
| '''[[Hadrian]]'''
| [[Quintus Tineius Rufus (consul 127)|Quintus Tineius Rufus]]
| [[Sextus Julius Severus]]
| [[Gaius Quinctius Certus Poblicius Marcellus|Gaius Poblicius Marcellus]]
| [[Titus Haterius Nepos (consul)|Titus Haterius Nepos]]
| [[Quintus Lollius Urbicus]]}}
| commander2 = {{unbulletedlist
| '''[[Simon bar Kokhba]]{{KIA}}'''
| [[Eleazar of Modi'im]]{{KIA}}
| [[Rabbi Akiva|Akiva ben Joseph]]{{Executed}}
| [[Yeshua ben Galgula]]{{KIA}}
| Yonatan ben Bai'in
| Masbelah ben Shimon
| Eleazar ben Khita
| Yehuda bar Menashe
| Shimon ben Matanya}}
| units1 = [[Legio III Cyrenaica]] [[Legio X Fretensis]] [[Legio VI Ferrata]] [[Legio III Gallica]] [[Legio XXII Deiotariana]] [[Legio II Traiana Fortis|Legio II Traiana]] [[Legio X Gemina]] [[Legio IX Hispana]]? [[Legio V Macedonica]] (partial) [[Legio XI Claudia]] (partial) [[Legio XII Fulminata]] (partial) [[Legio IV Flavia Felix]] (partial)
| units2 = Bar Kokhba's army • Bar Kokhba's guard • Local militias [[Samaritans|Samaritan]] Youth Bands
| strength1 = 2 legions – 20,000 (132–133) 5 legions – 80,000 (133–134) 6–7 full legions, cohorts of 5–6 more, 30–50 auxilary units – 120,000 (134–135)
| strength2 = 200,000–400,000 militiamen • 12,000 Bar Kokhba's guard force
| casualties1 = [[Legio XXII Deiotariana]] possibly destroyed<ref name=keppie>L. J. F. Keppie (2000) ''Legions and veterans: Roman army papers 1971–2000'' Franz Steiner Verlag, {{ISBN|3-515-07744-8}} pp. 228–229</ref> [[Legio IX Hispana]] possibly disbanded<ref>{{Citation |title=Two Legions: The Same Fate? |last=Menachem |first=Mor |date= |jstor=20186341 |url=https://www.jstor.org/stable/20186341}}</ref>{{efn|Legion was also possibly disbanded as a result of the campaigns in Brittania or [[Roman-Parthian War of 161-166]]}} [[Legio X Fretensis]] sustained heavy casualties<ref name=mor334>Mor, M. ''The Second Jewish Revolt: The Bar Kokhba War, 132-136 CE''. Brill, 2016. p. 334.</ref>
| casualties2 = 200,000–400,000 militiamen killed or enslaved
}}
{{Campaignbox Bar Kokhba revolt}}
{{Campaignbox Jewish-Roman wars}}
The '''Bar Kokhba revolt''' ({{lang-he|מֶרֶד בַּר כּוֹכְבָא}} {{Transliteration|he|Mereḏ Bar Kōḵəḇā}}) was a large-scale armed rebellion initiated by the [[Jews]] of [[Judaea (Roman province)|Judea]], led by [[Simon bar Kokhba]], against the [[Roman Empire]] in 132 CE.<ref name="Axelrod 2009 29">{{cite book |last=Axelrod |first=Alan |url=https://books.google.com/books?id=8x322-89x3MC&q=In+132%2C+a+revolt+led+by+Bar+Kokhba+quickly+spread+from+Modi%27in&pg=PA29 |title=Little-Known Wars of Great and Lasting Impact |publisher=Fair Winds Press |year=2009 |isbn=9781592333752 |page=29}}</ref> Lasting until 136, it was the third and final escalation of the [[Jewish–Roman wars]].<ref>for the year 136, see: W. Eck, ''The Bar Kokhba Revolt: The Roman Point of View'', pp. 87–88.</ref> Like the [[First Jewish–Roman War]] and the [[Kitos War|Second Jewish–Roman War]], the Bar Kokhba revolt resulted in a total Jewish defeat; Bar Kokhba himself was killed by Roman troops at [[Betar (ancient village)|Betar]] in 135 and the Jewish rebels who remained after his death were all killed or enslaved within the next year.

Roman rule in Judea was not well-received among the Jewish population, especially after the destruction of the [[Second Temple]] during the [[Siege of Jerusalem (70 CE)|Roman siege of Jerusalem]] in 70. The Romans had also continued to maintain a large military presence across the province; pushed unpopular changes in administrative and economic life;<ref>{{cite book |last1=Davies |first1=W. D. (William David) |last2=Finkelstein |first2=Louis |last3=Horbury |first3=William |last4=Sturdy |first4=John |last5=Katz |first5=Steven T. |last6=Hart |first6=Mitchell Bryan |last7=Michels |first7=Tony |last8=Karp |first8=Jonathan |title=The Cambridge history of Judaism |date=1984 |publisher=Cambridge ; New York : Cambridge University Press |isbn=978-0-521-21880-1 |page=106 |url=https://archive.org/details/cambridgehis_xxxx_1984_004_8494287/page/n11/mode/2up}}</ref> constructed the colony of [[Aelia Capitolina]] over the destroyed city of [[Jerusalem]]; and erected a place of worship for [[Jupiter (mythology)|Jupiter]] on Jerusalem's [[Temple Mount]], where the Jews' Second Temple had stood.<ref name="Eshel" /> [[Rabbinic literature]] and the [[Church Fathers]] emphasize the role of [[Quintus Tineius Rufus (consul 127)|Quintus Tineius Rufus]], the erstwhile Roman governor of Judea, in provoking the Bar Kokhba revolt.<ref name="William David Davies p. 35">{{cite book |last1=Katz |first1=Steven T. |title=The Cambridge history of Judaism |date=2006 |publisher=Cambridge University press |isbn=978-0-521-77248-8 |page=35 |url=9780521772488}}</ref> The charismatic and messianic nature of Bar Kokhba may have also been a factor in popularizing the uprising across all of Judea.<ref>Mor 2016, p. 11.</ref>

With the onset of the conflict, initial rebel victories established an independent Jewish enclave covering much of the province for several years. Bar Kokhba was appointed ''[[Nasi (Hebrew title)|nasi]]'' ({{Lang|he|נָשִׂיא|rtl=yes}}, {{Literal translation|prince}}) of the rebels' provisional state, and much of Judea's populace regarded him as the [[Messiah in Judaism|Messiah of Judaism]] who would restore [[History of the Jews and Judaism in the Land of Israel|Jewish national independence]].<ref>{{cite book |author=[[John S. Evans]] |title=The Prophecies of Daniel 2 |year=2008 |publisher=Xulon Press |quote=Known as the Bar Kokhba Revolt, after its charismatic leader, Simon Bar Kokhba, whom many Jews regarded as their promised messiah |url=https://books.google.com/books?id=_lfSnZiVLpsC&pg=PA174 |isbn=9781604779035}}</ref> This initial setback for the Romans, however, led [[Hadrian]] to assemble a large army — six full [[Roman legion|legions]] with [[Auxilia|auxiliaries]] and other elements from up to six additional legions, all under the command of [[Sextus Julius Severus]] — and launch an extensive military campaign across Judea in 134, ultimately crushing the revolt.<ref name="Israel-Tour-Day">{{cite web|archive-url=https://web.archive.org/web/20110616233531/http://www.rsy-netzer.org.uk/newsletters/mailing-view/65.html|archive-date=16 June 2011|title=Israel Tour Daily Newsletter|url=http://www.rsy-netzer.org.uk/newsletters/mailing-view/65.html|date=27 July 2010}}</ref> The killing of Bar Kokhba and the subsequent defeat of his rebels yielded disastrous consequences for Judea's Jewish populace, even more so than the crackdown that had taken place during and after the First Jewish–Roman War.<ref name="Taylor">{{cite book |last=Taylor |first=J. E. |publisher=Oxford University Press |quote=These texts, combined with the relics of those who hid in caves along the western side of the Dead Sea, tells us a great deal. What is clear from the evidence of both skeletal remains and artefacts is that the Roman assault on the Jewish population of the Dead Sea was so severe and comprehensive that no one came to retrieve precious legal documents, or bury the dead. Up until this date the Bar Kokhba documents indicate that towns, villages and ports where Jews lived were busy with industry and activity. Afterwards there is an eerie silence, and the archaeological record testifies to little Jewish presence until the Byzantine era, in En Gedi. This picture coheres with what we have already determined in Part I of this study, that the crucial date for what can only be described as genocide, and the devastation of Jews and Judaism within central Judea, was 135 CE and not, as usually assumed, 70 CE, despite the siege of Jerusalem and the Temple's destruction |url= https://books.google.com/books?id=XWIMFY4VnI4C&pg=PA243 |title=The Essenes, the Scrolls, and the Dead Sea|date= 15 November 2012 |isbn=9780199554485}}</ref> Based on archeological evidence and ancient sources, Judea was heavily depopulated as a result of many of the Jews being killed or expelled by Roman troops, and a significant number of captives were sold into slavery.{{sfn|Mor|2016|p=471}}<ref name="raviv2021">{{Cite journal|last1=Raviv|first1=Dvir|last2=Ben David|first2=Chaim|date=2021-05-27|title=Cassius Dio's figures for the demographic consequences of the Bar Kokhba War: Exaggeration or reliable account?|journal=Journal of Roman Archaeology|volume=34|issue=2|language=en|pages=585–607|doi=10.1017/S1047759421000271|s2cid=236389017|issn=1047-7594|doi-access=free}}</ref><ref>{{cite book | last1=Powell | first1=L. | last2=Dennis | first2=P. | title=The Bar Kokhba War AD 132–136: The last Jewish revolt against Imperial Rome | publisher=Bloomsbury Publishing | series=Campaign | year=2017 | isbn=978-1-4728-1799-0 | url=https://books.google.com/books?id=hhwrDwAAQBAJ | page=80}}</ref><ref name=":2">{{Cite book |last=Jones |first=A.H.M. |title=The Cities of the Eastern Roman Provinces |publisher=Oxford |year=1971 |edition=2nd |pages=277 |quote=This provoked the last Jewish war, which seems from our meager accounts [...] to have resulted in the desolation of Judaea and the practical extermination of its Jewish population.}}</ref> Roman casualties are also considered to have been heavy; the Roman army disbanded [[Legio XXII Deiotariana]] following the revolt, perhaps due to serious losses.<ref name="F. Keppie 2000 pp 228-229">L. J. F. Keppie (2000) ''Legions and Veterans: Roman Army Papers 1971-2000'' Franz Steiner Verlag, {{ISBN|3-515-07744-8}} pp 228–229</ref>

Following the failure of the Bar Kokhba revolt, the center of Jewish society shifted from Judea to [[Galilee]].<ref name="CambridgeJudaism">David Goodblatt, 'The political and social history of the Jewish community in the Land of Israel,' in William David Davies, Louis Finkelstein, Steven T. Katz (eds.) [[iarchive:cambridgehis_xxxx_1984_004_8494287/page/n437| ''The Cambridge History of Judaism: Volume 4, The Late Roman-Rabbinic Period'']], Cambridge University Press, 2006 pp.404-430, p.406.</ref> The Jews were also subjected to a series of religious edicts by the Romans, including an edict that barred all Jews from entering Jerusalem.<ref name="Eshel" /><ref name=":0">{{Cite book |last=Eshel |first=Hanan |url=https://www.worldcat.org/oclc/7672733 |title=The Cambridge History of Judaism |date=2006 |publisher=Cambridge |isbn=9780521772488 |editor-last=T. Katz |editor-first=Steven |volume=4. The Late Roman-Rabbinic Period |location=Cambridge |pages=105–127 |chapter=4: The Bar Kochba Revolt, 132 – 135 |oclc=7672733}}</ref> After Hadrian's death in 138, the Romans scaled back on their crackdown across Judea, but the ban on Jewish entry into Jerusalem remained in place, exempting only those Jews who wished to enter the city for [[Tisha B'Av]]. The Bar Kokhba revolt also had philosophical and religious ramifications; Jewish belief in the Messiah was abstracted and spiritualized, and rabbinical political thought became deeply cautious and conservative. The [[Talmud]] refers to Bar Kokhba as "Ben Koziva" ({{Lang|he|בֶּן כּוֹזִיבָא}}, {{Literal translation|Son of Deception}}), a derogatory term asserting that he was a [[List of Jewish messiah claimants|false Messiah]]. The rebellion was also among the events that helped differentiate [[Early Christianity]] from [[Judaism]].<ref>M. Avi-Yonah, ''The Jews under Roman and Byzantine Rule'', Jerusalem 1984 p. 143</ref>

==Naming==
The Bar Kokhba revolt was the last of three major [[Jewish–Roman wars]], so it is also known as the Third Jewish–Roman War or the Third Jewish Revolt. Some historians also refer to it as the Second Revolt of Judea,<ref>{{cite book |title=The Second Jewish Revolt |publisher=Brill |year=2016 |isbn=9789004314634 |pages=i-xxiv |chapter=Preliminary Material |doi=10.1163/9789004314634_001 |chapter-url=http://booksandjournals.brillonline.com/content/books/b9789004314634_001}}</ref> not counting the [[Kitos War]] (115–117 CE), which had only marginally been fought in Judea.

==Background==
{{Main|Jewish–Roman wars}}
{{multiple images
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| footer = The first coin issued at the mint of Aelia Capitolina about 130/132 CE. Reverse: COL[ONIA] AEL[IA] CAPIT[OLINA] COND[ITA] ('The founding of Colonia Aelia Capitolina').
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After the [[First Jewish–Roman War]] (66–73 CE), Roman authorities took measures to suppress the rebellious province of [[Judea (Roman province)|Roman Judea]]. Instead of a [[procurator (Roman)|procurator]], they installed a [[praetor]] as a [[Roman governor|governor]] and stationed an entire [[Roman legion|legion]], the [[Legio X Fretensis|X ''Fretensis'']], in the area. Tensions continued to build up in the wake of the [[Kitos War]], the second large-scale Jewish insurrection in the Eastern Mediterranean during 115–117, the final stages of which saw fighting in Judea. Mismanagement of the province during the early 2nd century might well have led to the proximate causes of the revolt, largely bringing governors with clear anti-Jewish sentiments to run the province. [[Quintus Coredius Gallus Gargilius Antiquus|Gargilius Antiques]] may have preceded Rufus during the 120s.<ref name="Antiques">{{cite news|title=Ancient Inscription Identifies Gargilius Antiques as Roman Ruler on Eve of Bar Kochva Revolt|url=http://www.jewishpress.com/news/breaking-news/ancient-inscription-identifies-gargilius-antiques-as-roman-ruler-on-eve-of-bar-kochva-revolt/2016/12/01/|publisher=The Jewish Press|date=December 1, 2016}}</ref> The Church Fathers and rabbinic literature emphasize the role of Rufus in provoking the revolt.<ref name="William David Davies p. 35"/>

Historians have suggested multiple reasons for the sparking of the Bar Kokhba revolt, long-term and proximate. Several elements are believed to have contributed to the rebellion; changes in administrative law, the widespread presence of legally-privileged [[Roman citizen]]s, alterations in agricultural practice with a shift from landowning to sharecropping, the impact of a possible period of economic decline, and an upsurge of nationalism, the latter influenced by similar revolts among the Jewish communities in Egypt, Cyrenaica and Mesopotamia during the reign of [[Trajan]] in the Kitos War.<ref name="Eshel" />

The proximate reasons seem to centre around the construction of a new city, [[Aelia Capitolina]], over the ruins of Jerusalem and the erection of a temple to Jupiter on the Temple mount.<ref name="Eshel" /> Until recently, some historians had tried to question the Colonia foundation event as one of the causes of the revolt, suggesting to rather time the Colonia establishment to the aftermath of the revolt as a punishment.<ref>"Jerusalem and the Bar Kokhba Revolt Again: A Note" by Eran Almagor, ELECTRUM Vol. 26 (2019): 141–157, http://www.ejournals.eu/electrum/2019/Volume-26/art/15133/ (abstract with link to full pdf article) which suggests Aelia Capitolina was founded during the last stage of the revolt which halted earlier reconstruction http://www.ejournals.eu/electrum/2019/Volume-26/art/15015/ and "Eusebius and Hadrian's Founding of Aelia Capitolina in Jerusalem" by Miriam Ben Zeev Hofman, ELECTRUM Vol. 26 (2019): 119–128 http://www.ejournals.eu/electrum/2019/Volume-26/art/15015/</ref> However, the 2014 archaeological finding of the Legio Fretensis inscription in Jerusalem dedicated to Hadrian and dated to 129/130 CE,<ref>{{cite news|url=https://www.jpost.com/not-just-news/watch-2000-year-old-commemorative-inscription-dedicated-to-roman-emperor-hadrian-unveiled-379384|title = WATCH: 2,000-year-old inscription dedicated to Roman emperor unveiled in Jerusalem| newspaper=The Jerusalem Post | Jpost.com }}</ref> as well as identification of Colonia Aelia Capitolina struck coins have since been largely accepted as confirmation to the sequence of events depicted in Jewish traditional literature. One interpretation involves the visit in 130 CE of [[Hadrian]] to the ruins of the Jewish Temple in Jerusalem. At first sympathetic towards the Jews, Hadrian promised to rebuild the Temple, but the Jews felt betrayed when they found out that he intended to build a temple dedicated to [[Jupiter (mythology)|Jupiter]] upon the ruins of the [[Second Temple]].<ref name="Dio">[[Cassius Dio]], Translation by [[Earnest Cary]]. ''Roman History'', book 69, 12.1-14.3. [[Loeb Classical Library]], 9 volumes, Greek texts and facing English translation: Harvard University Press, 1914 thru 1927. Online in [[LacusCurtius]]:[https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Cassius_Dio/69*.html#12]{{Dead link|date=December 2018|bot=InternetArchiveBot|fix-attempted=yes}} and livius.org:[https://www.livius.org/ja-jn/jewish_wars/bk05.html] {{Webarchive|url=https://web.archive.org/web/20160813220152/http://www.livius.org/ja-jn/jewish_wars/bk05.html|date=2016-08-13}}. Book scan in [[Internet Archive]]:[https://archive.org/stream/diosromanhistory08cassuoft#page/446/mode/2up].</ref> A rabbinic version of this story claims that Hadrian planned on rebuilding the Temple, but that a malevolent [[Samaritans|Samaritan]] convinced him not to. The reference to a malevolent Samaritan is, however, a familiar device of Jewish literature.<ref>{{cite book | title = The History of the Jews in the Greco-Roman World: The Jews of Palestine from Alexander the Great to the Arab Conquest | first = Peter | last = Schäfer | author-link = Peter Schäfer | publisher = Routledge | date = 2003 | page = 146 | others = Translated by David Chowcat}}</ref>

An additional legion, the [[Legio VI Ferrata|VI ''Ferrata'']], arrived in the province to maintain order. Works on Aelia Capitolina, as Jerusalem was to be called, commenced in 131 CE. The governor of Judea, Tineius Rufus, performed the foundation ceremony, which involved ploughing over the designated city limits.<ref>See {{cite encyclopedia | title = [[A Topographical Dictionary of Ancient Rome]] | chapter = Pomerium | chapter-url = https://penelope.uchicago.edu/Thayer/E/Gazetteer/Places/Europe/Italy/Lazio/Roma/Rome/_Texts/PLATOP*/Pomerium.html | first = Samuel Ball | last = Platner | author-link = Samuel Ball Platner | date = 1929 | via = LacusCurtius}} {{Cite book | publisher = Taylor & Francis | isbn = 9781136823282 | last = Gates | first = Charles | title = Ancient Cities: The Archaeology of Urban Life in the Ancient Near East and Egypt, Greece and Rome | date = 2011 | page = 335 }}</ref> "Ploughing up the Temple",<ref>The [[Mishnah]] has a segment: "[O]n the 9th of Ab...and the city was ploughed up." on mas. Taanith, Chapter 4, Mishnah no. 6. See:
* {{cite book | url = http://hebrewbooks.org/pdfpager.aspx?req=9670&st=&pgnum=432&hilite= | date = 1963 | publisher = Judaica | place = New York | page = 432 | title = MISHNAYOTH, VOLUME II, ORDER MOED | editor-last = Blackman | editor-first = Philip | language = he, en | via = HebrewBooks}}
* {{Cite book | publisher = [Palestine House] |author1-link=Albert William Greenup | last = Greenup | first = Albert William | title = The Mishna tractate Taanith (On the public fasts) | location = London | date = 1921 | url = https://archive.org/stream/mishnatractateta00greeiala#page/32/mode/2up | page = 32 | via = Internet Archive}}
* {{cite book | title = Eighteen Treatises from the Mishna | editor1-first = D. A. | editor1-last = Sola | editor2-first = M. J. | editor2-last = Raphall | date = 1843 | url = http://www.sacred-texts.com/jud/etm/etm093.htm | chapter = XX. Treatise Taanith, chapter IV, §6. | via = [[Internet Sacred Text Archive]]}}</ref><ref>The [[Babylonian Talmud]] and [[Jerusalem Talmud]] both explicate the segment refers to Rufus:
Babylonian: mas. Taanith 29a. See
* {{cite web | title = Shas Soncino: Taanith 29a | website = dTorah.com | access-date = 2014-06-28 | url = http://dtorah.com/otzar/shas_soncino.php?ms=Taanith&df=29a | archive-date = 2020-02-09 | archive-url = https://web.archive.org/web/20200209130750/http://dtorah.com/otzar/shas_soncino.php?ms=Taanith&df=29a | url-status = dead }}
* {{cite web | title = Bab. Taanith; ch.4.1-8, 26a-31a | publisher = RabbinicTraditions | url = http://instonebrewer.com/RabbinicTraditions/EngText.php?StandardRef=Taan.29a&GotoRef=b.Taan.29a&Words1UC=&Words2UC=&&Words1HA=&Words2HA=&Resource=b.EngSoncino&Reference=#b.Taan.29a | access-date = 2014-06-28 }}
* {{cite web | url = http://halakhah.com/rst/moed/19%20-%20Ta%27anis%20-%202a-31a.pdf | access-date = 2014-06-27 | pages = 92–93 | title = Ta'anis 2a-31a | translator = I Epstein | work = Soncino Babylonian Talmud | publisher = Halakhah.com | quote = AND THE CITY WAS PLOUGHED UP. It has been taught: When Turnus Rufus the wicked destroyed[note 20: Var lec.: ‘ploughed’.] the Temple,... }}.
See notes on {{cite web | url = http://www.steinsaltz.org/learning.php?pg=Daf_Yomi&articleId=530 | title = Ta'anit 29a-b | work = Daf Yomi series | publisher = The Aleph Society/[[Adin Steinsaltz]] | access-date = 2014-06-27 | archive-date = 2018-10-05 | archive-url = https://web.archive.org/web/20181005022855/http://halakhah.com/rst/moed/19%20-%20Ta%27anis%20-%202a-31a.pdf | url-status = dead }}</ref><ref>The Jerusalem Talmud relates it to the Temple, Taanith 25b:
* {{cite web | website = Mechon Mamre | language = he | title = דף כה,ב פרק ד | url = http://www.mechon-mamre.org/b/r/r2904.htm | at = הלכה ה גמרא | quote =ונחרשה העיר. חרש רופוס שחיק עצמות את ההיכל}}
* {{cite wikisource |wslink=ירושלמי תענית דף כה ב |wslanguage=he |title=ירושלמי תענית דף כה ב}}</ref> seen as a religious offence, turned many Jews against the Roman authorities. The Romans issued a coin inscribed ''Aelia Capitolina''.<ref>{{cite web | url=https://www.europeana.eu/resolve/record/08502/99CB80CA1CD7849AD1F0D9BDD8A004A1EB6C3BEA | archive-url=https://archive.today/20140702134337/http://www.europeana.eu/resolve/record/08502/99CB80CA1CD7849AD1F0D9BDD8A004A1EB6C3BEA | url-status=dead | archive-date=2014-07-02 | title=Roman provincial coin of Hadrian [image]| access-date=2014-07-01 | publisher= [[Israel Museum]] | via=[[Europeana]]}}</ref><ref>{{Cite book | publisher = Princeton University Press | isbn = 0691094934 | last = Boatwright | first = Mary Taliaferro | title = Hadrian and the Cities of the Roman Empire | date = 2003 | page = 199 }}</ref><ref>{{Cite book | publisher = Oxford University Press | isbn = 9780195305746 | last = Metcalf | first = William | title = The Oxford Handbook of Greek and Roman Coinage | date = 2012-02-23 | page = 492}}</ref>

A disputed tradition, based on the single source of the ''[[Augustan History|Historia Augusta]]'', regarded{{By whom|date=November 2015}} as 'unreliable and problematic,'<ref>Benjamin H. Isaac, Aharon Oppenheimer, 'The Revolt of Bar Kochba:Ideology and Modern Scholarship,' in [[Benjamin Isaac|Benjamin H. Isaac]], [https://books.google.com/books?id=jjcPG5UInzgC&pg=PA227 ''The Near East Under Roman Rule: Selected Papers ,''] BRILL (Volume 177 of Mnemosyne, bibliotheca classica Batava. 177: Supplementum), 1998 pp.220-252, 226-227</ref><ref>Aharon Oppenheimer, 'The Ban on Circumcision as a cause of the Revolt: A Reconsideration,' in [[Peter Schäfer]] (ed.) [https://books.google.com/books?id=1TA-Fg4wBnUC&pg=PA55 ''The History of the Jews in the Greco-Roman World: The Jews of Palestine from Alexander the Great to the Arab Conquest,''] Mohr Siebeck 2003 pp.55-69 pp.55f.</ref> states tensions rose after Hadrian banned [[History of male circumcision#Male circumcision in the Greco-Roman world|circumcision]], referred to as ''mutilare genitalia ''<ref>Craig A. Evans, [https://books.google.com/books?id=DRcQ2bkLxc8C&pg=PA185 ''Jesus and His Contemporaries: Comparative Studies,''] BRILL 2001 p.185:'moverunt ea tempestate et Iudaei bellum, quod vetabantur mutilare genitalia.'</ref><ref>Aharon Oppenheimer, ‘The Ban on Circumcision as a Cause of the Revolt: A Reconsideration,’ Aharon Oppenheimer, [https://books.google.com/books?id=BPkfQl2iDx8C&pg=PA243 ''Between Rome and Babylon,''] Mohr Siebeck 2005 pp.243-254 pp.</ref> taken to mean ''[[brit milah]]''.<ref>{{cite book | last1 = Schäfer | first1 = Peter | author-link1 = Peter Schäfer | title = Judeophobia: Attitudes Toward the Jews in the Ancient World | url = https://books.google.com/books?id=8jIhYBwkO80C | publisher = Harvard University Press | date = 1998 | pages = 103–105 | isbn = 9780674043213 | access-date = 2014-02-01 | quote = [...] Hadrian's ban on circumcision, allegedly imposed sometime between 128 and 132 CE [...]. The only proof for Hadrian's ban on circumcision is the short note in the ''Historia Augusta'': 'At this time also the Jews began war, because they were forbidden to mutilate their genitals (''quot vetabantur mutilare genitalia''). [...] The historical credibility of this remark is controversial [...] The earliest evidence for circumcision in Roman legislation is an edict by Antoninus Pius (138-161 CE), Hadrian's successor [...] [I]t is not utterly impossible that Hadrian [...] indeed considered circumcision as a 'barbarous mutilation' and tried to prohibit it. [...] However, this proposal cannot be more than a conjecture, and, of course, it does not solve the questions of when Hadrian issued the decree (before or during/after the Bar Kokhba war) and whether it was directed solely against Jews or also against other peoples.}}</ref> Were the claim true it has been conjectured that Hadrian, as a [[Philhellenism|Hellenist]], would have viewed circumcision as an undesirable form of [[mutilation]].<ref name=Mackay>Christopher Mackay, [https://books.google.com/books?id=6rLDy6qqi0UC&pg=PA230 ''Ancient Rome a Military and Political History''] Cambridge University Press 2007 p.230</ref> The claim is often considered suspect.<ref>Peter Schäfer, ''The Bar Kokhba War Reconsidered: New Perspectives on the Second Jewish Revolt Against Rome'', Mohr Siebeck 2003. p.68</ref><ref>Peter Schäfer, ''The History of the Jews in the Greco-Roman World: The Jews of Palestine from Alexander the Great to the Arab Conquest,'' Routledge, 2003 p. 146.</ref>

==Timeline of events==
===First phase===
====Eruption of the revolt====
Jewish leaders carefully planned the second revolt to avoid the numerous mistakes that had plagued the first [[First Jewish–Roman War]] sixty years earlier.<ref name="Haaretz:The Bar Kochba Revolt: A Disaster Celebrated by Zionists on Lag Ba'Omer">{{cite web |last1=Gilad |first1=Elon |title=The Bar Kochba Revolt: A Disaster Celebrated by Zionists on Lag Ba'Omer |url=https://www.haaretz.com/jewish/.premium-bar-kochba-revolt-utter-disaster-1.5358629 |publisher=Haaretz |access-date=14 May 2019 |date=6 May 2015}}</ref> In 132, the revolt, led by [[Simon bar Kokhba]] and [[Eleazar of Modi'im|Elasar]], quickly spread from [[Modi'in]] across the country, cutting off the Roman garrison in Jerusalem.<ref name="Axelrod 2009 29"/> Although Rufus was in charge during the early phase of the uprising, he disappears from the record after 132 for unknown reasons. Shortly after the eruption of the revolt, Bar Kokhba's rebels inflicted heavy casualties to [[Legio X Fretensis]], based in Aelia Capitolina (Jerusalem).

====Stalemate and reinforcements====
Given the continuing inability of Legio X and Legio VI to subdue the rebels, additional reinforcements were dispatched from neighbouring provinces. [[Gaius Quinctius Certus Poblicius Marcellus|Gaius Poblicius Marcellus]], the Legate of Roman Syria, arrived commanding [[Legio III Gallica]], while [[Titus Haterius Nepos (consul)|Titus Haterius Nepos]], the governor of [[Roman Arabia]], brought [[Legio III Cyrenaica]].<ref>{{cite journal|journal=The Journal of Roman Studies|volume=89|first=Werner|last=Eck|page=81|title=The bar Kokhba Revolt: The Roman Point of View}}</ref> Later on it is proposed by some historians{{vague|date=May 2022}} that [[Legio XXII Deiotariana]] was sent from [[Arabia Petraea]], but was [[Demise of Legio XXII Deiotariana|ambushed and massacred]] on its way to Aelia Capitolina (Jerusalem), and possibly disbanded as a result.<ref>{{cite journal|journal=The Journal of Roman Studies|volume=89|first=Werner|last=Eck|page=80|title=The bar Kokhba Revolt: The Roman Point of View}}</ref> [[Legio II Traiana Fortis]], previously stationed in Egypt, may have also arrived in Judea in this stage.

According to Rabbinic sources some 400,000 men were at the disposal of Bar Kokhba at the peak of the rebellion.<ref>{{cite web|url=https://www.sefaria.org/The_Jewish_Spiritual_Heroes%2C_Volume_I%3B_The_Creators_of_the_Mishna%2C_Rabbi_Akiba_ben_Joseph.32?ven=The_Jewish_spiritual_heroes,_by_Gershom_Bader._New_York,_N.Y._1940&lang=he&with=all&lang2=he|title=The Creators of the Mishna, Rabbi Akiba ben Joseph|website=www.sefaria.org.il}}</ref>

===Second phase===
====From guerilla warfare to open engagement====
The outbreak and initial success of the rebellion took the Romans by surprise. The rebels incorporated combined tactics to fight the Roman Army. According to some historians, Bar Kokhba's army mostly practiced [[guerrilla warfare]], inflicting heavy casualties. This view is largely supported by Cassius Dio, who wrote that the revolt began with covert attacks in line with preparation of hideout systems, though after taking over the fortresses Bar Kokhba turned to direct engagement due to his superiority in numbers.

==== Rebel Judean statehood ====
[[File:Barkokhba-silver-tetradrachm.jpg|left|thumb|Bar Kokhba's tetradrachm overstruck on a denarius. [[Obverse]]: the [[Temple in Jerusalem|Jewish Temple]] facade with the rising star. [[Obverse and reverse|Reverse]]: A [[lulav]], the text reads: "to the freedom of Jerusalem"]]
[[File:JUDAEA,_Bar_Kochba_Revolt._132-135_CE._Æ_(CNG_300483).jpg|left|thumb|Bar Kokhba's coin. [[Obverse]]: Grapes, the text reads: "year 1 to the redemption of Israel". [[Obverse and reverse|Reverse]]: a [[date palm]] with two branches of [[date (fruit)|dates]]; “Eleazar the Priest” (in Hebrew) around]]
[[File:Israel under Bar Kokhba.jpg|right|thumb|300px|Territory held by the rebels in blue.]]
Simon bar Kokhba took the title ''[[Nasi (Hebrew title)|Nasi Israel]]''<ref>{{cite journal |last1=Bourgel |first1=Jonathan |title=Ezekiel 40–48 as a Model for Bar Kokhba's Title "Nasi Israel"? |journal=Journal of Ancient Judaism |date=23 March 2023 |volume=14 |issue=3 |pages=446–481 |doi=10.30965/21967954-bja10037|s2cid=257812293 |doi-access=free }}</ref> and ruled over an entity named ''Israel'' that was virtually independent for over two and a half years. The Jewish sage [[Rabbi Akiva]], who was the spiritual leader of the revolt,{{sfn|Mor|2016|p=466}} identified Simon Bar Koziba as the [[Jewish messiah]], and gave him the [[Patronymic#Aramaic|Aramaic patronymic]] ''bar Kokhba'', meaning "Son of a Star", a reference to the [[Star Prophecy]] in [[Book of Numbers|Numbers]] {{bibleverse-nb|Numbers|24:17}}: "A star rises from [[Jacob]]".<ref>{{cite web |title=Numbers 24:17 |url=https://www.sefaria.org/Numbers.24.17?lang=bi&with=all&lang2=en |website=www.sefaria.org|quote=What I see for them is not yet, What I behold will not be soon: A star rises from Jacob, A scepter comes forth from Israel; It smashes the brow of Moab, The foundation of all children of [[Seth]].}}</ref> The name Bar Kokhba does not appear in the [[Talmud]] but in ecclesiastical sources.<ref>{{Cite encyclopedia |chapter=BAR KOKBA AND BAR KOKBA WAR | chapter-url=http://jewishencyclopedia.com/articles/2464-bar-cochba-bar-cochbah |encyclopedia=[[The Jewish Encyclopedia]] |date=1906 |editor-last=Singer |editor-first=Isidore |first=S. |last=Krauss |volume=2 |pages=506–507 |quote=Bar Kokba, the hero of the third war against Rome, appears under this name only among ecclesiastical writers: heathen authors do not mention him; and Jewish sources call him Ben (or Bar) Koziba or Kozba...}}</ref> The era of the [[Salvation#Judaism|redemption of Israel]] was announced, contracts were signed and a large quantity of [[Bar Kokhba Revolt coinage]] was struck over foreign coins.

====From open warfare to rebel defensive tactics====
With the slowly advancing Roman army cutting supply lines, the rebels engaged in long-term defense. The defense system of Judean towns and villages was based mainly on hideout caves, which were created in large numbers in almost every population center. Many houses utilized underground hideouts, where Judean rebels hoped to withstand Roman superiority by the narrowness of the passages and even ambushes from underground. The cave systems were often interconnected and used not only as hideouts for the rebels but also for storage and refuge for their families.<ref>{{Cite book|url=https://books.google.com/books?id=1TA-Fg4wBnUC&dq=bar+kokhba+hideout+systems&pg=PA184|title=The Bar Kokhba War Reconsidered: New Perspectives on the Second Jewish Revolt Against Rome|first=Peter|last=Schäfer|date=September 10, 2003|publisher=Isd|isbn=9783161480768 |via=Google Books}}</ref> Hideout systems were employed in the Judean hills, the Judean desert, northern Negev, and to some degree also in Galilee, Samaria and Jordan Valley. As of July 2015, some 350 hideout systems have been mapped within the ruins of 140 Jewish villages.<ref name=nrg/>

===Third phase===
====Julius Severus' campaign====
Following a series of setbacks, Hadrian called his general [[Sextus Julius Severus]] from [[Roman Britain|Britannia]],{{sfn|Mor|2016|p=491}} and troops were brought from as far as the [[Danube]]. In 133/4, Severus landed in Judea with a massive army, bringing three legions from Europe (including [[Legio X Gemina]] and possibly also [[Legio IX Hispana]]), cohorts of additional legions and between 30 and 50 auxiliary units.

The size of the Roman army amassed against the rebels was much larger than that commanded by [[Titus]] sixty years earlier - nearly one third of the Roman army took part in the campaign against Bar Kokhba. It is estimated that forces from at least 10 legions participated in Severus' campaign in Judea, including [[Legio X Fretensis]], [[Legio VI Ferrata]], [[Legio III Gallica]], [[Legio III Cyrenaica]], [[Legio II Traiana Fortis]], [[Legio X Gemina]], cohorts of [[Legio V Macedonica]], cohorts of [[Legio XI Claudia]], cohorts of [[Legio XII Fulminata]] and cohorts of [[Legio IV Flavia Felix]], along with 30–50 auxiliary units, for a total force of 60,000–120,000 Roman soldiers facing Bar Kokhba's rebels. It is plausible that [[Legio IX Hispana]] was among the legions Severus brought with him from Europe, and that its demise occurred during Severus' campaign, as its disappearance during the second century is often attributed to this war.<ref name="livius.org">{{cite web |url=https://www.livius.org/articles/legion/legio-viiii-hispana/ |title=Legio VIIII Hispana |publisher=livius.org |access-date=2014-06-26}}</ref>{{unreliable?|date=October 2022}}

====Battle of Tel Shalem (theory)====
According to some views one of the crucial battles of the war took place near Tel Shalem in the [[Beit She'an]] valley, near what is now identified as the legionary camp of [[Legio VI Ferrata]]. This theory was proposed by Werner Eck in 1999, as part of his general maximalist work which did put the Bar Kokhba revolt as a very prominent event on the course of the Roman Empire's history.<ref>Journal of Roman Archaeology , Volume 12 , 1999 , pp. 294 - 313
DOI: https://doi.org/10.1017/S1047759400018043</ref> Next to the camp, archaeologists unearthed the remnants of a triumphal arch, which featured a dedication to Emperor Hadrian, which most likely refers to the defeat of Bar Kokhba's army.<ref>Mohr Siebek et al. Edited by Peter Schäfer. ''The Bar Kokhba War reconsidered''. 2003. P172.</ref> Additional finds at Tel Shalem, including a bust of Emperor Hadrian, specifically link the site to the period. The theory for a major decisive battle in Tel Shalem implies a significant extension of the area of the rebellion, with Werner Eck suggesting the war encompassed also northern Valleys together with Galilee.<ref>{{Cite journal|url=https://www.ceeol.com/search/article-detail?id=81536|title=What Does Tel Shalem Have To Do with the Bar Kokhba Revolt?|first=Menahem|last=Mor|date=September 10, 2013|journal=Scripta Judaica Cracoviensia|issue=11|via=www.ceeol.com}}</ref>

====Judean highlands and desert====
[[File:PikiWiki Israel 20047 Archeological sites of Israel.jpg|right|thumb|200px|Remains of [[Hurvat Itri]], destroyed during the Bar Kokhba revolt]]
{{main|Herodium}}
[[Simon bar Kokhba]] declared Herodium as his secondary headquarters. Its commander was [[Yeshua ben Galgula]], likely Bar Kokhba's second or third line of command. Archaeological evidence for the revolt was found all over the site, from the outside buildings to the water system under the mountain.

===Fourth phase===
The last phase of the revolt is characterized by Bar Kokhba's loss of territorial control, with the exception of the surroundings of the Betar fortress, where he made his last stand against the Romans.

====Siege of Betar====
[[File:Beitar-169.jpg|thumb|200px|Ruined walls of the Beitar fortress, the last stand of Bar Kokhba]]
{{main|Betar (fortress)}}

After losing many of their strongholds, Bar Kokhba and the remnants of his army withdrew to the fortress of [[Betar (fortress)|Betar]], which subsequently came under siege in the summer of 135. [[Legio V Macedonica]] and [[Legio XI Claudia]] are said to have taken part in the siege.<ref>Charles Clermont-Ganneau, ''Archaeological Researches in Palestine during the Years 1873-1874'', London 1899, pp. 463-470</ref> According to Jewish tradition, the fortress was breached and destroyed on the fast of [[Tisha B'av]], the ninth day of the lunar month Av, a day of mourning for the destruction of the First and the Second Jewish Temple. Rabbinical literature ascribes the defeat to Bar Kokhba killing his maternal uncle, Rabbi [[Eleazar of Modi'im|Elazar Hamudaʻi]], after suspecting him of collaborating with the enemy, thereby forfeiting Divine protection.<ref>Jerusalem Talmud [[Ta'anit (Talmud)|Ta'anit]] iv. 68d; [[Lamentations Rabbah]] ii. 2</ref> The horrendous scene after the city's capture could be best described as a [[massacre]].<ref>Jerusalem Talmud, ''Taanit'' 4:5 (24a); Midrash Rabba (Lamentations Rabba 2:5).</ref> The Jerusalem Talmud relates that the number of dead in Betar was enormous, that the Romans "went on killing until their horses were submerged in blood to their nostrils."<ref>[[Ta'anit (Talmud)|Ta'anit]] 4:5</ref>
[[File:Roman Inscription found near Bettir in 19th century.jpg|thumb|right|Roman Inscription found near Battir mentioning the 5th and 11th Roman Legions]]

====Final accords====
{{see|Cave of Horror|Cave of Letters|Ten Martyrs}}

According to a rabbinic [[midrash]], the Romans executed eight leading members of the [[Sanhedrin]] (The list of [[Ten Martyrs]] includes two earlier rabbis): [[Rabbi Akiva]]; [[Haninah ben Teradion]]; the interpreter of the Sanhedrin, Rabbi Huspith; [[Eleazar ben Shammua]]; [[Hanina ben Hakinai]]; [[Jeshbab the Scribe]]; [[Judah ben Dama]]; and [[Judah ben Bava]]. The precise date of Akiva's execution is disputed, some dating it to the beginning of the revolt based on the midrash, while others link it to final phases. The rabbinic account describes agonizing tortures: Akiva was [[Flaying|flayed]] with iron combs, Ishmael had the skin of his head pulled off slowly, and Haninah was [[burned at a stake]], with wet wool held by a [[Torah scroll]] wrapped around his body to prolong his death.<ref>{{JewishEncyclopedia|title=10447-martyrs-the-ten|quote=The fourth martyr was Hananiah ben Teradion, who was wrapped in a scroll of the Law and placed on a pyre of green brushwood; to prolong his agony, wet wool was placed on his chest.}}</ref> Bar Kokhba's fate is not certain, with two alternative traditions in the Babylonian Talmud ascribing the death of Bar Kokhba either to a snakebite or other natural causes during the Roman siege or possibly killed on the orders of the Sanhedrin, as a [[List of Jewish messiah claimants|false messiah]]. According to [[Lamentations Rabbah]], the head of Bar Kokhba was presented to Emperor Hadrian after the Siege of Betar.

Following the Fall of Betar, the Roman forces went on a rampage of systematic killing, eliminating all remaining Jewish villages in the region and seeking out the refugees. [[Legio III Cyrenaica]] was the main force to execute this last phase of the campaign. Historians disagree on the duration of the Roman campaign following the fall of Betar. While some claim further resistance was broken quickly, others argue that pockets of Jewish rebels continued to hide with their families into the winter months of late 135 and possibly even spring 136. By early 136 however, it is clear that the revolt was defeated.<ref>Mohr Siebek et al. Edited by Peter Schäfer. ''The Bar Kokhba War reconsidered''. 2003. P160. "Thus it is very likely that the revolt ended only in early 136."</ref>

==Aftermath==

=== Impact on the Jewish population ===
The Bar Kokhba Revolt had catastrophic consequences for the Jewish population in [[Judaea (Roman province)|Judaea]], with profound loss of life, extensive forced displacements, and widespread enslavement. The scale of suffering surpassed even the aftermath of the [[First Jewish–Roman War]], leaving [[Judea|central Judea]] in a state of desolation.<ref name="Taylor" /><ref name=":2" /> Some scholars characterize these consequences as an act of [[genocide]].<ref name="Taylor" /><ref name="google.co.il">Totten, S. ''Teaching about genocide: issues, approaches and resources.'' p24. [https://books.google.com/books?id=LoQo50YPzTUC&pg=PA23]</ref> Jewish religious and political authority was suppressed far more brutally than before, and the province of Judaea was renamed [[Syria Palaestina]].

==== Casualties and widespread destruction ====
In his account of the revolt, Roman historian [[Cassius Dio]] ({{circa|155}}–235) wrote that:<ref name="raviv2021" /><blockquote>{{Blockquote|text="50 of their most important outposts and 985 of their most famous villages were razed to the ground. 580,000 men were slain in the various raids and battles, and the number of those that perished by famine, disease and fire was past finding out, Thus nearly the whole of Judaea was made desolate."|author=Cassius Dio|title=History of Rome|source=69.14.1-2}}</blockquote>Every village in the region of Judea whose remains have been excavated so far had been destroyed in the revolt.<ref name=":0" /> The majority of Roman-period settlements in Judea that have been excavated exhibit [[Destruction layer|destruction]] or abandonment layers, and there is a settlement gap above these layers. It appears that Jewish settlement in Judea had been almost completely eradicated by the end of the revolt.<ref name="raviv2021" />

In 1981, Schäfer suggested that Dio exaggerated his numbers.<ref name="schafer-1981">{{cite book |last1=Schäfer |first1=P. |title=Der Bar Kochba-Aufstand |date=1981 |location=Tübingen |pages=131ff}}</ref> On the other hand, in 2003 Cotton considered Dio's figures highly plausible, in light of accurate Roman census declarations.<ref>Mohr Siebek et al. Edited by Peter Schäfer. ''The Bar Kokhba War reconsidered''. 2003. P142-3.</ref> In 2021, an ethno-archaeological comparison analysis by Dvir Raviv and Chaim Ben David was published, in which the two scholars assert of sufficient accuracy in Dio's depopulation claims, and describe it as a "reliable account, which he based on contemporaneous documentation".<ref name="raviv2021" />

==== Displacement ====
Eusebius writes that:

{{Blockquote|text="[...] all the families of the Jewish nation have suffered pain worthy of wailing and lamentation because God's hand has struck them, delivering their mother-city over to strange nations, laying their Temple low, and driving them from their country, to serve their enemies in a hostile land."|author=Eusebius of Caesarea|title=Demonstratio Evangelica|source=VIII, 4, 23}}

[[Jerome]] provides a similar account:

{{Blockquote|text="in Hadrian's reign, when Jerusalem was completely destroyed and the Jewish nation was massacred in large groups at a time, with the result that they were even expelled from the borders of Judaea."|author=Jerome|title=Commentary on Daniel (translated by Gleason L. Archer)|source=III, ix, 24}}

Jews were expelled from the area of Jerusalem.<ref>{{Cite journal |last=Bar |first=Doron |date=2005 |title=Rural Monasticism as a Key Element in the Christianization of Byzantine Palestine |url=https://www.jstor.org/stable/4125284 |journal=The Harvard Theological Review |volume=98 |issue=1 |pages=49–65 |doi=10.1017/S0017816005000854 |issn=0017-8160 |jstor=4125284 |s2cid=162644246 |quote=The phenomenon was most prominent in Judea, and can be explained by the demographic changes that this region underwent after the second Jewish revolt of 132-135 C.E. The expulsion of Jews from the area of Jerusalem following the suppression of the revolt, in combination with the penetration of pagan populations into the same region, created the conditions for the diffusion of Christians into that area during the fifth and sixth centuries. [...] This regional population, originally pagan and during the Byzantine period gradually adopting Christianity, was one of the main reasons that the monks chose to settle there. They erected their monasteries near local villages that during this period reached their climax in size and wealth, thus providing fertile ground for the planting of new ideas.}}</ref> Mor writes that Jews were expelled from the districts of [[Jifna|Gophna]], [[Herodium|Herodion]], and [[Aqraba, Nablus|Aqraba]].<ref>{{harvnb|Mor|2016|pp=483–484}}: "Land confiscation in Judaea was part of the suppression of the revolt policy of the Romans and punishment for the rebels. But the very claim that the sikarikon laws were annulled for settlement purposes seems to indicate that Jews continued to reside in Judaea even after the Second Revolt. There is no doubt that this area suffered the severest damage from the suppression of the revolt. Settlements in Judaea, such as Herodion and Bethar, had already been destroyed during the course of the revolt, and Jews were expelled from the districts of Gophna, Herodion, and Aqraba. However, it should not be claimed that the region of Judaea was completely destroyed. Jews continued to live in areas such as Lod (Lydda), south of the Hebron Mountain, and the coastal regions. In other areas of the Land of Israel that did not have any direct connection with the Second Revolt, no settlement changes can be identified as resulting from it."</ref>

Artistic, [[Epigraphy|epigraphic]] and [[Numismatics|numismatic]] findings from post-revolt Judea, in Klein's assessment, indicates that the Roman authorities replaced the departing and slain Jews with a mixed population that was made up of a mixture of Roman veterans and immigrants from the western parts of the empire who settled in Aelia Capitolina, its surroundings, administrative centers, and along the main roads, as well as immigrants from the coastal plain and neighboring provinces from [[Roman Syria|Syria]], [[Phoenicia under Roman rule|Phoenicia]], and [[Arabia Petraea|Arabia]] who settled in the Judean countryside.<ref name="Klein2010">Klein, E. (2010), “The Origins of the Rural Settlers in Judean Mountains and Foothills during the Late Roman Period”, In: E. Baruch., A. Levy-Reifer and A. Faust (eds.), ''New Studies on Jerusalem'', Vol. 16, Ramat-Gan, pp. 321-350 (Hebrew).</ref><ref>קליין, א' (2011). ''היבטים בתרבות החומרית של יהודה הכפרית בתקופה הרומית המאוחרת'' ''(135–324 לסה"נ)''. עבודת דוקטור, אוניברסיטת בר-אילן. עמ' 314–315. (Hebrew)</ref><ref>שדמן, ע' (2016). ''בין נחל רבה לנחל שילה: תפרוסת היישוב הכפרי בתקופות ההלניסטית, הרומית והביזנטית לאור חפירות וסקרים''. עבודת דוקטור, אוניברסיטת בר-אילן. עמ' 271–275. (Hebrew)</ref>

In the vicinity of Jerusalem, villages were depopulated, and arable land owned by Jews was confiscated. The lack of an alternative population to fill the empty villages led Roman and later Byzantine authorities to seek a different approach to benefit the nobles and finally the church by constructing estate farms and monasteries on the empty village lands.<ref>Seligman, J. (2019). Were There Villages in Jerusalem's Hinterland During the Byzantine Period? In. Peleg- Barkat O. et.al. (Eds.) ''Between Sea and Desert: On Kings, Nomads, Cities and Monks. Essays in Honor of Joseph Patrich''. Jerusalem; Tzemach. Pp. 167-179.</ref> The Roman legionary tomb at [[Malha|Manahat]], the ruins of [[Roman villa|Roman villas]] at Ein Yael, Khirbet er-Ras, Rephaim Valley and [[Ramat Rachel]], and the Tenth Legion's [[Kiln|kilns]] discovered near [[Givat Ram|Giv'at Ram]] are all indications that the rural area surrounding [[Aelia Capitolina]] underwent a romanization process, with Roman citizens and Roman veterans settling in the area during the Late Roman period.<ref name="ZissuKlein">{{cite journal |last1=Zissu |first1=Boaz |author-link1=:he:בועז זיסו |last2=Klein |first2=Eitan |year=2011 |title=A Rock-Cut Burial Cave from the Roman Period at Beit Nattif, Judaean Foothills |url=http://lisa.biu.ac.il/files/lisa/shared/Zissu-Klein-IEJ_61-2011.pdf |url-status=dead |journal=[[Israel Exploration Journal]] |volume=61 |issue=2 |pages=196–216 |archive-url=https://web.archive.org/web/20140816223801/http://lisa.biu.ac.il/files/lisa/shared/Zissu-Klein-IEJ_61-2011.pdf |archive-date=2014-08-16 |access-date=2014-08-16}}</ref> Indications for the settlement of Roman veterans in other parts of Judea proper includes a magnificent marble sarcophagus showing [[Dionysus]] discovered in [[Turmus Ayya]], Latin-inscribed stone discovered at [[Khirbet Tibnah]], a statue of [[Minerva]] discovered at [[Khirbat al-Mafjar]], a tomb of a [[centurion]] at [[Bayt Nattif|Beit Nattif]] and a Roman mansion with western elements discovered at Arak el-Khala, near [[Beit Guvrin-Maresha National Park|Beit Guvrin]].<ref name="Klein2010" />

In [[Perea]], a Roman military presence in the middle of the second century CE suggests that the Jews there were also victims of the revolt. The name of a Roman veteran from the village of Meason in Perea appears on a papyrus that was signed in [[Caesarea Maritima|Caesarea]] in the year 151 CE, implying that lands there had been expropriated and given to Roman settlers. A building inscription of the Sixth Legion from the second century CE was discovered at [[as-Salt]], which is identified as Gadara, one of the principal Jewish settlements in Perea, and provides more proof of the Roman military presence there.<ref name="raviv2021" />

==== Enslavement ====
Sources indicate that Jewish captives were sold into slavery and sent to various parts of the empire.{{sfn|Mor|2016|p=471}} A chronicle written in the 7th century CE, which was based on lost ancient sources, states that "Jewish captives were sold for the price of one ration of food for a horse."<ref name="Powell, p.812">Powell, ''The Bar Kokhba War AD 132-136'', Osprey Publishing, Oxford, ç2017, p.81</ref> This number indicates that the slave market was flooded with new slaves. According to Harris, the overall number of enslaved captives taken in the revolt must have been much higher than 100,000.<ref>{{Cite journal |last=Harris |first=William V. |date=1980 |title=Towards a Study of the Roman Slave Trade |url=https://www.jstor.org/stable/4238700 |journal=Memoirs of the American Academy in Rome |volume=36 |pages=117–140 |doi=10.2307/4238700 |issn=0065-6801 |jstor=4238700}}</ref> Captives who were not sold as slaves were deported to Gaza, Egypt and elsewhere, greatly adding to the [[Jewish diaspora]].<ref name="Powell, p.812" />

==== Punitive measures against Jews ====
[[File:Expulsion of the Jews in the Reign of the Emperor Hadrian AD 135 How Heraclius turned the Jews out of Jerusalem Fac simile of a Miniature in the Histoire des Empereurs Manuscript of the Fifteenth Century.png|right|thumb|Expulsion of the Jews from Jerusalem during the reign of Hadrian. A miniature from the 15th-century manuscript "Histoire des Empereurs".]]After the suppression of the revolt, Hadrian promulgated a series of religious edicts aimed at uprooting the Jewish nationalism in Judea.<ref name="Eshel" /><ref name=":0" /> He prohibited [[Torah]] law and the [[Hebrew calendar]] and executed Judaic scholars. The sacred scrolls of Judaism were ceremonially burned at the large Temple complex for Jupiter which he built on the [[Temple Mount]]. At this Temple, he installed two statues, one of [[Jupiter (god)|Jupiter]], another of himself. These proclamations remained in effect until Hadrian’s death in 138, which marked a significant relief to the surviving Jewish communities.<ref name=":0" />

A further, more lasting punishment was also implemented by the Romans.<ref name=":0" /> In an attempt to erase any memory of Judea or [[Kingdom of Israel (united monarchy)|Ancient Israel]], the name Judaea was dropped from the provincial name, and [[Judaea (Roman province)|Provincia Iudaea]] was renamed [[Syria Palaestina]].<ref name="H.H. Ben-Sasson, 1976, page 334">H.H. Ben-Sasson, ''A History of the Jewish People'', Harvard University Press, 1976, {{ISBN|0-674-39731-2}}, page 334: "In an effort to wipe out all memory of the bond between the Jews and the land, Hadrian changed the name of the province from Judaea to Syria-Palestina, a name that became common in non-Jewish literature."</ref><ref name="Ariel Lewin p. 33">Ariel Lewin. ''The archaeology of Ancient Judea and Palestine''. Getty Publications, 2005 p. 33. "It seems clear that by choosing a seemingly neutral name - one juxtaposing that of a neighboring province with the revived name of an ancient geographical entity (Palestine), already known from the writings of Herodotus - Hadrian was intending to suppress any connection between the Jewish people and that land." {{ISBN|0-89236-800-4}}</ref><ref name="The Bar Kokhba War Reconsidered">[https://books.google.com/books?id=1TA-Fg4wBnUC&dq=intentionally+suppressed+jewish+national+Aelia+Capitolina+Palaestina&pg=PA33 ''The Bar Kokhba War Reconsidered''] by Peter Schäfer, {{ISBN|3-16-148076-7}}</ref> Despite such name changes taking place elsewhere, rebellions have never resulted in a nation's name being expunged.<ref name=":0" /> Similarly, under the argument to ensure the prosperity of the newly founded [[Colonia (Roman)|Roman colony]] of [[Aelia Capitolina]], Jews were forbidden to enter, except on the day of [[Tisha B'Av]].<ref>H.H. Ben-Sasson, ''A History of the Jewish People'', page 334: "Jews were forbidden to live in the city and were allowed to visit it only once a year, on the Ninth of Ab, to mourn on the ruins of their holy Temple."</ref> By destroying the association of Jews with Judea and forbidding the practice of the Jewish faith, Hadrian aimed to root out a nation that had inflicted heavy casualties on the Roman Empire.

==== Cultural aspects ====
After the victorious defeat of the Jews in the Bar Kokhba revolt and the destruction of Judea, not many years passed and the [[Hebrew language]], which before that was still used as a living language among a very significant part of the Jewish population in this region of the country, disappeared from daily use. In the 3rd century sages no longer knew how to identify the Hebrew names of many plants mentioned in the [[Mishnah]]. Only a small number of sages who resided in the south still spoke Hebrew. The [[Jerusalem Talmud]] and the classic legend midrashes (in which the majority of the acts and stories are in [[Jewish Palestinian Aramaic|Aramaic]]) both demonstrate that Hebrew was used mostly as a literary and artificial language. Hebrew is only found on a small percentage of cemeteries and synagogues.<ref name=":3">{{Cite book |last=הר |first=משה דוד |title=ארץ-ישראל בשלהי העת העתיקה: מבואות ומחקרים |publisher=יד יצחק בן-צבי |year=2022 |isbn=978-965-217-444-4 |editor-link=Moshe David Herr |volume=1 |publication-place=ירושלים |pages=218–219 |language=he |trans-title=Eretz Israel in Late Antiquity: Introductions and Studies |chapter=היהודים בארץ-ישראל בימי האימפריה הרומית הנוצרית |trans-chapter=The Jews in the Land of Israel in the Days of the Christian Roman Empire}}</ref>

==== Continuity ====
[[File:Ancient Galilee.jpg|thumb|right|225px|The [[Galilee]] in [[late antiquity]]]]
While Jewish presence in the region significantly dwindled after the failure of the Bar Kokhba revolt,<ref>Oppenheimer, A'haron and Oppenheimer, Nili. ''Between Rome and Babylon: Studies in Jewish Leadership and Society''. Mohr Siebeck, 2005, p. 2.</ref> there was a continuous small Jewish presence, and [[Galilee]] became its religious center.<ref>{{cite book |last=Cohn-Sherbok |first=Dan |title=Atlas of Jewish History |publisher=Routledge |year=1996 |isbn=978-0-415-08800-8 |page=58}}</ref><ref>{{cite web |last=Lehmann |first=Clayton Miles |date=18 January 2007 |title=Palestine |url=http://sunburst.usd.edu/~clehmann/erp/Palestine/palestin.htm |url-status=dead |archive-url=https://archive.today/20130407005423/http://sunburst.usd.edu/~clehmann/erp/Palestine/palestin.htm |archive-date=7 April 2013 |access-date=9 February 2013 |website=Encyclopedia of the Roman Provinces |publisher=University of South Dakota}}</ref> Some of the Judean survivors resettled in Galilee, with some rabbinical families gathering in [[Sepphoris]].<ref>Miller, 1984, p. [https://books.google.com/books?id=KcsUAAAAIAAJ&pg=PA132 132]</ref> The [[Mishnah]] and part of the [[Jerusalem Talmud|Talmud]], central Jewish texts, were composed during the 2nd to 4th centuries CE in Galilee.<ref>{{Harvard citation no brackets|Morçöl|2006|p=304}}</ref>

Jewish communities continued to live on the edges of Judea, including [[Bayt Jibrin|Eleutheropolis]],<ref>Zissu, B., Ecker, A., and Klein, E, 2017, "Archaeological Explorations North of Bet Guvrin (Eleutheropolis)", in: ''Speleology and Spelestology, Proceedings of the VIII International Scientific Conference''. Nabereznye Chelny, pp. 183-203.</ref> [[Ein Gedi]]<ref>Hirschfeld, Y. (2004). Ein Gedi: A Large Jewish Village1. ''Qadmoniot'', ''37'', 62-87. "The consequences of the Second Revolt were infinitely more catastrophic for the Jewish population than were those of the First Revolt. The chilling evidence found in the caves of Nahal Hever illustrates the scale of the killing and suffering. However, the Jewish settlement at Ein Gedi survived. As suggested above, relatives of refugees who had fled to the caves traveled to those sites at some point after the revolt to give the deceased a proper burial. The results of the excavations at Ein Gedi indicate a continuity of settlement during the transition from the Late Roman (Stratum III) to the Byzantine (II) period."</ref> and the southern [[Hebron Hills]]. There were also Jewish communities along the coastal plain, in [[Caesarea Maritima|Caesarea]], [[Beit She'an]] and on the [[Golan Heights]].<ref name="CambridgeJudaism2">David Goodblatt, 'The political and social history of the Jewish community in the Land of Israel,' in William David Davies, Louis Finkelstein, Steven T. Katz (eds.) [[iarchive:cambridgehis xxxx 1984 004 8494287/page/n437| ''The Cambridge History of Judaism: Volume 4, The Late Roman-Rabbinic Period'']], Cambridge University Press, 2006 pp.404-430, p.406.</ref><ref name=":02">{{harvnb|Mor|2016|pp=483–484}}: "Land confiscation in Judaea was part of the suppression of the revolt policy of the Romans and punishment for the rebels. But the very claim that the sikarikon laws were annulled for settlement purposes seems to indicate that Jews continued to reside in Judaea even after the Second Revolt. There is no doubt that this area suffered the severest damage from the suppression of the revolt. Settlements in Judaea, such as Herodion and Bethar, had already been destroyed during the course of the revolt, and Jews were expelled from the districts of Gophna, Herodion, and Aqraba. However, it should not be claimed that the region of Judaea was completely destroyed. Jews continued to live in areas such as Lod (Lydda), south of the Hebron Mountain, and the coastal regions. In other areas of the Land of Israel that did not have any direct connection with the Second Revolt, no settlement changes can be identified as resulting from it."</ref>

In the aftermath of the defeat, the maintenance of Jewish settlement in Palestine became a major concern of the rabbis.<ref name="smelik2">Willem F. Smelik, [https://books.google.com/books?id=tju-IreyEDMC&pg=PA434 ''The Targum of Judges,''] BRILL 1995 p.434.</ref> They endeavored to halt Jewish [[Jewish diaspora|dispersal]], and even banned emigration from Palestine, branding those who settled outside its borders as idolaters.<ref name="smelik2" />

=== Roman losses ===
[[File:Reconstruction drawing of the triumphal arch dedicated to Hadrian near the camp of the Sixth Legion at Tel Shalem, Israel Museum, Jerusalem (15472504477).jpg|thumb|220px|Schematic reconstruction of the Arch of Hadrian in Tel Shalem, dedicated to the Emperor for defeating the Jewish revolt of 132–135]]

Cassius Dio wrote that "Many Romans, moreover, perished in this war. Therefore, Hadrian, in writing to the Senate, did not employ the opening phrase commonly affected by the emperors: 'If you and your children are in health, it is well; I and the army are in health.'"<ref name="Cassius">Cassius Dio, ''Roman History''</ref> Some argue that the exceptional number of preserved Roman veteran diplomas from the late 150s and 160s CE indicate an unprecedented conscription across the Roman Empire to replenish heavy losses within military legions and auxiliary units between 133 and 135, corresponding to the revolt.<ref>E. Werner. ''The bar Kokhba Revolt: The Roman Point of View. The Journal of Roman Studies Vol. 89 (1999), pp. 76-89. ''[https://www.jstor.org/stable/300735]</ref>

As noted above, [[Legio XXII Deiotariana|XXII ''Deiotariana'']] may have been disbanded after serious losses.<ref name="F. Keppie 2000 pp 228-229" /><ref name="livius.org account">[https://www.livius.org/le-lh/legio/xxii_deiotariana.html livius.org account] {{Webarchive|url=https://web.archive.org/web/20150317020539/http://www.livius.org/le-lh/legio/xxii_deiotariana.html |date=2015-03-17 }}(Legio XXII Deiotariana)</ref> In addition, some historians argue that [[Legio IX Hispana]]'s disbandment in the mid-2nd century could have been a result of this war.<ref name="livius.org" /> Previously it had generally been accepted that the Ninth disappeared around 108 CE, possibly suffering its demise in Britain, according to [[Theodor Mommsen|Mommsen]]; but archaeological findings in 2015 from [[Nijmegen]], dated to 121 CE, contained the known inscriptions of two senior officers who were deputy commanders of the Ninth in 120 CE, and lived on for several decades to lead distinguished public careers. It was concluded that the Legion was disbanded between 120 and 197 CE—either as a result of fighting the Bar Kokhba revolt, or in [[Cappadocia (Roman province)|Cappadocia]] (161), or at the Danube (162).<ref>{{Cite web|url=https://www.livius.org/articles/legion/legio-viiii-hispana/|title=Legio VIIII Hispana - Livius|website=www.livius.org}}</ref>{{unreliable?|date=October 2022}} [[Legio X Fretensis]] sustained heavy casualties during the revolt.<ref name="mor334" />

=== Impact on other communities ===
[[Eusebius]] of Caesarea wrote that [[Jewish Christian|Christian]]s were killed and suffered "all kinds of persecutions" at the hands of rebel Jews when they refused to help Bar Kokhba against the Roman troops.<ref>{{Cite web|url=https://www.livius.org/ja-jn/jewish_wars/bk03.html#Justin.|title=Texts on Bar Kochba: Eusebius}}</ref><ref>Bourgel, Jonathan, ″The Jewish-Christians in the storm of the Bar Kokhba Revolt″, in: ''From One Identity to Another: The Mother Church of Jerusalem Between the Two Jewish Revolts Against Rome (66-135/6 EC)''. Paris: Éditions du Cerf, collection Judaïsme ancien et Christianisme primitive, (French), pp. 127-175.</ref> Although Christians regarded Jesus as the Messiah and did not support Bar Kokhba,<ref>Justin, "Apologia", ii.71, compare "Dial." cx; Eusebius "Hist. Eccl." iv.6,§2; Orosius "Hist." vii.13</ref> they were barred from Jerusalem along with the Jews.<ref>{{cite book |last=Davidson |first=Linda |url=https://books.google.com/books?id=YVYkrNhPMQkC&q=For+the+next+two+centuries+the+Romans+barred+both+Jewish+and+Christian+access+to+the+holy+sites+of+the+city&pg=PA279 |title=Pilgrimage: From the Ganges to Graceland: an Encyclopedia, Volume 1 |publisher=ABC-CLIO |year=2002 |isbn=1576070042 |page=279}}</ref>

===Later relations between the Jews and the Roman Empire===
{{Main|History of the Jews in the Roman Empire}}
Relations between the Jews in the region and the Roman Empire continued to be complicated. [[Constantine the Great and Judaism|Constantine I]] allowed Jews to mourn their defeat and humiliation once a year on [[Tisha B'Av]] at the [[Western Wall]]. In 351–352 CE, the Jews of Galilee launched [[Jewish revolt against Constantius Gallus|yet another revolt]], provoking heavy retribution.<ref>Bernard Lazare and Robert Wistrich, Antisemitism: Its History and Causes, University of Nebraska Press, 1995, I, pp.46-7.</ref> The Gallus revolt came during the rising influence of early Christians in the Eastern Roman Empire, under the [[Constantinian dynasty]]. In 355, however, the relations with the Roman rulers improved, upon the rise of Emperor [[Julian (emperor)|Julian]], the last of the Constantinian dynasty, who, unlike his predecessors, defied Christianity. In 363, not long before Julian left Antioch to launch his campaign against Sassanian Persia, he ordered the Jewish Temple rebuilt in his effort to foster religions other than Christianity.<ref>Ammianus Marcellinus, ''Res Gestae'', 23.1.2–3.</ref> The failure to rebuild the Temple has mostly been ascribed to the dramatic [[Galilee earthquake of 363]], and traditionally also to the Jews' ambivalence about the [[Third Temple|project]]. Sabotage is a possibility, as is an accidental fire, though Christian historians of the time ascribed it to divine intervention.<ref name="Solomon">See [http://www.fordham.edu/halsall/jewish/julian-jews.html "Julian and the Jews 361–363 CE"] (Fordham University, The Jesuit University of New York) and [https://web.archive.org/web/20051020130904/http://www.gibsoncondo.com/~david/convert/history.html "Julian the Apostate and the Holy Temple"].</ref> Julian's support of [[Judaism]] caused Jews to call him "Julian the [[Hellenes (religion)|Hellene]]".<ref>A Psychoanalytic History of the Jews, [[Avner Falk]]</ref> Julian's fatal wound in the Persian campaign put an end to Jewish aspirations, and Julian's successors embraced Christianity through the entirety of [[Byzantine Empire|Byzantine rule]] of Jerusalem, preventing any Jewish claims.

In 438 CE, when the Empress [[Licinia Eudoxia|Eudocia]] removed the ban on Jews' praying at the [[Temple Mount|Temple site]], the heads of the Community in Galilee issued a call "to the great and mighty people of the Jews" which began: "Know that the end of the exile of our people has come!" However, the Christian population of the city saw this as a threat to their primacy, and a riot erupted which chased Jews from the city.<ref>Avraham Yaari, ''Igrot Eretz Yisrael'' (Tel Aviv, 1943), p. 46.</ref><ref>{{Cite book|url=https://books.google.com/books?id=8O95ErDSZQgC&dq=eudocia+jews+temple+mount&pg=PA157|title=Remains of the Jews: The Holy Land and Christian Empire in Late Antiquity|first=Andrew S.|last=Jacobs|date=September 10, 2004|publisher=Stanford University Press|isbn=9780804747059 |via=Google Books}}</ref>

During the fifth and sixth centuries, a series of [[Samaritan revolts]] broke out across the [[Palaestina Prima]] province. Especially violent were the third and the fourth revolts, which resulted in near annihilation of the Samaritan community.<ref>Shalev-Hurvitz, V. Oxford University Press 2015. p235</ref> It is likely that the [[Samaritan revolts#556 Samaritan revolt|Samaritan revolt of 556]] was joined by the Jewish community, which had also suffered brutal suppression of their religion under Emperor Justinian.<ref>Weinberger, p. 143</ref><ref>{{Cite journal|url=http://www.jstor.org/stable/41443760|author=Brewer, Catherine|title=The Status of the Jews in Roman Legislation: The Reign of Justinian 527-565 Ce |year=2005|journal=European Judaism: A Journal for the New Europe|volume=38|issue=2|pages=127–139|jstor=41443760 |via=JSTOR}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=xDNv6qZ_I-IC&dq=oppression+jews+justinian&pg=PR30|title=The Emperor Justinian and the Byzantine Empire|first=James Allan Stewart|last=Evans|date=September 10, 2005|publisher=Greenwood Publishing Group|isbn=9780313325823 |via=Google Books}}</ref>

In the belief of restoration to come, in the early seventh century, the Jews made an [[Jewish revolt against Heraclius|alliance]] with the [[Sasanian Empire]], joining the invasion of Palaestina Prima in 614 to overwhelm the [[Byzantine Empire|Byzantine]] garrison, and gaining autonomous rule over Jerusalem.<ref name = "Phoenicia">{{cite book |title= Itineraria Phoenicia|url = https://books.google.com/books?id=SLSzNfdcqfoC&q=Opusculum+de+Persica+captivitate&pg=PA542| author = Edward Lipiński |publisher = Peeters Publishers |pages = 542–543 |year = 2004 |isbn = 9789042913448|access-date=11 March 2014}}</ref> However, their autonomy was brief: the [[Nehemiah ben Hushiel|Jewish leader]] was shortly assassinated during a Christian revolt and, though Jerusalem was reconquered by Persians and Jews within 3 weeks, it fell into anarchy. With the subsequent withdrawal of Persian forces, Jews surrendered to the Byzantines in 625 CE or 628 CE. Byzantine control of the region was finally lost to Muslim Arab armies in 637 CE, when [[Umar]] completed the conquest of [[Akko]].

==Archaeology==
===Destroyed Jewish villages and fortresses===
Several archaeological excavations have been performed during the 20th and 21st centuries in ruins of Roman-period Jewish villages across Judea and Samaria, as well in the Roman-dominated cities on the [[Israeli coastal plain|coastal plain]]. Most of the villages in Judea's larger region show signs of devastation or abandonment that dates to the Bar-Kokhba revolt. Buildings and underground installations carved out beneath or close to towns, such as hiding complexes, burial caves, storage facilities, and field towers, have both been found to have [[Destruction layer|destruction layers]] and abandonment deposits. Furthermore, there is a gap in settlement above these levels. Fragmentary material from Transjordan and the Galilee adds to the discoveries from Judea.<ref name="raviv2021" />
[[File:Hurvat-Itri-4713.jpg|thumb|The ruins of [[Hurvat Itri]] display a [[destruction layer]] dating to the revolt, along with a mass grave containing the remains of 15 individuals, including one with signs of beheading]]
Excavations at archaeological sites such as [[Hurvat Itri]] and [[Kiryat Sefer|Khirbet Badd ‘Isa]] have demonstrated that these Jewish villages were destroyed in the revolt, and were only resettled by pagan populations in the third century.<ref>{{Cite journal |last=Bar |first=Doron |date=2005 |title=Rural Monasticism as a Key Element in the Christianization of Byzantine Palestine |url=https://www.jstor.org/stable/4125284 |journal=The Harvard Theological Review |volume=98 |issue=1 |pages=64 |doi=10.1017/S0017816005000854 |issn=0017-8160 |jstor=4125284 |s2cid=162644246 |quote=}}</ref><ref>Yitzhak Magen, Yoav Zionit, and Erna Sirkis, "Kiryat Sefer‒A Jewish Village and Synagogue of the Second Temple Period" (in Hebrew) ''Qadmoniot'' 117. Vol 32 (1999) 25-32.</ref><ref>Boaz Zisu, Amir Ganor, "Horvat 'Etri‒The Ruins of a Second Temple Period Jewish Village on the Coastal Plain" (in Hebrew). ''Qadmoniot'' 132, vol. 35. (2000). 18-27</ref> Discoveries from towns like Gophna, known to be Jewish before the revolt, demonstrate that pagans of Hellenistic and Roman culture lived there during the Late Roman period.<ref>Klein, E, 2011, “Gophna during the Late Roman Period in Light of Artistic and Epigraphic Finds”, in: A. Tavger., Z. Amar and M. Billig (eds.), ''In the Highland’s Depth: Ephraim Range and Binyamin Research Studies'', Beit-El, pp. 119-134 (Hebrew).</ref>

Herodium was excavated by archaeologist Ehud Netzer in the 1980s, publishing results in 1985. According to findings, during the later Bar-Kokhba revolt, complex tunnels were dug, connecting the earlier cisterns with one another.<ref>Netzer E. and Arzi S., 1985. “Herodium Tunnels”, Qadmoniot 18, Pp. 33–38. (in Hebrew)</ref> These led from the Herodium fortress to hidden openings, which allowed surprise attacks on Roman units besieging the hill.

The ruins of [[Betar (ancient village)|Betar]], the last standing stronghold of Bar Kokhba, can be found at ''Khirbet al-Yahud'', an archeological site located in the vicinity of [[Battir]] and [[Beitar Illit]]. A stone inscription bearing Latin characters and discovered near the site shows that the [[Legio V Macedonica|Fifth Macedonian Legion]] and the [[Legio XI Claudia|Eleventh Claudian Legion]] took part in the siege.<ref name="C. Clermont-Ganneau, 1899, pp. 263-270">C. Clermont-Ganneau, ''Archaeological Researches in Palestine during the Years 1873-74'', London 1899, pp. 263-270.</ref>

=== Hideout systems ===
[[File:PikiWiki Israel 19975 Archeological sites of Israel.jpg|200px|thumb|Entrance to a hideout system dating from the revolt which was discovered in [[Adullam Grove Nature Reserve#Archaeology|Hurvat Midras]]]]The Bar Kokhba revolt has been better understood thanks to the discovery of artificially carved hideout systems under many sites across Judea. Their discovery is consistent with [[Cassius Dio]]'s writings, which reported that the rebels used underground networks as part of their tactics to avoid direct confrontations with the Romans. Many were hewn in earlier times and were utilized by rebels during the revolt as indicated by the usage of [[Bar Kokhba Revolt coinage|Bar Kokhba revolt coinage]] and other archeological findings.<ref name=":1">Zissu, B., & Kloner, A. (2010). The Archaeology of the Second Jewish Revolt against Rome (The Bar Kokhba Revolt)–Some New Insights. ''Bollettino di Archeologia online I Volume speciale F'', ''8'', 40-52.</ref><ref>Kloner, A., Zissu, B., (2003). Hiding Complexes in Judaea: An Archaeological and Geographical Update on the Area of the Bar Kokhba Revolt. In P. SCHÄFER (ed), ''The Bar Kokhba War Reconsidered: New Perspectives on the Second Jewish Revolt against Rome''. Tübingen, 181–216</ref>

Hideout systems were found at more than 130 archaeological sites in Judea; most of them in the [[Shephelah|Judaean Lowlands]], but also in the [[Judaean Mountains]], and even in the Galilee.<ref name=":1" /><ref>Kloner A., and Zissu B., 2009, Underground Hiding Complexes in Israel and the Bar Kokhba Revolt, ''Opera Ipogea'' 1/2009, pp. 9-28</ref> Examples include: [[Hurvat Midras]], [[Tell ej-Judeideh|Tel Goded]], [[Maresha]], [[Aboud]] and others.

===Refuge caves===
{{see|Cave of Horror|Cave of Letters|}}
[[File:Cave_of_Letters_(1).jpg|thumb|The [[Cave of Letters]], where several documents of the period, including letters from Simeon bar Kokhba to the people of Ein Gedi, were discovered]]
Near the end of the uprising, many Jews fleeing for their life sought asylum in refuge caves, the most of which are found in Israel's [[Judaean Desert]] on high cliffs overlooking the [[Dead Sea]] and the [[Jordan Valley]]. The majority of these caves are large natural caverns (with few man-made modifications) that are situated in nearly inaccessible vertical cliffs. <ref name=":1" />

They carried luxury goods, cash, arms, papers and deeds, and even the keys to their homes as a hint that they intended to return there once the fighting was over. These items were frequently discovered with their owners' bones in caverns, which is evidence of their tragic fate. The [[Cave of Letters]] in [[Nahal Hever]] and the caverns in [[Wadi Murabba'at]], which yielded a plethora of written records from the time of the revolt, are among the best-known refuge caves.<ref name=":1" />

The [[Cave of Letters]] was surveyed in explorations conducted in 1960–1961, when letters and fragments of papyri were found dating back to the period of the Bar Kokhba revolt.[[File:BabathaScroll.jpg|thumb|A scroll found in the cave, part of the [[Babatha]] archive]]
[[Cave of Horror]] is the name given to Cave 8, where the skeletons of 40 Jewish refugees from the Bar Kokhba revolt, including men, women and children, were discovered.<ref>{{Cite journal |author=AHARONI, Y. |year=1962 |title=Expedition B — The Cave of Horror |url=http://www.jstor.org/stable/27924906 |journal=Israel Exploration Journal |volume=12 |issue=3/4 |pages=186–199 |jstor=27924906 |via=JSTOR}}</ref><ref>{{Cite news |date=March 16, 2021 |title=Rare ancient scroll found in Israel Cave of Horror |work=BBC News |url=https://www.bbc.com/news/world-middle-east-56405090}}</ref> Three potsherds with the names of three of the deceased were also found alongside the skeletons in the cave.

In 2023, archaeologists discovered a cache consisting of four Roman swords and a [[pilum]] concealed within a crevice in a cave located within the [[Ein Gedi]] nature reserve. Analysis of the sword types and the discovery of a Bar Kokhba revolt coin within the cave strongly support the hypothesis put forth by archaeologists, which suggests that these items were concealed by Jewish rebels during the Bar Kokhba revolt, serving as a precautionary measure to elude detection by Roman authorities.<ref>{{Cite web |last=Guy |first=Jack |date=2023-09-06 |title=Four 1,900-year-old Roman swords found in cave in Israel |url=https://www.cnn.com/style/article/ancient-weapons-cache-judean-desert-scli-intl-scn/index.html |access-date=2023-09-07 |website=CNN |language=en}}</ref>

=== Coinage ===
As of 2023, twenty-four coins from the Bar Kokhba revolt have been discovered outside of Judaea in various parts of Europe, including what was then the provinces of [[Roman Britain|Britannia]], [[Pannonia Superior|Pannonia]], [[Roman Dacia|Dacia]], and [[Dalmatia (Roman province)|Dalmatia]]. The bulk of the coins were discovered near Roman military locations, including multiple legionary and auxiliary camps, though not necessarily in a strict military context. It has been suggested to attribute these findings to Roman soldiers who took part in the uprising and brought the coins as souvenirs or commemorative relics, or to Jewish captives, slaves or immigrants who arrived in those areas in the aftermath of the revolt.<ref name="EZB2009">Eshel, H., Zissu, B., & Barkay, G. (2009). Sixteen Bar Kokhba Coins from Roman Sites in Europe. ''Israel Numismatic Journal'', ''17'', 91-97.</ref><ref name="Grull2023">Grull, T. (2023), ''Bar Kokhba Coins from Roman Sites in Europe: A Reappraisal.''</ref><ref name="CFK2018">Cesarik, N., Filipčić, D., Kramberger, V. (2018). "[https://www.researchgate.net/publication/331320253_Bar_Kokhba's_bronze_coin_from_Kolovare_Beach_in_Zadar Bar Kokhba’s bronze coin from Kolovare Beach in Zadar]". ''Journal of the Archaeological Museum in Zadar'', Vol. 32. No. 32.</ref>

==== Hoards ====
One [[Baraita]] contains a rabbinic depiction of a widespread archeological phenomenon: the discovery of [[Hoard|hoards]] of [[Bar Kokhba Revolt coinage|Bar Kokhba coinage]] all over Judea. The Jews who hid those hoards were unable to collect them due to the presence of Roman garrisons, or because they were killed during the revolt's suppression. It is reasonable to believe that the extensive destruction played a part in the loss of the hiding locations as well. Thirty hoards from this era have been found, more than any other decade.<ref>{{Cite book |last=ספראי |first=זאב |title=מחקרי יהודה ושומרון |publisher=מו"פ אזורי השומרון ובקעת הירדן; המרכז האוניברסיטאי אריאל בשומרון |editor-last=ביליג |editor-first=מרים |volume=יט |location=אריאל |pages=70 |language=Hebrew |chapter=הר המלך עדיין חידה |issn=0792-8416 |author-link=Ze'ev Safrai}}</ref>

===Roman legionary camps===
A number of locations have been identified with Roman Legionary camps in the time of the Bar Kokhba War, including in Tel Shalem, Jerusalem, Lajjun and more.

====Jerusalem inscription dedicated to Hadrian (129/30 CE)====
In 2014, one half of a Latin inscription was discovered in Jerusalem during excavations near the Damascus Gate.<ref name="i24news">Jerusalem Post. 21 October 2014 [https://www.jpost.com/Not-Just-News/WATCH-2000-year-old-commemorative-inscription-dedicated-to-Roman-emperor-Hadrian-unveiled-379384 WATCH: 2,000-YEAR-OLD INSCRIPTION DEDICATED TO ROMAN EMPEROR UNVEILED IN JERUSALEM]</ref> It was identified as the right half of a complete inscription, the other part of which was discovered nearby in the late 19th century and is currently on display in the courtyard of Jerusalem's Studium Biblicum Franciscanum Museum. The complete inscription was translated as follows:
:: To the Imperator Caesar Traianus Hadrianus Augustus, son of the deified Traianus Parthicus, grandson of the deified Nerva, high priest, invested with tribunician power for the 14th time, consul for the third time, father of the country (dedicated by) the 10th legion Fretensis Antoniniana.
The inscription was dedicated by Legio X Fretensis to the emperor Hadrian in the year 129/130 CE. The inscription is considered to greatly strengthen the claim that indeed the emperor visited Jerusalem that year, supporting the traditional claim that Hadrian's visit was among the main causes of the Bar Kokhba Revolt, and not the other way around.<ref name="i24news" />

====Tel Shalem triumphal arc and Hadrian's statue====
The location was identified as a Roman military post during the 20th century, with archaeological excavation performed in the late 20th century following an accidental discovery of Hadrian's bronze statue in the vicinity of the site in 1975.<ref>{{Cite journal |last1=Gergel |first1=Richard A. |year=1991 |title=The Tel Shalem Hadrian Reconsidered |url=https://www.jstor.org/stable/505724 |journal=American Journal of Archaeology |volume=95 |issue=2 |pages=231–251 |doi=10.2307/505724 |jstor=505724 |s2cid=193092889}}</ref> Remains of a large Roman military camp and fragments of a triumphal arc dedicated to Emperor Hadrian were consequently discovered at the site.

==Geographic extent of the revolt==
Over the years, two schools formed in the analysis of the Revolt. One of them is ''maximalists'', who claim that the revolt spread through the entire Judea Province and beyond it into neighboring provinces. The second one is that of the ''minimalists'', who restrict the revolt to the area of the Judaean hills and immediate environs.<ref name="menahem2013">M. Menahem. ''WHAT DOES TEL SHALEM HAVE TO DO WITH THE BAR KOKHBA REVOLT?''. U-ty of Haifa / U-ty of Denver. SCRIPTA JUDAICA CRACOVIENSIA. Vol. 11 (2013) pp. 79–96.</ref>

===Judea proper===
It is generally accepted that the Bar Kokhba revolt encompassed all of Judea, namely the villages of the [[Judean hills]], the Judean desert, and northern parts of the [[Negev]] desert. It is not known whether the revolt spread outside of Judea.{{sfn|Mor|2016|p=152}}

====Jerusalem====
Until 1951, [[Bar Kokhba Revolt coinage]] was the sole archaeological evidence for dating the revolt.<ref name="Eshel" /> These coins include references to "Year One of the redemption of Israel", "Year Two of the freedom of Israel", and "For the freedom of Jerusalem". Despite the reference to Jerusalem, as of early 2000s, archaeological finds, and the lack of revolt coinage found in Jerusalem, supported the view that the revolt did not capture Jerusalem.<ref name="Eshel-2">{{harvnb|Eshel|2003|pp=95–96}}: "Returning to the Bar Kokhba revolt, we should note that up until the discovery of the first Bar Kokhba documents in Wadi Murabba'at in 1951, Bar Kokhba coins were the sole archaeological evidence available for dating the revolt. Based on coins overstock by the Bar Kokhba administration, scholars dated the beginning of the Bar Kokhba regime to the conquest of Jerusalem by the rebels. The coins in question bear the following inscriptions: "Year One of the redemption of Israel", "Year Two of the freedom of Israel", and "For the freedom of Jerusalem". Up until 1948 some scholars argued that the "Freedom of Jerusalem" coins predated the others, based upon their assumption that the dating of the Bar Kokhba regime began with the rebel capture Jerusalem." L. Mildenberg's study of the dies of the Bar Kokhba definitely established that the "Freedom of Jerusalem" coins were struck later than the ones inscribed "Year Two of the freedom of Israel". He dated them to the third year of the revolt.' Thus, the view that the dating of the Bar Kokhba regime began with the conquest of Jerusalem is untenable. lndeed, archeological finds from the past quarter-century, and the absence of Bar Kokhba coins in Jerusalem in particular, support the view that the rebels failed to take Jerusalem at all."</ref>

In 2020, the fourth Bar Kokhba minted coin and the first inscribed with the word "Jerusalem" was found in Jerusalem Old City excavations.<ref>https://www.israelhayom.com/2020/05/11/rare-bar-kochba-era-coin-discovered-at-foot-of-temple-mount/ {{Bare URL inline|date=August 2022}}</ref> Despite this discovery, the Israel Antiques Authority still maintained the opinion that Jerusalem was not taken by the rebels, due to the fact that of thousands of Bar Kokhba coins had been found outside Jerusalem, but only four within the city (out of more than 22,000 found within the city). The Israel Antiques Authority's archaeologists Moran Hagbi and Dr. Joe Uziel speculated that "It is possible that a Roman soldier from the Tenth Legion found the coin during one of the battles across the country and brought it to their camp in Jerusalem as a souvenir."<ref>{{cite web|url=https://www.timesofisrael.com/year-2-of-freedom-ancient-coin-from-bar-kochba-revolt-found-near-temple-mount/|title = 'Year 2 of freedom': Ancient coin from Bar Kochba revolt found near Temple Mount|website = [[The Times of Israel]]}}</ref>

===Galilee===
Among those findings are the rebel hideout systems in the Galilee, which greatly resemble the Bar Kokhba hideouts in Judea, and though are less numerous, are nevertheless important. The fact that Galilee retained its Jewish character after the end of the revolt has been taken as an indication by some that either the revolt was never joined by Galilee or that the rebellion was crushed relatively early there compared to Judea.<ref name="Harkabi1983">{{cite book|url=https://books.google.com/books?id=rb00WGipKCIC&pg=RA1-PA37|title=The Bar Kokhba Syndrome: Risk and Realism in International Politics|author=Yehoshafat Harkabi|publisher=SP Books|year=1983|isbn=978-0-940646-01-8|pages=1–}}</ref>

===Northern valleys===
Several historians, notably W. Eck of the University of Cologne, theorized that the Tel Shalem arch depicted a major battle between Roman armies and Bar Kokhba's rebels in Bet Shean valley,<ref name="menahem2013" /> thus extending the battle areas some 50 km northwards from Judea. The 2013 discovery of the [[Legio|military camp]] of [[Legio VI Ferrata]] near [[Tel Megiddo]].<ref>{{cite web|url=http://www.israelnationalnews.com/News/News.aspx/197910#.VdAs6vmqqko|title=Roman Legion Camp Unearthed in Megiddo - Inside Israel - News - Arutz Sheva|work=Arutz Sheva|date=9 July 2015 |access-date=2016-03-02}}</ref> However, Eck's theory on battle in Tel Shalem is rejected by M. Mor, who considers the location implausible given Galilee's minimal (if any) participation in the Revolt and distance from the main conflict flareup in Judea proper.<ref name="menahem2013" />

===Samaria===
A 2015 archaeological survey in Samaria identified some 40 hideout cave systems from the period, some containing Bar Kokhba's minted coins, suggesting that the war raged in Samaria at high intensity.<ref name="nrg">{{lang-he|התגלית שהוכיחה: מרד בר כוכבא חל גם בשומרון}} [http://www.nrg.co.il/online/1/ART2/709/420.html] NRG. 15 July 2015.</ref>

===Transjordan===
Jews from [[Perea|Peraea]] are thought to have taken part in the revolt. This is demonstrated by a [[destruction layer]] dating from the early second century at Tel Abu al-Sarbut in the [[Deir Alla|Sukkoth]] Valley,<ref>Steiner, M., Mulder-Hymans, N., and Boertien, J.. 2013. “Een joods huishouden in Perea? De resultaten van de eerste opgravingscampagne op Tell Abu Sarbut in 2012.” ''Tijdschrift voor Mediterrane Archeologie'' 50: 38–44</ref> and by abandonment deposits from the same period that were discovered at al-Mukhayyat<ref>Sagiv, N. 2013. “Jewish Finds from Peraea (Transjordan) from the Second Temple Period until the Bar-Kokhba Revolt.” Jerusalem and Eretz-Israel 8–9: 191–210. (Hebrew)</ref> and [[Callirrhoe (Jordan)|Callirrhoe]].<ref>Gerber, Y. 1998. Review of ''Fouilles archéologiques de ʿAïn ez-Zâra/Callirrhoé, villégiature hérodienne'', by C. Clamer. BASOR 312: 86–89</ref> There is also evidence for Roman military presence in Perea in the middle of the century, as well as evidence of the settlement of Roman veterans in the area.<ref name="raviv2021" />

This view is supported by a destruction layer in [[Heshbon|Tel Hesban]] that dates to 130 CE,<ref>Mitchel, L. A. 1992. Hesban 7: Hellenistic and Roman Strata. Berrien Springs, MI: Institute of Archaeology. p. 62-63</ref> and a decline in settlement from the Early Roman to the Late Roman periods discovered in the survey of the [[Iraq al-Amir]] region.<ref>Ji, C. C., and Lee, J. K.. 2002. “The survey in the regions of 'Iraq al-Amir and Wadi al-Kafrayn, 2000.” ''Annual of the Department of Antiquities of Jordan'' 46: 179–95</ref> However, it is still unclear whether this decline was caused by the [[First Jewish–Roman War]] or the Bar Kokhba revolt.<ref name="raviv2021" />

Bowersock suggested of linking the [[Nabataeans|Nabateans]] to the revolt, claiming "a greater spread of hostilities than had formerly been thought... the extension of the Jewish revolt into northern [[Transjordan (region)|Transjordan]] and an additional reason to consider the spread of local support among [[Safaitic]] tribes and even at [[Jerash|Gerasa]]."<ref name="The Bar Kokhba War Reconsidered" />

==Sources==
The revolt is mostly still shrouded in mystery, and only one brief historical account of the rebellion survives.<ref name="Eshel">Hanan Eshel,[https://archive.org/details/cambridgehis_xxxx_1984_004_8494287/page/n1082  'The Bar Kochba revolt, 132-135,'] in William David Davies, Louis Finkelstein, Steven T. Katz (eds.) ''The Cambridge History of Judaism: Volume 4, The Late Roman-Rabbinic Period,'' pp.105-127, p.105.</ref>

=== Dio Cassius ===
The best recognized source for the revolt is [[Cassius Dio]], ''Roman History'' (book 69),<ref name="Dio" /><ref>Mordechai, Gihon. ''New insight into the Bar Kokhba War and a reappraisal of Dio Cassius 69.12-13''. University of Pennsylvania Press. The Jewish Quarterly Review Vol. 77, No. 1 (Jul., 1986), pp. 15-43. {{doi|10.2307/1454444}}</ref> even though the writings of the Roman historian concerning the Bar Kokhba revolt survived only as fragments. The account extends on about two pages and is largely an historical perspective with the general course of the rebellion and its disastrous results, without mentioning specific names and locations.

=== Eusebius of Caesarea ===
The Christian author [[Eusebius]] of Caesarea wrote a brief account of the revolt within the [[Church History (Eusebius)]] compilation, notably mentioning Bar Chochebas (which means “star” according to Eusebius) as the leader of the Jewish rebels and their last stand at Beththera (i.e., [[Betar (fortress)|Betar]]). Though Eusebius lived one and a half centuries after the revolt and wrote the brief account from the Christian theological perspective, his account provides important details on the revolt and its aftermath in Judea.

=== Jerusalem Talmud ===
The Jerusalem Talmud contains descriptions of the results of the rebellion, including the Roman executions of Judean leaders and religious persecution.

=== Primary sources ===
[[File:Bar Kokhba's papyrus.png|thumb|200px|A cluster of [[papyrus]] containing Bar Kokhba's orders during the last year of the revolt, found at the [[Cave of Letters]] in the Judean desert by Israeli archaeologist [[Yigael Yadin]].]]
The discovery of the [[Cave of Letters]] in the Dead Sea area, dubbed as "Bar Kokhba archive",<ref>Peter Schäfer. ''The Bar Kokhba War reconsidered''. 2003. p184.</ref> which contained letters actually written by Bar Kokhba and his followers, has added much new primary source data, indicating among other things that either a pronounced part of the Jewish population spoke only Greek or there was a foreign contingent among Bar Kokhba's forces, accounted for by the fact that his military correspondence was, in part, conducted in Greek.<ref>Mordechai Gichon, [https://www.jstor.org/stable/1454444 'New Insight into the Bar Kokhba War and a Reappraisal of Dio Cassius 69.12-13,'] [[The Jewish Quarterly Review]], Vol. 77, No. 1 (Jul., 1986), pp. 15-43, p.40.</ref> Close to the Cave of Letters is the [[Cave of Horror]], where the remains of Jewish refugees from the rebellion were discovered along with fragments of letters and writings. Several briefer sources have been uncovered in the area over the past century, including references to the revolt from Nabatea and Roman Syria.

== Legacy ==

=== In Rabbinic Judaism ===
The disastrous end of the revolt occasioned major changes in Jewish religious thought. [[Jewish messianism]] was abstracted and spiritualized, and rabbinical political thought became deeply cautious and conservative. The Talmud, for instance, refers to Bar Kokhba as "Ben-Kusiba", a derogatory term used to indicate that he was a [[List of messiah claimants|false Messiah]]. The deeply ambivalent rabbinical position regarding Messianism, as expressed most famously in [[Maimonides]] "Epistle to Yemen," would seem to have its origins in the attempt to deal with the trauma of a failed Messianic uprising.<ref>Wikisource: "[[wikisource:Epistle to Yemen|Epistle to Yemen]]"</ref>

=== In Zionism and modern Israel ===
In the post-rabbinical era, the Bar Kokhba Revolt became a symbol of valiant national resistance. The [[Zionist youth movement]] [[Betar]] took its name from Bar Kokhba's traditional last stronghold, and [[David Ben-Gurion]], Israel's first prime minister, took his Hebrew last name from one of Bar Kokhba's generals.<ref>"[https://www.knesset.gov.il/vip/BenGurion/eng/BenGurion_Bio_eng.html]"</ref>

A popular children's song, included in the curriculum of Israeli kindergartens, has the refrain "Bar Kokhba was a Hero/He fought for Liberty," and its words describe Bar Kokhba as being captured and thrown into a lion's den, but managing to escape riding on the lion's back.<ref name="The-military-and-militarism-in-Israeli-society">''[https://books.google.com/books?id=tRwTElL0pKcC&dq=%22Bar+Kokhba+was+a+Hero%22&pg=PA150 The military and militarism in Israeli society]'' by Edna Lomsky-Feder, Eyal Ben-Ari]." Retrieved on September 3, 2010</ref>

==See also==
* [[List of conflicts in the Near East]]
* [[Sicaricon (Jewish law)]]

==References==
{{Reflist|30em}}

== Bibliography ==
{{refbegin|2}}
* {{cite journal |last1=Feldman |first1=Louis H. |title=Some Observations on the Name of Palestine |journal=Hebrew Union College Annual |date=1990 |volume=61 |pages=1–23 |jstor=23508170 |issn=0360-9049}}
* {{cite journal |first=David |last=Jacobson |year=2001 |url=http://cojs.org/when_palestine_meant_israel-_david_jacobson-_bar_27-03-_may-jun_2001/ |title=When Palestine Meant Israel |journal=Biblical Archaeology Review |volume=27 |issue=3 |url-status=live |archive-url=https://web.archive.org/web/20220407183723/http://cojs.org/when_palestine_meant_israel-_david_jacobson-_bar_27-03-_may-jun_2001/ |archive-date=7 April 2022}}
* {{cite book|first=Menahem|last=Mor|title=The Second Jewish Revolt: The Bar Kokhba War, 132-136 CE|url=https://books.google.com/books?id=T8wJDAAAQBAJ|date=4 May 2016|publisher=BRILL|isbn=978-90-04-31463-4}}
* {{cite book|last=Eshel|first=Hanan|author-link=Hanan Eshel|editor=Peter Schäfer|title=The Bar Kokhba War Reconsidered: New Perspectives on the Second Jewish Revolt Against Rome|chapter-url=https://books.google.com/books?id=1TA-Fg4wBnUC&pg=PA95|year=2003|publisher=Mohr Siebeck|isbn=978-3-16-148076-8|pages=95–96|chapter=The Dates used during the Bar Kokhba Revolt}}
* Yohannan Aharoni & Michael Avi-Yonah, ''The MacMillan Bible Atlas'', Revised Edition, pp. 164–65 (1968 & 1977 by Carta Ltd.)
* ''The Documents from the Bar Kokhba Period in the Cave of Letters (Judean Desert studies)''. Jerusalem: Israel Exploration Society, 1963–2002.
** Vol. 2, "Greek Papyri", edited by Naphtali Lewis; "Aramaic and Nabatean Signatures and Subscriptions", edited by [[Yigael Yadin]] and [[Jonas C. Greenfield]]. ({{ISBN|9652210099}}).
** Vol. 3, "Hebrew, Aramaic and Nabatean–Aramaic Papyri", edited Yigael Yadin, Jonas C. Greenfield, Ada Yardeni, [[BaruchA. Levine]] ({{ISBN|9652210463}}).
* W. Eck, 'The Bar Kokhba Revolt: the Roman point of view' in the ''Journal of Roman Studies'' 89 (1999) 76ff.
* Peter Schäfer (editor), ''Bar Kokhba reconsidered'', Tübingen: Mohr: 2003
* Aharon Oppenheimer, 'The Ban of Circumcision as a Cause of the Revolt: A Reconsideration', in ''Bar Kokhba reconsidered'', Peter Schäfer (editor), Tübingen: Mohr: 2003
* Faulkner, Neil. ''Apocalypse: The Great Jewish Revolt Against Rome''. Stroud, Gloucestershire, UK: Tempus Publishing, 2004 (hardcover, {{ISBN|0-7524-2573-0}}).
* Goodman, Martin. ''The Ruling Class of Judaea: The Origins of the Jewish Revolt against Rome, A.D. 66–70''. Cambridge: Cambridge University Press, 1987 (hardcover, {{ISBN|0-521-33401-2}}); 1993 (paperback, {{ISBN|0-521-44782-8}}).
* Richard Marks: ''The Image of Bar Kokhba in Traditional Jewish Literature: False Messiah and National Hero'': University Park: Pennsylvania State University Press: 1994: {{ISBN|0-271-00939-X}}
* {{cite book |title=Handbook of Decision Making |last=Morçöl |first=Göktuğ |isbn=978-1-57444-548-0 |publisher=CRC Press |year=2006}}
* David Ussishkin: "Archaeological Soundings at Betar, Bar-Kochba's Last Stronghold", in: ''Tel Aviv. Journal of the Institute of Archaeology of Tel Aviv University'' 20 (1993) 66ff.
* Yadin, Yigael. ''Bar-Kokhba: The Rediscovery of the Legendary Hero of the Second Jewish Revolt Against Rome''. New York: Random House, 1971 (hardcover, {{ISBN|0-394-47184-9}}); London: Weidenfeld and Nicolson, 1971 (hardcover, {{ISBN|0-297-00345-3}}).
* Mildenberg, Leo. ''The Coinage of the Bar Kokhba War''. Switzerland: Schweizerische Numismatische Gesellschaft, Zurich, 1984 (hardcover, {{ISBN|3-7941-2634-3}}).
{{refend}}

==External links==
* [https://www.livius.org/articles/concept/roman-jewish-wars/roman-jewish-wars-8/ Wars between the Jews and Romans: Simon ben Kosiba (130-136 CE)], with English translations of sources.
* [https://web.archive.org/web/20080117142713/http://www.yadinproductions.com/yadin_archeology.html Photographs from Yadin's book ''Bar Kokhba'']
* [https://www.haaretz.com/1.4895760 Archaeologists find tunnels from Jewish revolt against Romans] by the Associated Press. ''[[Haaretz]]'' March 13, 2006
* [http://jewishencyclopedia.com/view.jsp?artid=237&letter=B&search=Bar%20Kokba Bar Kokba and Bar Kokba War] ''Jewish Encyclopedia''
* [https://www.youtube.com/watch?v=r0qjv3nP3Ig Sam Aronow - The Bar Kochba Revolt | 132 - 136]

{{Tabernacle and Jerusalem Temples}}
{{Authority control}}

[[Category:Bar Kokhba revolt| ]]
[[Category:132]]
[[Category:133]]
[[Category:134]]
[[Category:135]]
[[Category:136]]
[[Category:130s in the Roman Empire]]
[[Category:130s conflicts]]
[[Category:2nd-century rebellions]]
[[Category:Genocide of indigenous peoples]]
[[Category:Jewish nationalism]]
[[Category:Jewish rebellions]]
[[Category:Jewish refugees]]
[[Category:Jews and Judaism in the Roman Empire]]
[[Category:Judea (Roman province)]]
[[Category:Religion-based wars]]
[[Category:Genocides in Asia]]
<references group="lower-alpha" />{{notelist}}

Talk:Four color theorem

2023-05-24T21:58:07Z

IntegralPython: /* Re: the recent added images of colorings of different countries */ new section

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== Simple Proof ==
{{hat|Discussion closed because it is not about improvements to the article based on reliably-published sources; see [[WP:TALK]]}}
On a square grid start with a single square. Add a layer of squares around it so that it becomes a 3x3 square. Each square you add will touch at most 3 other squares, so will only require at most 4 colors to map it. Add layer after layer to infinity, similarly the criterion for 4 colors is met.
Is this what was said to be the proof?[[User:GuildCompounder|GuildCompounder]] ([[User talk:GuildCompounder|talk]]) 03:14, 15 December 2020 (UTC)
: You are only colouring a particular map. Most maps don't consist of layers like this. [[User:McKay|McKay]] ([[User talk:McKay|talk]]) 04:42, 15 December 2020 (UTC)
:[[File:Fussball 1936.jpg|100px|right]] {{re|GuildCompounder}} Consider this ball. When you come to adding the last segment, it will touch five segments previously added. Does your method guarantee those five use no more than three colors? --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 15:47, 2 March 2021 (UTC)
::The four color theorem only applies to planar maps/graphs. It's well-known (and discussed in the article) that other topologies can need more colors. I don't see the relevance to this section. --[[User:Mfb|mfb]] ([[User talk:Mfb|talk]]) 04:18, 3 March 2021 (UTC)
:::Sphere is equivalent to plane, actually. But McKay's original reply is correct -- building from layers of squares doesn't deal with any of the interesting maps. (And of course the observation that it sat unsolved for a century, and then required computer assistance, should be a good indication that any "simple" proof attempt is quite likely to be wrong!) [[User:Joule36e5|Joule36e5]] ([[User talk:Joule36e5|talk]]) 05:47, 3 March 2021 (UTC)
:::[[File:Fussball 1936 map.png|100px|thumb|A map of the ball's surface]]
:::{{re|Mfb}} Yes, other topologies can need more colors. But this is the same topology. If you exclude any point of a sphere (which may be an interior point of any "country" region, hence meaningless in map coloring), then the rest of the sphere is[[homeomorphism| homeomorphic]] with a plane (see the [[Stereographic projection]] for an example of a continuous bijection between a punctured sphere and a plane), so any result of coloring a map on a sphere applies verbatim to a plane and ''vice versa''. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 10:46, 3 March 2021 (UTC)
:::{{re|Mfb}} I have added a planar map corresponding to the ball's surface structure. Hopefully it makes it clear how the 'planar' theorem applies to the ball. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 18:54, 3 March 2021 (UTC)
::::I still don't see any relevance to the original question in this section. --[[User:Mfb|mfb]] ([[User talk:Mfb|talk]]) 04:51, 4 March 2021 (UTC)
:::::So look closer. Can you apply the proposed algorithm to the ball? Can you color it the way described? If not, then the reasoning does not guarantee this particular map can be properly filled with four colors, hence it's not a proof of the theorem (as the theorem applies to ''all'' planar maps, to this particular one among them). --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 07:06, 4 March 2021 (UTC)
::::::Of course you cannot, but why use a ball? The first reply already pointed out that the algorithm only works for specific maps - without needing to introduce a ball. --[[User:Mfb|mfb]] ([[User talk:Mfb|talk]]) 10:48, 5 March 2021 (UTC)
:::::::Nope, the first reply pointed out that [[User:GuildCompounder]] '''applied''' the algorithm to '''a specific map''', but [[User:McKay]] did not prove "that the algorithm '''only works for specific maps'''", i.e. the algorithm can't be applied to other kinds of maps. And it actually '''can''', for example it works perfectly well also for [[hexagonal tiling]], and for [[rhombitrihexagonal tiling]], too. So I gave the ball as an example of another simple map, so OP can explain how their method applies to it, or try to strengthen their intended 'proof' by expanding the presented method so that it handles the ball, too. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 18:37, 6 March 2021 (UTC)
It recently occurred to me that layers can be added inwards instead of outwards. Start for example with a 9x9 layer, add a 7x7...3x3. All the squares in the layers touch at most 3 other squares making them 4 colorable. The exception is the 1x1 centre square which touches 4 squares. However, if opposite sides of the centre square touch each other, that would separate the other opposite sides of the centre square which could then be the same colour. That is why what works out for the 2 dimensional map does not work for the 3 dimensional map (which has no limit to the number of colours required).[[User:GuildCompounder|GuildCompounder]] ([[User talk:GuildCompounder|talk]]) 17:59, 13 March 2021 (UTC)
: Doesn't work, either. Assume two-'layers' honey-comb pattern. The central piece touches six outer pieces. No part of your proposal guarantees those six use three colors only. And if they use more, you can't color the central one. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 21:13, 13 March 2021 (UTC)
::It seems to me that if the squares are relatively infinitesimals, they can cover patterns like the honey comb.[[User:GuildCompounder|GuildCompounder]] ([[User talk:GuildCompounder|talk]]) 20:43, 15 March 2021 (UTC)
:While interesting, this discussion does not really conform to the purpose of a talk page, which is confined to discussions of how to improve the article. [[User:Paul August|Paul August]] [[User_talk:Paul August|☎]] 12:25, 14 March 2021 (UTC)
::Sorry but maybe there is some relevance? One more little thing...[[User:GuildCompounder|GuildCompounder]] ([[User talk:GuildCompounder|talk]]) 20:43, 15 March 2021 (UTC)
THe "opposites cutoff" theorem I mentioned above is valid for convex objects like spheres and cylinders, but is blown for toroids. We can visualize 6 colours for the toroid by applying a diagonal slash through the failed theorem rectangle. Then delete the original 5 colour region allowing a 2 colour loop to touch in 2 places when it only needs 1 connection. This allows a gap at the far end of the loop. Now the 3 five colour regions all touch each other, requiring 7 colours. [[User:GuildCompounder|GuildCompounder]] ([[User talk:GuildCompounder|talk]]) 20:43, 15 March 2021 (UTC)
{{hab}}

== Web Maps ==

Perhaps add an example, like "Bing Maps political layer" one day when there is finally something out there users can zoom into to their hometown to see. [[User:Jidanni|Jidanni]] ([[User talk:Jidanni|talk]]) 18:29, 14 December 2022 (UTC)

== flaw in colors of the map at the top of the four-color theorem article ==

In the picture at the top of the article, take the orang colored squarish item. to the upper left there is a blue section that is triangle ish shaped that should be yellow as it is now the blue is adjacent at a point to two other blue sections that are spikey. A point counts as they are touching. pls fix. [[Special:Contributions/73.180.167.41|73.180.167.41]] ([[User talk:73.180.167.41|talk]]) 12:26, 24 February 2023 (UTC)

:Adjacency at a single point does not count as adjacency for the purposes of this theorem. As the article already clearly states: "{{tq|regions are adjacent if they share a boundary segment; two regions that share only isolated boundary points are not considered adjacent.}}" —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 16:01, 24 February 2023 (UTC)

== Use "colour" not "color" when quoting British mathematicians ==

There are several quotations here from British mathematicians which are not spelled as they would have originally been made, and which I think should be emended for veracity. [[Special:Contributions/92.27.162.236|92.27.162.236]] ([[User talk:92.27.162.236|talk]]) 09:41, 16 May 2023 (UTC)

:Context: the above comment is an opening to the ''discuss'' step of [[WP:BRD]], following my revert [[Special:Diff/1154889964]] and my explanation at the IP editor's talk page [[User talk:92.27.162.236#A change of 'color' to 'colour' in Four color theorem]]. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 12:14, 16 May 2023 (UTC)
::There is no doubt that British mathematicians used the word "colour" in everything they wrote on the subject, not "color". Just because an American work is used as a source should not imply that the word was spelled "color" by any of the original British authors.
::There must be original British sources which should be quoted first hand. "Color" is factually incorrect, and should be changed. [[Special:Contributions/92.27.162.236|92.27.162.236]] ([[User talk:92.27.162.236|talk]]) 03:09, 24 May 2023 (UTC)
::Indeed, by looking at the facsimile of De Morgan's letter you will be able to read the words "coloured" and "colours" for yourselves. [[Special:Contributions/92.27.162.236|92.27.162.236]] ([[User talk:92.27.162.236|talk]]) 03:16, 24 May 2023 (UTC)
:::We should certainly use the original spelling, regardless of ENGVAR, as we do also for certain quotations using archaic spellings rather than attempting to modernize them. (Not that I think UK spelling is in any way archaic, but I think the precedent is similar.) —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:51, 24 May 2023 (UTC)

== Re: the recent added images of colorings of different countries ==

I agree with {{u|David Eppstein}} that adding an entire section as a gallery to simply present different countries with their colorings is a bit overkill (especially with the map of America's at the beginning). I was thinking, though, of re-adding at least one of them to the section "Use outside of mathematics," since it seems apt. I was originally just going to place the Germany one in, but maybe adding a picture of a real-life exclave forcing 5-colors would be nice. In any case, I don't think it hurts to add a second picture illustrating how it works, since non-math inclined people are probably going to be interested in (more than one) examples of these type. I didn't want to just make the edit immediately though since the similar one had just been reverted. What do other people think? '''[[User:IntegralPython| Integral Python]]''[[User talk:IntegralPython| click here to argue with me]]''''' 21:58, 24 May 2023 (UTC)

Peter Ozsváth

2023-02-08T04:51:37Z

IntegralPython: Added citation

{{short description|American mathematician}}
{{Infobox scientist
| name = Peter Ozsváth
| image = Peter Ozsváth.jpg
| image_size =
| caption = Peter Ozsváth in [[Berkeley, California|Berkeley]], 2005
| birth_date = October 20, 1967
| birth_place = [[Dallas]], [[Texas]]
| death_date =
| death_place =
| nationality = American
| fields = [[Mathematics]]
| workplaces = [[Princeton University]] [[Massachusetts Institute of Technology]] [[Columbia University]] [[Yale University]] [[University of California, Berkeley]]
| alma_mater = [[Princeton University]]
| doctoral_advisor = [[John Morgan (mathematician)|John Morgan]]
| doctoral_students = [[Elisenda Grigsby]]
| known_for =
| awards = [[Oswald Veblen Prize in Geometry]] {{small|(2007)}}{{br}}[[Guggenheim Fellow]] {{small|(2008)}}{{br}}Member of the [[National Academy of Sciences]] {{small|(2018)}}
}}
'''Peter Steven Ozsváth''' (born October 20, 1967) is a professor of mathematics at [[Princeton University]]. He created, along with [[Zoltán Szabó (mathematician)|Zoltán Szabó]], [[Heegaard Floer homology]], a [[homology theory]] for [[3-manifold]]s.

==Education==
Ozsváth received his Ph.D. from Princeton in 1994 under the supervision of [[John Morgan (mathematician)|John Morgan]]; his dissertation was entitled ''On Blowup Formulas For SU(2) [[Donaldson theory|Donaldson Polynomials]]''.

==Awards==

In 2007, Ozsváth was one of the recipients of the [[Oswald Veblen Prize in Geometry]].<ref>{{citation|url=https://www.ams.org/notices/200704/comm-veblen-web.pdf|title=2007 Veblen Prize|journal=[[Notices of the AMS]]|date=April 2007|volume=54|issue=4|pages=527–530}}.</ref> In 2008 he was named a [[Guggenheim Fellow]].<ref>{{cite web|url=http://www.gf.org:80/newfellow.html|archive-url=https://web.archive.org/web/20080920050536/http://www.gf.org/newfellow.html|url-status=dead|archive-date=20 September 2008|title=2008 Fellows|publisher=Guggenheim Foundation|accessdate=12 Jan 2019}} from Wayback Machine</ref> In July 2017, he was a plenary lecturer in the Mathematical Congress of the Americas.<ref>{{cite web|url=https://mca2017.org|title=Mathematical Congress of the Americas 2017}}</ref> He was elected a member of the [[National Academy of Sciences]] in 2018.{{citation needed|date=February 2021}}

==Selected publications==
*{{Cite journal |last1=Ozsváth |first1=Peter |last2=Szabó |first2=Zoltán |title=Holomorphic disks and topological invariants for closed three-manifolds |journal=[[Annals of Mathematics|Ann. of Math.]] |volume=159 |year=2004 |issue=3 |pages=1027–1158 |doi=10.4007/annals.2004.159.1027|arxiv=math/0101206 |s2cid=119143219 }}
*{{Cite journal |last1=Ozsváth |first1=Peter |last2=Szabó |first2=Zoltán |title=Holomorphic disks and three-manifold invariants: properties and applications |journal=Ann. of Math. |volume=159 |year=2004 |issue=3 |pages=1159–1245 |doi=10.4007/annals.2004.159.1159|doi-access=free }}
*[https://www.ams.org/bookstore-getitem/item=surv-208''Grid Homology for Knots and Links''], American Math Society, (2015)

==References==
{{Reflist}}

==External links==
*[https://web.archive.org/web/20121221003816/http://www.math.columbia.edu/~petero/ Personal homepage]
*{{MathGenealogy |id=18896 }}
{{Veblen Prize recipients}}
{{Authority control}}

{{DEFAULTSORT:Ozsvath, Peter}}
[[Category:Living people]]
[[Category:1967 births]]
[[Category:20th-century American mathematicians]]
[[Category:20th-century Hungarian mathematicians]]
[[Category:21st-century American mathematicians]]
[[Category:21st-century Hungarian mathematicians]]
[[Category:Princeton University faculty]]
[[Category:Columbia University faculty]]
[[Category:Topologists]]
[[Category:Mathematicians from Texas]]
[[Category:People from Dallas]]
[[Category:Princeton University alumni]]
[[Category:Members of the United States National Academy of Sciences]]

{{US-mathematician-stub}}

User:IntegralPython

2023-01-27T16:54:50Z

IntegralPython:

[[File:WikiProject Mathematics AD.gif|center]]
<table style="float: right; margin-left: 1em; margin-bottom: 0.5em; width: 250px; border: #99B3FF solid 1px">
<tr><td>{{Template:User WP Mathematics}}</td>
<td>{{Template:User interest mathematics}}</td></tr>
</table>

Hi! I'm a [[Christians|Christian]] [[Wikipedia]] browser and recreational mathematician. My interests mainly lie in [[math]] and [[Science]], particularly in [[fractal]] analysis, [[quaternion]]s, [[hyperoperation]]s, and [[quantum physics]]. If I do anything stupid, please leave a long and angry comment on my talk page; make sure to include as many strongly worded critiques of me and my poor intelligence. Thanks!

==My Articles==
*[[Quota rule]]
*[[Hand eye calibration problem]]
*[[Open set condition]]
===Other articles===
Articles I have spent a significant amount of effort on
*[[Genetic use restriction technology]]
*[[internet meme]]
*[[tetration]]

==Helpful links==
*[[User:IntegralPython/sandbox|My Sandbox]]
*[https://en.wikipedia.org/w/index.php?hidebots=1&hidecategorization=1&hideWikibase=1&tagfilter=coi-spam&limit=50&days=30&title=Special:RecentChanges&urlversion=2| conflict of interest pages]

===Math links===
*{{Random page in category|Mathematics|text=Random Math page}}
*{{Random page in category|Mathematics_stubs|text=Random Math stub}}

==Uploaded Pictures==
[[File:Koch Snowflake.svg|200px]]
[[File:GURT process diagram.png|200px]]
[[File:Approximations of 0.5 tetratrated to the x.png|200px]]
[[File:Open set condition.png|200px]]
[[File:Pentation.jpg|200px]]
[[File:Superpermutations.jpg|200px]]
[[File:Superpermutation distribution.png|200px]]
[[File:Kempe Chain.png|200px]]
[[File:Cube super root.png|200px]]

Scratch (programming language)

2022-10-12T17:01:11Z

IntegralPython: Copy/pasting a FAQ onto the page is not appropriate for wikipedia

{{Short description|Programming language learning environment}}
{{about|the programming language|other uses|Scratch (disambiguation)}}
{{primary sources|date=February 2022}}
{{Use dmy dates|date=February 2022}}

{{Infobox programming language
| logo = Scratchlogo.svg
| logo_alt = Scratch logo
| screenshot = Scratch 3.0 GUI.png
| screenshot caption = Scratch 3.0 editor
| paradigm = [[Event-driven programming|Event-driven]], [[Visual programming language|block-based]] programming language
| year = {{Start date and age|2003}} (first prototype) {{Start date and age|2004}} (second prototype) {{Start date and age|df=yes|15 May 2007}} (public launch)<ref>{{cite web|url=https://en.scratch-wiki.info/wiki/Scratch_Timeline#May|title=Scratch Timeline – Scratch Wiki|website=en.scratch-wiki.info}}</ref> {{Start date and age|df=yes|9 May 2013}} (Scratch 2.0) {{Start date and age|df=yes|2 January 2019}} (Scratch 3.0)|
| influenced by = [[Logo (programming language)|Logo]], [[Smalltalk]], [[HyperCard]], [[StarLogo]], [[AgentSheets]], [[AgentCubes]], [[Etoys (programming language)|Etoys]]
| influenced = [[Catrobat]],<ref>{{cite web|url=https://catrobat.org/|title=Catrobat Home|website=catrobat.org}}</ref>
[[ScratchJr]],<ref>{{cite web|url=https://scratchjr.org/|title=ScratchJr – Home|website=scratchjr.org}}</ref> [[Snap! (programming language)|Snap''!'']],<ref>{{cite web|url=https://snap.berkeley.edu/|title=Snap! Build Your Own Blocks|website=snap.berkeley.edu}}</ref> [[mBlock]], [[Turtlestitch]]
| programming language = [[Squeak]] (Scratch 0.x, 1.x) [[ActionScript]] (Scratch 2.0) [[HTML5]] and [[JavaScript]] (Scratch 3.0)<ref>{{cite web|url=https://scratch.mit.edu/discuss/post/3431979/|title=Converting Scratch Projects to HTML5 - Discuss Scratch|website=scratch.mit.edu|access-date=May 16, 2022}}</ref>
| operating system = [[Microsoft Windows]], [[macOS]], [[Linux]] (via renderer), [[HTML5]], [[iOS]], [[iPadOS]], and [[Android (operating system)|Android]].
| license = [[BSD licenses|BSD 3-Clause]], [[GNU General Public License|GPLv2]] and Scratch Source Code License
| file_ext = .scratch (Scratch 0.x) .sb, .sprite (Scratch 1.x) .sb2, .sprite2 (Scratch 2.0) .sb3, .sprite3 (Scratch 3.0)
| website = {{URL|https://scratch.mit.edu/}}
| latest_release_version = {{unbulleted list|Scratch 3.0 (online editor) / {{Start date and age|2019|01|02}}|Scratch 3.29.1 (offline editor) / {{Start date and age|2022|02|27}}}}
}}

'''Scratch''' is a [[High-level programming language|high-level]] block-based [[visual programming language]] and website aimed primarily at children as an educational tool for programming, with a target audience of ages 8 to 16 though many adults use it<ref name=":3" />.<ref name=":3">{{cite web |last=scratch |first=scratch |date=11 October 2022 |title=Scratch – About |url=https://scratch.mit.edu/about#:~:text=but%20is%20used%20by%20people%20of%20all%20ages. |url-status=live |website=scratch.mit.edu}}</ref> Users on the site, called Scratchers, can create projects on the website using a block-like interface. Projects can be exported to [[HTML5]], [[JavaScript]], [[Android (operating system)|Android]] [[Application software|apps]], [[.app|Bundle (macOS)]] and [[.exe|EXE]] files using external tools. The service is developed by the [[MIT Media Lab]], has been translated into 70+ languages, and is used in most parts of the world.<ref name="statistics">{{cite web |url=https://scratch.mit.edu/statistics/ |title=Community statistics at a glance |website=scratch.mit.edu |access-date=18 May 2019|archive-url=https://web.archive.org/web/20160406023520/https://scratch.mit.edu/statistics/ |archive-date=6 April 2016 |url-status=live }}</ref> Scratch is taught and used in after-school centers, schools, and colleges, as well as other public knowledge institutions. As of May 8, 2022, community statistics on the language's official website show more than 104 million projects shared by over 90 million users, over 686 million total projects ever created (including unshared projects), and more than 100 million monthly website visits.<ref name="statistics" />

Scratch takes its name from a technique used by [[disk jockey]]s called "[[scratching]]", where vinyl records are clipped together and manipulated on a turntable to produce different sound effects and music. Like scratching, the website lets users mix together different media (including graphics, sound, and other programs) in creative ways by creating and 'remixing' projects, like [[video games]], [[animations]], and [[simulations]].<ref name="ScholarWorks">{{cite journal |last1=Lamb |first1=Annette |last2=Johnson |first2=Larry |date=April 2011 |title=Scratch: Computer Programming for 21st Century Learners |url=https://scholarworks.iupui.edu/bitstream/handle/1805/8622/38-4.pdf?sequence=1 |format=PDF |journal=Teacher Librarian |volume=38 |issue=4 |pages=64–68 |access-date=18 May 2019}}</ref><ref>{{cite news |url=https://news.mit.edu/2007/resnick-scratch |title=Creating from Scratch |newspaper=MIT News |first=Stephanie |last=Schorow |date=14 May 2007 |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20181013032644/http://news.mit.edu/2007/resnick-scratch |archive-date=13 October 2018 |url-status=live }}</ref>

== Scratch 3.0 ==
=== User interface ===
The Scratch interface is divided into three main sections: a ''stage area'', ''block palette'', and a coding area to place and arrange the blocks into scripts that can be run by pressing the green flag or clicking on the code itself. Users may also create their own code blocks and they will appear in "My Blocks".

The ''stage area'' features the results (e.g., animations, [[turtle graphics]], either in a small or normal size, with a full-screen option also available) and all sprites thumbnails being listed in the bottom area. The stage uses x and y [[Coordinate system|coordinates]], with 0,0 being the stage center.<ref name="LearnToProgram" />

[[File:Dialogo scratch Best Friends 1.png|thumb|A program to change the background and make a character speak, when clicked]]
With a sprite selected at the bottom of the staging area, blocks of commands can be applied to it by dragging them from the block palette into the coding area. The Costumes tab allows users to change the look of the sprite with a [[Vector graphics|vector]] and [[bitmap]] editor in order to create various effects, including animation.<ref name="LearnToProgram" /> The Sounds tab allows attaching sounds and music to a sprite.<ref name="sciencebuddies.org" />

When creating sprites and also backgrounds, users can draw their own sprite manually,<ref name="LearnToProgram" /> choose a Sprite from the library, or upload an image.<ref name="sciencebuddies.org">{{cite web |title=Science Buddies: Scratch User Guide: Installing & Getting Started with Scratch |url=https://www.sciencebuddies.org/science-fair-projects/references/installing-getting-started-with-scratch |website=ScienceBuddies.org |access-date=2019-05-18 |archive-url=https://web.archive.org/web/20190518123252/https://www.sciencebuddies.org/science-fair-projects/references/installing-getting-started-with-scratch |archive-date=2019-05-18 |url-status=live }}</ref>

The table below shows the categories of the programming blocks:
{| class="wikitable" style="text-align: left"
|-
! colspan="2" style="background: #efefef;" | Category !! Notes !! rowspan="6" style="background:white;" | !! colspan="2" style="background: #efefef;" | Category !! Notes
|- valign="top"
| style="background:#4c97ff;"| || Motion || Movements of sprites like angles and directions. || style="background:#5cb1d6;" | || Sensing || Sprites can interact with other sprites, the mouse pointer, and the backdrop.
|- valign="top"
| style="background:#9966ff;"| || Looks || Controls the visuals of the sprite|| style="background:#52b55a;"| || Operators || Mathematical operators and comparisons
|- valign="top"
| style="background:#cf5acf;"| || Sound || Plays [[audio files]] and effects|| style="background:#ff8c1a;"| || Variables || Variables and Lists of usage and assignment. Variables can be put on the cloud (variables that can be accessed by all running versions of the project)
|- valign="top"
| style="background:#ffbf00;"| || Events || Event handlers and broadcasters. || style="background:#ff4d88;" | || My Blocks || Certain [[Subroutine|functions]] created by the user defined by other blocks in defining scripts.
Can have option to run without screen refresh.
|- valign="top"
| style="background:#ffab19;"| || Control || Conditionals, loops and cloning. || style="background:#0fbd8c;" | || Extensions || Explained [[#Extensions|below]]
|}

=== Offline editing ===
An offline "Desktop editor" for Scratch 3.0 is available for Microsoft Windows 10 in the Microsoft Store and Apple's macOS 10.13;<ref>{{cite web |title=Scratch Desktop |url=https://scratch.mit.edu/download |access-date=19 September 2019}}</ref> this allows the creation and playing of Scratch programs offline. The offline editor can also be downloaded in previous versions, such as Scratch 2.0 and Scratch 1.4.

=== Extensions ===
In Scratch, extensions add extra blocks and features that can be used in projects. In Scratch 2.0 and 3.0, the extensions were all hardware-based. Software-based extensions were added in Scratch 3.0, such as text-to-speech voices, along with some new hardware-based extensions like the [[Micro Bit|micro:bit]]. The extensions are listed below.

* Music
* Pen
* [[Video sensor technology|Video Sensing]]
* Text to Speech
* Translate
* [[BBC micro:bit|BBC Micro:bit]]
* [[Lego Mindstorms EV3|LEGO Mindstorms EV3]]
* LEGO WeDo 2.0
* Makey Makey
* LEGO SPIKE Prime
* LEGO BOOST
* Go Direct Force & Acceleration
*Text To Speech

==== Physical ====
* Lego Mindstorms EV3 – control motors and receive sensor data from the Lego Mindstorms EV3
* [[Makey Makey]] – use the Makey Makey to control projects
* [[LEGO|Lego]] Education WeDo 2.0 – control motors and receive sensor data from the Lego WeDo
*[[Lego]] Education SPIKE Prime—The main programming language for the Lego SPIKE Prime, including motor control and receiving sensor data
* BBC micro:bit – use of a BBC micro:bit to control projects
* Lego BOOST – bring robotic creations to life
* Go Direct Force & Acceleration – Sense pull, push, motion, and spin
* Text To Speech - Make your sprites talk

==== Digital ====
Many of the digital extensions in Scratch 3.0 used to be regular block categories that were moved to the extensions section to reduce clutter. These include:
* Music – Play digital instruments (drums, trumpets, violins, pianos, and more)
* Pen – Draw on the Stage with a variety of thicknesses and color
* Video Sensing – Detect motion with the camera

New digital extensions have also been added in collaborations with commercial companies. These include:
* Text to Speech – Converts words in a text into voice output (variety of voices, supplied by Amazon)
* Translate – Uses [[Google Translate]] to translate text from one language into a variety of other languages, including Arabic, Chinese, Dutch, English, French, Greek, Norwegian, and Japanese

=== Scratch Lab ===
The scratch lab has experimental extensions that they might add in the future, intended to explore whether the blocks may be added to the full website in the future. Experimental blocks include:
* Face Sensing – Make animated costumes and games that interact with your face.
* Animated text – Bring words to life with colors, fonts, and animations.
Users can also create their own extensions for Scratch 3.0 using [[JavaScript]].<ref>{{cite web |title=Scratch 3.0 Extensions |url=https://github.com/LLK/scratch-vm/blob/develop/docs/extensions.md |access-date=19 September 2019 |website=Github |publisher=MIT}}</ref>

=== Code base ===
Scratch 3.0 is a completely new JavaScript-based codebase made up of multiple components such as "Scratch-GUI,"<ref name="https://github.com/llk/scratch-gui">{{cite web |title=GitHub: Scratch-GUI |website=[[GitHub]] |url=https://github.com/llk/scratch-gui|url-status=live }}</ref> now based on a library from [[Blockly]],<ref>{{cite news |last1=Pasternak |first1=Erik |title=Scratch 3.0's new programming blocks, built on Blockly |url=https://developers.googleblog.com/2019/01/scratch-30s-new-programming-blocks.html |access-date=2 October 2019 |date=17 January 2019}}</ref> "Scratch-VM,"<ref name="https://github.com/llk/scratch-vm">{{cite web |title=GitHub: Scratch-VM |website=[[GitHub]] |url=https://github.com/llk/scratch-vm|url-status=live }}</ref> which interprets code, and "Scratch-Render,"<ref name="https://github.com/llk/scratch-render">{{cite web |title=GitHub: Scratch-Render |website=[[GitHub]] |url=https://github.com/llk/scratch-render|url-status=live }}</ref> the rendering engine.<ref>{{cite news |last1=Frang |first1=Corey |title=Porting Scratch from Flash to Javascript |url=https://bocoup.com/blog/porting-scratch-from-flash-to-javascript-performance-interoperability-and-extensions |access-date=21 September 2019 |date=28 February 2019}}</ref> The Scratch Blocks are made using Blockly.<ref>{{cite web|url=https://developers.google.com/blockly|title=Blockly|website=Google Developers}}</ref>

A paper published in 2019 by NYU argues and illustrates, for coding music with Scratch, "that the music and sound blocks as currently implemented in Scratch may limit and frustrate meaningful music-making for children, the core user base for Scratch."<ref>{{cite web|url=https://jitp.commons.gc.cuny.edu/music-making-in-scratch-high-floors-low-ceilings-and-narrow-walls/ |title=Music Making in Scratch: High Floors, Low Ceilings, and Narrow Walls? / |publisher=Jitp.commons.gc.cuny.edu |date= |accessdate=2022-02-27}}</ref>

== Community of users ==
[[File:Scratch (programming language) 2007.PNG|thumb|The Scratch website after the release of public project sharing in late 2007]]
Scratch is used in many different settings: schools, [[museum]]s, [[library|libraries]], [[Community centre|community centers]], and homes.<ref>{{cite news |url=https://www.ctvnews.ca/sci-tech/canadian-schools-starting-to-teach-computer-coding-to-kids-1.1799365 |title=Canadian schools starting to teach computer coding to kids |publisher=[[CTV.ca]] |access-date=2019-05-18 |date=2014-04-30 |first=Michael |last=Oliveira |archive-url=https://web.archive.org/web/20190518123251/https://www.ctvnews.ca/sci-tech/canadian-schools-starting-to-teach-computer-coding-to-kids-1.1799365 |archive-date=2019-05-18 |url-status=live }}</ref><ref>{{cite web |url=http://www.smm.org/ltc/scratchday |title=Scratch Day |website=Science Museum of Minnesota |access-date=18 May 2019 |url-status=dead |archive-url=https://web.archive.org/web/20130408060603/http://www.smm.org/ltc/scratchday |archive-date=2013-04-08}}</ref><ref name="ScholarWorks" /> Scratch is designed primarily for users aged 8–16, but it is used by all ages and has a sizeable adult userbase as of 2009.<ref name=":3" /><ref name="all">{{cite journal|last1=Resnick|first1=Mitchel|last2=Maloney|first2=John|last3=Hernández|first3=Andrés|last4=Rusk|first4=Natalie|author-link4=Natalie Rusk|last5=Eastmond|first5=Evelyn|last6=Brennan|first6=Karen|last7=Millner|first7=Amon|last8=Rosenbaum|first8=Eric|last9=Silver|first9=Jay|last10=Silverman|first10=Brian|last11=Kafai|first11=Yasmin|year=2009|title=Scratch: Programming for All|url=https://web.media.mit.edu/~mres/papers/Scratch-CACM-final.pdf|journal=[[Communications of the ACM]]|volume=52|issue=11|pages=60–67|doi=10.1145/1592761.1592779|s2cid=229934947 }}</ref> This wide outreach has created many surrounding communities, both physical and digital.<ref name="statistics" /> In April 2020, the Tiobe ranking of the world's programming languages included Scratch into the top 20. According to Tiobe, there are 50 million projects written in Scratch, and every month one million new projects are added.<ref>{{cite web|url=https://devclass.com/2020/04/06/kids-programming-language-scratch-nails-top-20-in-latest-dev-rankings/|title=Kids programming language Scratch nails top 20 in latest dev rankings • DEVCLASS|last=Fay|first=Joe|date=6 April 2020|website=DEVCLASS|language=en-GB|access-date=2020-04-27}}</ref>

=== Educational users ===
Scratch is used as the introductory language because the creation of interesting programs is relatively easy, and skills learned can be applied to other programming languages such as [[Python (programming language)|Python]] and [[Java (programming language)|Java]].
[[File:Comparison of Scratch 1.4 and Scratch 2.png|thumb|Comparison of Scratch 1.4 and Scratch 2.0]]

Scratch is not exclusively for creating games. With the provided visuals, programmers can create animations, text, stories, music, and more. There are already many programs that students can use to learn topics in math, history, and even photography. Scratch allows teachers to create conceptual and visual lessons and science lab assignments with animations that help visualize difficult concepts. Within the social sciences, instructors can create quizzes, games, and tutorials with interactive elements. Using Scratch allows young people to understand the logic of programming and how to creatively build and collaborate.<ref>{{cite web |url=https://www.avinteractive.com/features/blogs/scratch-av-25-06-2015/ |title=What is Scratch? Is it AV or IT? |date=25 June 2015 |access-date=18 May 2019 |website=AV Magazine |last=Martin |first=Neil |archive-url=https://web.archive.org/web/20190518123255/https://www.avinteractive.com/features/blogs/scratch-av-25-06-2015/ |archive-date=18 May 2019 |url-status=live }}</ref>

Scratch is taught to more than 800 schools and 70 colleges of [[D.A.V. College Managing Committee|DAV organization]] in India and across the world.<ref>{{cite web |title=DAV CS Syllabus |url=http://davnewpanvel.com/File/5651/Syllabus%20Std-VII%202018-19.pdf |access-date=2019-05-18 |url-status=dead |archive-url=https://web.archive.org/web/20180713122300/http://davnewpanvel.com/File/5651/Syllabus%20Std-VII%202018-19.pdf |archive-date=2018-07-13}}</ref><ref>{{cite web |title=DAV Jharkhand Syllabus |url=https://drive.google.com/file/d/0BzYkgDPSlegKbFFncEdwV3czVU0/view |access-date=18 May 2019}}</ref>

In higher education, Scratch is used in the first week of Harvard University's [[CS50]] introductory computer science course.<ref>{{cite news |title=Fun, Not Fear, Is at the Heart of Scratch, a New Programming Language |url=https://www.chronicle.com/article/Fun-Not-Fear-Is-at-the-Heart/34008/ |newspaper=The Chronicle of Higher Education |date=20 July 2007 |access-date=18 May 2019 |issn=0009-5982 |first=Jeffrey R. |last=Young |archive-url=https://web.archive.org/web/20190518123249/https://www.chronicle.com/article/Fun-Not-Fear-Is-at-the-Heart/34008/ |archive-date=18 May 2019 |url-status=live }}</ref><ref>{{cite web |title=CS50 Syllabus |url=https://cdn.cs50.net/2015/x/references/syllabus/syllabus.html |access-date=2019-05-18 |archive-url=https://web.archive.org/web/20150317075307/http://cdn.cs50.net/2015/x/references/syllabus/syllabus.html |archive-date=17 March 2015 |url-status=live }}</ref>

=== Online community ===
[[File:Jumper platformer.png|thumb|"Jumper", an example of a game created with Scratch 2.0]]
[[File:Abyss scratch 3.0.png|thumb|"Abyss", an example of a game created with Scratch 3.0]]
Users of Scratch are called 'Scratchers'. Scratchers have the capability to share their projects and get feedback. Projects can be uploaded directly from the development environment to the Scratch website and any member of the community can download the full source code to study or to remix into new projects.<ref>{{cite book |title=Proceedings of the 29th International Conference on Human Factors in Computing Systems (CHI '11) |last1=Monroy-Hernandez |first1=Andres |last2=Hill |first2=Benjamin Mako |last3=Gonzalez-Rivero |first3=Jazmin |last4=Boyd |first4=Danah |publisher=ACM |year=2011 |pages=3421–30 |chapter=Computers Can't Give Credit: How Automatic Attribution Falls Short in an Online Remixing Community |doi=10.1145/1978942.1979452 |arxiv=1507.01285|s2cid=7494330 }}</ref><ref>{{cite book |title=ICWSM 2010 : Proceedings of the 4th International Conference on Weblogs and Social Media, May 23–26, 2010 |last1=Hill |first1=B.M |last2=Monroy-Hernández |first2=A. |last3=Olson |first3=K.R. |publisher=AAAI Press |year=2010 |isbn=978-1-57735-445-1 |location=Washington, D.C. |chapter=Responses to remixing on a social media sharing website |oclc=844857775 |arxiv=1507.01284 |bibcode=2015arXiv150701284M}}</ref> Scratchers can also create project studios, comment, favorite, and "love" others' projects, follow other members to see their projects and activity, and share ideas. Projects range from games and animations to practical tools. Additionally, to encourage creation and sharing amongst users, the website frequently establishes "Scratch Design Studio" challenges.<ref>{{cite web |url=https://en.scratch-wiki.info/wiki/Scratch_Design_Studio |title=Scratch Design Studio |website=wiki.scratch.mit.edu |access-date=2019-05-18 |archive-url=https://web.archive.org/web/20190518123252/https://en.scratch-wiki.info/wiki/Scratch_Design_Studio |archive-date=18 May 2019 |url-status=live }}</ref>

The MIT Scratch Team works to ensure that this community maintains a friendly and respectful environment for all people.<ref name="parents">{{cite web |url=https://scratch.mit.edu/parents/ |title=For Parents |website=scratch.mit.edu |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190404210527/https://scratch.mit.edu/parents |archive-date=4 April 2019 |url-status=live }}</ref><ref>{{cite web |url=https://scratch.mit.edu/community_guidelines |title=Scratch Community Guidelines |website=scratch.mit.edu |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190407171658/https://scratch.mit.edu/community_guidelines/ |archive-date=7 April 2019 |url-status=live }}</ref>

Educators have their own online community called ScratchEd, developed and supported by the Harvard Graduate School of Education. In this community, Scratch educators share stories, exchange resources, and ask questions.<ref>{{cite web |url=https://scratch.mit.edu/educators |title=Scratch for Educators |website=scratch.mit.edu |access-date=18 May 2010 |archive-url=https://web.archive.org/web/20081005234300/http://scratch.mit.edu/educators |archive-date=5 October 2008 |url-status=live }}</ref>

=== Scratch Wiki ===
The Scratch Wiki is a support resource for Scratch and its website, history, and phenomena surrounding it. Although supported by the Scratch Team (developers of Scratch), it is primarily written by Scratchers (users of Scratch) for information regarding the program and website.<ref name=":1">{{cite web |url=https://en.scratch-wiki.info/wiki/Scratch_Wiki |title=Scratch Wiki |website=en.scratch-wiki.info |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190512210211/https://en.scratch-wiki.info/wiki/Scratch_Wiki |archive-date=12 May 2019 |url-status=live }}</ref>

====April Fools' Day====
Every year on [[April Fools' Day]] since 2014, the wiki shows joke versions of many pages instead of the real one, even though the real version of the page can still be accessed at the bottom of the prank version. Although it's not the 1st of April, you can still access the joke version of the pages [https://en.scratch-wiki.info/wiki/Scratch_Wiki:April_Fools here]. Additionally, a semicolon has also appeared on the bottom of every page on the wiki on April Fools' Day since 2019.<ref>{{Cite web |title=April Fools Day - Scratch Wiki |url=https://en.scratch-wiki.info/wiki/April_Fools%27_Day#Wiki_.28since_2014.29 |access-date=2022-05-07 |website=en.scratch-wiki.info}}</ref>

=== Developers ===
Both Scratch 2.0 and Scratch 3.0's GUIs are open source on GitHub,<ref>{{cite web|url=https://github.com/LLK/scratch-gui|title=LLK/scratch-gui|date=9 January 2021|via=GitHub}}</ref> and developers may contribute to Scratch.<ref>{{cite web|url=https://scratch.mit.edu/|title=Scratch – Developers|website=scratch.mit.edu}}</ref>

=== Roles ===
There are 3 roles. The "'''New Scratcher'''" is obtained by creating an account. As they continue on loving;favoriting;sharing stuff, they will be asked if they want the "'''Scratcher'''" role. The "'''Scratch Team'''" role is only given out to people in the scratch team. Scratch team roles will have a "'''*'''" in front of their name. An example of a user with the scratch team role can be found [https://scratch.mit.edu/users/ceebee/ here].

== Events ==
Scratch Educators can gather in person at Scratch Educator Meetups. At these gatherings, Scratch Educators learn from each other and share ideas and strategies that support computational creativity.<ref>{{cite web |url=https://www.meetup.com/pro/scratched/ |title=Scratch Educator |website=Meetup.com |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190421144408/https://www.meetup.com/pro/scratched |archive-date=21 April 2019 |url-status=live }}</ref>

An annual "Scratch Week", formerly known as "Scratch Day", is declared in May each year. Community members are encouraged to host an event on or around this day, large or small, that celebrates Scratch. These events are held worldwide, and a listing can be found on the Scratch Day website. Scratch Week is a series focusing on Scratch activities on the Scratch website.<ref>{{cite web |url=https://day.scratch.mit.edu/ |title=Scratch Week|access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190407171614/https://day.scratch.mit.edu/ |archive-date=7 April 2019 |url-status=live }}</ref>

Every [[April Fools' Day]], the Scratch Team will play pranks on users and add Easter eggs such as turning all the event blocks into cat versions of the same blocks.

== History ==
In the early 2000s, the [[MIT Media Lab]]'s ''Lifelong Kindergarten'' group (LLK) was developing visual programming languages targeted towards children.<ref>{{cite web|title=LLK – Projects – Building-Block Programming|url=http://llk.media.mit.edu:80/projects/summaries/bbp.shtml|access-date=19 December 2021|website=llk.media.mit.edu|archive-url=https://web.archive.org/web/20010430112136/http://llk.media.mit.edu:80/projects/summaries/bbp.shtml|archive-date=30 April 2001}}</ref> In 2003, [[Mitchel Resnick]], [[Yasmin Kafai]], and [[John Maeda]] were awarded a [[National Science Foundation]] grant for the development of a new programming environment for children to express themselves with code.<ref name=":0">{{cite web|title=NSF Award Search: Award # 0325828 – ITR: A Networked, Media-Rich Programming Environment to Enhance Informal Learning and Technological Fluency at Community Technology Centers|url=https://www.nsf.gov/awardsearch/showAward?AWD_ID=0325828|access-date=15 April 2021|website=www.nsf.gov}}</ref> The LLK, led by Mitchel Resnick, in partnership with Yasmin Kafai's team at [[University of California, Los Angeles|UCLA]] worked closely with [[The Clubhouse Network|Computer Clubhouses]] in Boston and Los Angeles to develop Scratch, grounding its design in the practices and social dynamics of these after-school youth centers.<ref name=":0" /> It started as a basic programming language, with no labeled categories and no green flag.<ref name="Dev1.0">{{cite web |url=https://en.scratch-wiki.info/wiki/Development_of_Scratch_1.0 |title=Development of Scratch 1.0 |website=en.scratch-wiki.info |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190103004838/https://en.scratch-wiki.info/wiki/Development_of_Scratch_1.0 |archive-date=3 January 2019 |url-status=live }}</ref> Similar to AgentSheets<ref name="AgentSheets">{{cite web |url=https://www.researchgate.net/publication/3651308 |title=Tactile Programming: A Unified Manipulation Paradigm Supporting Program Comprehension, Composition and Sharing|access-date=15 October 2021}}</ref> Scratch employed concepts of Tactile Programming later known as blocks-based programming. Scratch was made with the intention to teach kids to program.<ref name="Dev1.0" />

The philosophy of Scratch encourages the sharing, reuse, and combination of code, as indicated by the team slogan, "Imagine, Program, Share".<ref>{{cite web |url=https://scratch.mit.edu/ |title=Scratch – Imagine, Program, Share |website=scratch.mit.edu |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20110222201702/http://scratch.mit.edu/ |archive-date=22 February 2011 |url-status=live }}</ref> Users can make their own projects, or they may choose to "[[remix]]" someone else's project. Projects created and remixed with Scratch are licensed under the [[Creative Commons Attribution-Share Alike License]].<ref>{{cite web |url=https://en.scratch-wiki.info/wiki/Creative_Commons_License |title=Creative Commons License |website=wiki.scratch.mit.edu |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123252/https://en.scratch-wiki.info/wiki/Creative_Commons_License |archive-date=18 May 2019 |url-status=live }}</ref> Scratch automatically gives credit to the user who created the original project and program in the top part of the project page.<ref name="ScholarWorks" />

Scratch was developed based on ongoing interaction with youth and staff at Computer Clubhouses. The use of Scratch at Computer Clubhouses served as a model for other after-school centers demonstrating how informal learning settings can support the development of technological fluency.<ref>{{cite web |title=ITR: A Networked, Media-Rich Programming Environment to Enhance Informal Learning and Technological Fluency at Community Technology Centers |url=https://www.nsf.gov/awardsearch/showAward?AWD_ID=0325828 |website=National Science Foundation |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20151230194131/https://www.nsf.gov/awardsearch/showAward?AWD_ID=0325828 |archive-date=30 December 2015 |url-status=live }}</ref>

[[File:Scratch.mit.edu Homepage.JPG|thumb|The 2.0 Scratch Homepage, before Scratch 3.0 comes out]]
Scratch 2.0 was released on 9 May 2013.<ref name="LearnToProgram">{{cite book |title=Learn to Program with Scratch |last=Marji |first=Majed |publisher=No Starch Press |year=2014 |isbn=978-1-59327-543-3 |location=San Francisco, California |pages=xvii, 1–9, 13–15}}</ref> The update changed the look of the site and included both an online project editor and an offline editor.<ref>{{cite web |url=https://scratch.mit.edu/download |title=Scratch Desktop |website=scratch.mit.edu |access-date=2019-05-18 |archive-url=https://web.archive.org/web/20190406112722/https://scratch.mit.edu/download |archive-date=6 April 2019 |url-status=live }}</ref> Custom blocks could now be defined within projects, along with several other improvements.<ref>{{cite web |url=https://techcrunch.com/2013/05/10/kids-programming-tool-scratch-now-runs-in-the-browser/ |title=Kids' Programming Tool Scratch Now Runs in the Browser |first=John |last=Biggs |website=[[TechCrunch]] |date=10 May 2013 |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20170709215339/https://techcrunch.com/2013/05/10/kids-programming-tool-scratch-now-runs-in-the-browser/ |archive-date=9 July 2012 |url-status=live }}</ref> The Scratch 2.0 Offline editor could be downloaded for Windows, Mac and Linux directly from Scratch's website, although support for Linux was later dropped. The unofficial [[mobile phone|mobile]] version had to be downloaded from the Scratch forums.<ref>{{cite web |url=https://scratch.mit.edu/discuss/topic/14690/ |title=Updated Scratch 2.0 Offline (Beta) is now available! |website=Scratch |date=2013-08-29 |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123257/https://scratch.mit.edu/discuss/topic/14690/ |archive-date=18 May 2019 |url-status=live }}</ref><ref>{{cite web |url=https://www.youtube.com/watch?v=qDFY4O2JU9U |title=Scratch 2.0 Preview |website=YouTube |date=2013-05-01 |publisher=MITScratchTeam |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20140124200207/http://www.youtube.com/watch?v=qDFY4O2JU9U |archive-date=24 January 2014 |url-status=live }}</ref>

[[File:Scratch MIT Homepage.png|thumb|The 3.0 Scratch Homepage]]
Scratch 3.0 was first announced by the Scratch Team in 2016. Several public alpha versions were released between then and January 2018, after which the pre-beta "Preview" versions were released.<ref>{{cite web |url=https://en.scratch-wiki.info/wiki/Scratch_3.0 |title=Scratch 3.0 |website=en.scratch-wiki.info |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190509215315/https://en.scratch-wiki.info/wiki/Scratch_3.0 |archive-date=2019-05-09 |url-status=live }}</ref> A beta version of Scratch 3.0 was released on 1 August 2018.<ref>{{cite web |url=https://medium.com/scratchteam-blog/3-things-to-know-about-scratch-3-0-18ee2f564278 |title=3 Things To Know About Scratch 3.0 |website=Medium.com |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190512230529/https://medium.com/scratchteam-blog/3-things-to-know-about-scratch-3-0-18ee2f564278 |archive-date=12 May 2019 |url-status=live }}</ref> for use on most browsers; with the notable exception of [[Internet Explorer]].<ref name="3.0 FAQ">{{cite web |url=https://scratch.mit.edu/info/faq#scratch3 |title=Scratch 3.0 |website=scratch.mit.edu |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190406112722/https://scratch.mit.edu/info/faq/#scratch3 |archive-date=6 April 2019 |url-status=live }}</ref>

Scratch 3.0, the first 3.x release version, was released on 2 January 2019.<ref>{{cite web|title=Scratch 3.0 – Scratch Wiki|url=https://en.scratch-wiki.info/wiki/Scratch_3.0|access-date=27 April 2021|website=en.scratch-wiki.info}}</ref>

== Filetypes ==
In Scratch 1.4, an *{{Not a typo|.sb}} file was the file format used to store projects.<ref>{{cite web|url = http://en.scratch-wiki.info/wiki/.sb|title = Scratch Wiki – *.sb|date = 4 October 2015|access-date = 7 November 2015}}</ref>

An *{{Not a typo|.sb}} file is divided into four sections:
* "header", this 10-byte header contains the ASCII string 'ScratchV02' in versions higher than Scratch 1.2, and 'ScratchV01' in Scratch 1.2 and below
* "infoSize", encodes the length of the project's infoObjects. A 4-byte long, 32-bit, [[Endianness|big-Endian]] integer.
* "infoObjects", a dictionary-format data section. It contains: "thumbnail", a thumbnail of the project's stage; "author", the username of the project's creator; "comment", the Project Notes; "history", the save and upload log; "scratch-version", the version of Scratch used to save the file;
* "contents", an object table with the Stage as the root.<ref>{{cite web|title=Scratch File Format (1.4)/Object Table – Scratch Wiki|url=https://en.scratch-wiki.info/wiki/Scratch_File_Format/Object_Table|access-date=2022-02-19|website=en.scratch-wiki.info}}</ref> All objects in the program are stored here as references.<ref>{{cite web|title=Scratch File Format (1.4)/Object Table – Scratch Wiki|url=https://en.scratch-wiki.info/wiki/Scratch_File_Format/References|access-date=2022-02-19|website=en.scratch-wiki.info}}</ref>

Scratch 2.0 uses the *{{Not a typo|.sb2}} file format. These are zip files containing a [[JSON|.json]] file as well as the contents of the Scratch project including sounds (stored as {{Not a typo|.wav}}) and images (stored as {{Not a typo|.png}}).<ref>{{cite web |title=Scratch File Format (2.0) |url=https://en.scratch-wiki.info/wiki/Scratch_File_Format_(2.0) |website=Scratch Wiki |access-date=2 October 2019}}</ref> Each filetype, excluding the {{Not a typo|project.json}}, is stored as a number, starting at 0 and counting up with each additional file. The image file labeled '{{Not a typo|0.png}}' is always a 480x360 white image, but '{{Not a typo|0.wav}}' will still be the earliest non-deleted file.

The ScratchX experimental version of Scratch used the {{Not a typo|.sbx}} file format.<ref>{{cite web|url=https://github.com/LLK/scratchx|title=LLK/scratchx|website=GitHub}}</ref>

Scratch 3.0 uses the *{{Not a typo|.sb3}} format, which is very similar to *{{Not a typo|.sb2}}, one difference being the sound.<ref>{{cite web |title=Scratch File Format |url=https://en.scratch-wiki.info/wiki/Scratch_File_Format |website=Scratch Wiki |access-date=2 October 2019}}</ref>

== Older versions ==
[[File:Scratch 2.0 Default screen.png|thumb|Scratch 2.0 development environment and its different areas at startup]]

Although the main Scratch website now runs only the current version (Scratch 3.0), the offline editors for Scratch 2.0 (and the earlier Scratch 1.4) are still available for download<ref>{{cite web |title=Scratch 2.0 Offline Editor |url=https://scratch.mit.edu/download/scratch2 |publisher=MIT |access-date=21 September 2019}}</ref> and can be used to create and run games locally.
<ref>{{cite web |title=3 Things To Know About Scratch 3.0 |date=31 January 2019 |url=https://medium.com/scratchteam-blog/3-things-to-know-about-scratch-3-0-18ee2f564278 |publisher=The Scratch Team |access-date=21 September 2019}}</ref> It is still possible to upload projects from the Scratch 2.0 launcher, which are immediately converted into Scratch 3.0 when uploaded to the main site.<ref>{{cite web|title=Offline Editor (2.0) – Scratch Wiki|url=https://en.scratch-wiki.info/wiki/Offline_Editor_(2.0)#Uploading|access-date=2021-04-27|website=en.scratch-wiki.info}}</ref> There is also an offline version of Scratch 3.0.

=== Technology ===
The editor of Scratch 1.4 and below was written in [[Squeak]], while its online project viewer was written in [[Java (programming language)|Java]], and a player written in [[Adobe Flash]] was later added.<ref name="Squeak">{{cite web |title=Scratch |url=https://wiki.squeak.org/squeak/5833 |website=Squeak/Smalltalk |access-date=7 March 2021}}</ref><ref>{{cite web |url=https://scratcharchive.asun.co/forums/viewtopic.php?id=57148 |website=Scratch Archived Forums |access-date=7 March 2021|title=Scratch Forums / Beta Flash player }}</ref> Scratch 2.0 relied on Adobe Flash for the online version, and [[Adobe AIR]] for the offline editor. These have fallen out of favor, and Adobe has dropped support for them at the end of 2020.<ref>{{cite news |last1=O'Donnell |first1=Lindsey |title=Mozilla Kills Default Support for Adobe Flash in Firefox 69 |url=https://threatpost.com/flash-default-mozilla-firefox-69/140814/ |access-date=21 September 2019 |date=14 January 2019}}</ref><ref>{{cite news |last1=Adobe Corporate Communications |title=The Future of Adobe AIR |url=https://theblog.adobe.com/the-future-of-adobe-air/ |access-date=21 September 2019 |date=30 May 2019}}</ref>

=== Interface ===
[[File:Scratch Hello World.png|thumb|A script that lets the sprite say [[Hello, World!]] then stops the script in Scratch 2.0]]
In Scratch 2.0, the stage area is on the left side, with the programming blocks palette in the middle the coding area on the right. Extensions are in the "More blocks" section of the palette.<ref name=all />

The blocks palette in Scratch 2.0 is made of discrete sections that are not scrollable from one to the next; the table below shows the different sections:
{| class="wikitable" style="text-align: left"
|-
! colspan="2" style="background: #efefef;" | Category !! Notes !! style="background:white;" | !! colspan="2" style="background: #efefef;" | Category !! Notes
|- valign="top"
| style="background:#4a6cd4;"| || Motion || Moves and changes position of sprites || rowspan="5" style="background:white;"| || style="background:#c88340;"| || Events || Event handlers
|- valign="top"
| style="background:#8a55d7;"| || Looks || Controls the visuals of the sprite || style="background:#e1a91a;"| || Control || Conditionals and loops
|- valign="top"
| style="background:#bb42c3;"| || Sound || [[Audio files]], sequences || style="background:#2ca5e2;"| || Sensing || Sprite interaction
|- valign="top"
| style="background:#0e9a6c;"| || Pen || Draw on the canvas || style="background:#5cb712;"| || Operators || Mathematical operators
|- valign="top"
| style="background:#ee7d16;"| || Data || Variables and arrays || style="background:#632d99;"| || More Blocks || Functions, return value is always <code>void</code>
|}

=== 1.4 sounds ===
Scratch 2.0 changed how sounds were imported, so many Scratch 1.4 sounds stopped working. (The project file was changed from *{{Not a typo|.sb}} to *{{Not a typo|.sb2}}).

== Extensions ==
[[File:Scratch Extensions Page.jpeg|thumb|389x389px|An example of the Scratch 3 Extensions Page.]]
In Scratch 2.0, extensions were all hardware-based.

=== Features and derivatives ===
Scratch uses [[event-driven programming]] with multiple active objects called ''[[Sprite (computer graphics)|sprites]]''.<ref name="LearnToProgram" /> Sprites can be drawn, as [[vector graphics|vector]] or [[bitmap]] graphics, from scratch in a simple editor that is part of Scratch, or can be imported from external sources. Scratch 3.0 only supports one-dimensional [[array data structure|arrays]], known as "lists", and floating-point [[scalar (computing)|scalars]] and [[string (computer science)|strings]] are supported, but with limited string manipulation ability. There is a strong contrast between the powerful multimedia functions and multi-threaded programming style and the rather limited scope of the Scratch programming language.

Scratch 2.0 does not treat procedures as [[First-class object|first class structures]] and has limited [[input/output|file I/O]] options with Scratch 2.0 Extension Protocol, an experimental extension feature that allows interaction between Scratch 2.0 and other programs.<ref>{{cite web |title=Scratch Extension |publisher=MIT |url=https://en.scratch-wiki.info/wiki/Scratch_Extension |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123255/https://en.scratch-wiki.info/wiki/Scratch_Extension |archive-date=18 May 2019 |url-status=live }}</ref> The Extension protocol allows interfacing with hardware boards such as [[Lego Mindstorms]]<ref>{{cite web |title=EV3+Scratch Extension |work=Scratch extension GitHub |publisher=Code & Circuit |url=https://kaspesla.github.io/ev3_scratch/ |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20160120204258/http://kaspesla.github.io/ev3_scratch/ |archive-date=20 January 2016 |url-status=live }}</ref> or [[Arduino]].<ref>{{cite web |title=Preliminary Scratch extension for talking to Arduino boards running Firmata |work=Scratch extension GitHub |publisher=Damellis |url=https://github.com/damellis/A4S |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20180116143517/https://github.com/damellis/A4S |archive-date=16 January 2018 |url-status=live }}</ref> Scratch 2.0 was implemented in [[ActionScript]], with an experimental JavaScript-based interpreter being developed in parallel.<ref>{{cite web |url=https://scratch.mit.edu/discuss/topic/19132/ |title=We're seeking contributors to help finish our HTML5 Scratch player (now open sourced!) |website=Scratch |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123251/https://scratch.mit.edu/discuss/topic/19132/ |archive-date=18 May 2019 |url-status=live }}</ref>

Scratch 1.4 was based on Squeak, which is based on [[Smalltalk]]-80.<ref name="Squeak" /> A number of Scratch derivatives<ref>{{cite web |title=Scratch Modification |website=Scratch Wiki |publisher=Lifelong Kindergarten Group at the MIT Media Lab |url=https://en.scratch-wiki.info/wiki/Scratch_Modification |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123253/https://en.scratch-wiki.info/wiki/Scratch_Modification |archive-date=18 May 2019 |url-status=live }}</ref> called Scratch Modifications have been created using the source code of Scratch 1.4. These programs are a variant of Scratch that normally include a few extra blocks or changes to the [[Graphical user interface|GUI]].<ref>{{cite web |title=Blocks |website=Scratch Wiki |url=https://en.scratch-wiki.info/wiki/Blocks |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123256/https://en.scratch-wiki.info/wiki/Blocks |archive-date=18 May 2019 |url-status=live }}</ref> TurboWarp is an unofficial modification of Scratch 3.0 that improves the performance of Scratch by compiling projects to JavaScript and allows the loading of external extensions.<ref>{{Cite web |title=TurboWarp - Run Scratch projects faster |url=https://turbowarp.org/ |access-date=2022-04-17 |website=turbowarp.org}}</ref>

==== Snap''!'' (Build Your Own Blocks) ====
A more advanced visual programming language inspired by Scratch is [[Snap! (programming language)|Snap''!'']], featuring [[first class function|first class]] procedures (their mathematical foundations are called also ''[[lambda calculus]]''), first class lists (including lists of lists), and first class truly object oriented sprites with prototyping inheritance, and nestable sprites, which are not part of Scratch.<ref>{{cite web |title=Snap''!'' – Build Your Own Blocks |publisher=University of California, Berkeley |url=https://snap.berkeley.edu/ |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190516160757/https://snap.berkeley.edu/ |archive-date=16 May 2019 |url-status=live }}</ref> Snap''!'' (previously "BYOB") was developed by Jens Mönig<ref>{{cite web |url=https://scratch.mit.edu/users/Jens/ |title=Jens on Scratch |website=Scratch |first=Jens |last=Mönig |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123251/https://scratch.mit.edu/users/Jens/ |archive-date=18 May 2019 |url-status=live }}</ref><ref>{{cite web |url=http://www.chirp.scratchr.org/blog/?m=201105 |title=BYOB 3.1 – Prototypal Inheritance for Scratch |date=31 May 2011 |website=Chirp Blog |first=Jens |last=Mönig |access-date=18 May 2019 |url-status=dead |archive-url=https://web.archive.org/web/20131206131246/http://www.chirp.scratchr.org/blog/?m=201105 |archive-date=6 December 2013}}</ref> with documentation provided by [[Brian Harvey (lecturer)|Brian Harvey]]<ref>{{cite web |url=https://people.eecs.berkeley.edu/~bh/ |title=Brian Harvey |website=Electrical Engineering and Computer Sciences |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190403045124/https://people.eecs.berkeley.edu/~bh/ |archive-date=3 April 2019 |url-status=live }}</ref><ref>{{cite web |url=https://scratch.mit.edu/users/bharvey/ |title=bharvey |website=Scratch |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20190518123251/https://scratch.mit.edu/users/bharvey/ |archive-date=18 May 2019 |url-status=live }}</ref> from [[University of California, Berkeley]] and has been used to teach "The Beauty and Joy of Computing" introductory course in CS for non-CS-major students.<ref>{{cite web |url=http://inst.eecs.berkeley.edu/~cs10/fa11/ |title=CS10 : The Beauty and Joy of Computing |website=EECS Instructional Support Group Home Page |access-date=18 May 2019 |archive-url=https://web.archive.org/web/20140123111318/http://inst.eecs.berkeley.edu/~cs10/fa11/ |archive-date=23 January 2014 |url-status=live }}</ref> Both of them were members of the Scratch Team before designing "Snap''!''".<ref>{{cite web|url=https://forum.snap.berkeley.edu/t/relationship-with-the-scratch-team/1277|title=Relationship With the Scratch Team}}</ref><ref name=all />

==== ScratchJr ====
In July 2014, [[ScratchJr]] was released for [[iPad]], and in 2016, ScratchJr for Android. Although heavily inspired by Scratch and co-led by Mitch Resnick, it is nonetheless a complete rewrite designed for younger children—targeting ages 5 through 8.<ref>{{cite web |title=About ScratchJr |url=https://www.scratchjr.org/about/info |publisher=scratchjr.org |access-date=19 September 2019}}</ref>

== Censorship ==
In August 2020, [[GreatFire]] announced that the [[Internet censorship in China|Chinese government had blocked access]] to the Scratch website. At the time, it was estimated that more than three million people in China were using it.<ref name=":2" /><ref>{{cite web|date=7 September 2020|title=China appears to be blocking access to children's programming language Scratch – Computer – News|url=https://www.world-today-news.com/china-appears-to-be-blocking-access-to-childrens-programming-language-scratch-computer-news/|access-date=19 November 2020|website=World Today News|language=en-US}}</ref> The outlet cited the fact that [[Macau]], [[Hong Kong]] and [[Taiwan]] were listed as countries on the website.<ref name=":2">{{cite web|last=Liao|first=Rita|date=7 September 2020|title=China bans Scratch, MIT's programming language for kids|url=https://social.techcrunch.com/2020/09/07/scratch-ban-in-china/|url-status=live|access-date=27 April 2021|website=TechCrunch|language=en-US}}</ref><ref>{{cite web|date=8 September 2020|title=China blocks MIT's kid-friendly programming language Scratch|url=https://developer-tech.com/news/2020/sep/08/china-blocks-mit-kid-friendly-programming-language-scratch/|access-date=19 November 2020|website=Developer Tech News|language=en-GB}}</ref>

== See also ==
{{Portal|Free and open-source software|Computer programming}}
* [[Blockly]], interface used by Scratch to make the code blocks
* [[Swift Playgrounds]]
* [[Alice (software)]]
* [[Twine (software)]]
* [[Lego Mindstorms EV3]]
* [[Kodu Game Lab]]
* [[Code.org]]
* [[Programmable Cricket]]
* [[PWCT]]
* [[Visual programming language]]
* [https://lab.scratch.mit.edu/ Scratch Lab]

== References ==
{{reflist}}

== External links ==
* {{Official website|https://scratch.mit.edu}}
* {{Curlie|Computers/Programming/Languages/Smalltalk/Squeak/Scratch|Scratch}}

{{Wikibooks-inline|Scratch}}

{{Commons category-inline}}{{Authority control}}

{{Video game engines |state=collapsed}}

[[Category:Dynamically typed programming languages]]
[[Category:Visual programming languages]]
[[Category:Educational programming languages]]
[[Category:Free educational software]]
[[Category:MIT Media Lab]]
[[Category:Pedagogic integrated development environments]]
[[Category:Smalltalk programming language family]]
[[Category:Video game development software]]
[[Category:Video game engines]]
[[Category:Video game IDE]]
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User:IntegralPython

2022-03-03T15:56:10Z

IntegralPython: /* My Articles */ added bluelink

[[File:WikiProject Mathematics AD.gif|center]]
<table style="float: right; margin-left: 1em; margin-bottom: 0.5em; width: 250px; border: #99B3FF solid 1px">
<tr><td>{{Template:User WP Mathematics}}</td>
<td>{{Template:User interest mathematics}}</td></tr>
<tr><td>{{Template:User 4-D}}</td></tr>
</table>

Hi! I'm a [[Christians|Christian]] [[Wikipedia]] browser and recreational mathematician. My interests mainly lie in [[math]] and [[Science]], particularly in [[fractal]] analysis, [[quaternion]]s, [[hyperoperation]]s, and [[quantum physics]]. If I do anything stupid, please leave a long and angry comment on my talk page; make sure to include as many strongly worded critiques of me and my poor intelligence. Thanks!

==My Articles==
*[[Quota rule]]
*[[Hand eye calibration problem]]
*[[Open set condition]]
===bonus===
Articles I have spent a significant amount of effort on
*[[Genetic use restriction technology]]
*[[internet meme]]
*[[tetration]]

==Helpful links==
*[[User:IntegralPython/sandbox|My Sandbox]]
*[https://en.wikipedia.org/w/index.php?hidebots=1&hidecategorization=1&hideWikibase=1&tagfilter=coi-spam&limit=50&days=30&title=Special:RecentChanges&urlversion=2| conflict of interest pages]

===''[[Fortnite|Math]] links''===
*{{Random page in category|Mathematics|text=Random Math page}}
*{{Random page in category|Mathematics_stubs|text=Random Math stub}}

==Uploaded Pictures==
[[File:Koch Snowflake.svg|200px]]
[[File:GURT process diagram.png|200px]]
[[File:Approximations of 0.5 tetratrated to the x.png|200px]]
[[File:Open set condition.png|200px]]
[[File:Pentation.jpg|200px]]
[[File:Superpermutations.jpg|200px]]
[[File:Superpermutation distribution.png|200px]]
[[File:Kempe Chain.png|200px]]
[[File:Cube super root.png|200px]]

Cantor set

2022-02-13T21:19:28Z

IntegralPython: /* Measure and probability */ added hausdorff measure of the cantor set

{{Short description|Set of points on a line segment}}
{{Distinguish|Cantor space}}
In [[mathematics]], the '''Cantor set''' is a set of points lying on a single [[line segment]] that has a number of unintuitive properties. It was discovered in 1874 by [[Henry John Stephen Smith]]<ref>{{cite journal | first=Henry J.S. | last=Smith | date=1874 | title=On the integration of discontinuous functions | journal=Proceedings of the London Mathematical Society | series=First series | volume=6 | pages=140–153| url=https://zenodo.org/record/1932560 }}</ref><ref>The “Cantor set” was also discovered by [[Paul du Bois-Reymond]] (1831–1889). See {{cite journal | at=footnote on p. 128 | first=Paul | last=du Bois-Reymond | date=1880 | url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002245256 | title=Der Beweis des Fundamentalsatzes der Integralrechnung | journal=Mathematische Annalen | volume=16 | language=de | mode=cs2}}. The “Cantor set” was also discovered in 1881 by Vito Volterra (1860–1940). See: {{cite journal | first=Vito | last=Volterra | date=1881 | title=Alcune osservazioni sulle funzioni punteggiate discontinue | trans-title=Some observations on point-wise discontinuous function | journal=Giornale di Matematiche | volume=19 | pages=76–86 | language=it | mode=cs2}}.</ref><ref>{{cite book | first=José | last=Ferreirós | title=Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics | url=https://archive.org/details/labyrinthofthoug0000ferr | url-access=registration | location=Basel, Switzerland | publisher=Birkhäuser Verlag | date=1999 | pages=[https://archive.org/details/labyrinthofthoug0000ferr/page/162 162]–165 | isbn=9783034850513 }}</ref><ref>{{cite book | first=Ian | last=Stewart | author-link=Ian Stewart (mathematician) | title=Does God Play Dice?: The New Mathematics of Chaos | date=26 June 1997 | publisher=Penguin | isbn=0140256024}}</ref> and introduced by German mathematician [[Georg Cantor]] in 1883.<ref>{{cite journal | first=Georg | last=Cantor | date=1883 | url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002247461 | title=Über unendliche, lineare Punktmannigfaltigkeiten V | trans-title=On infinite, linear point-manifolds (sets), Part 5 | journal=Mathematische Annalen | volume=21 | pages=545–591 | language=de | doi=10.1007/bf01446819 | s2cid=121930608 | access-date=2011-01-10 | archive-url=https://web.archive.org/web/20150924114632/http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002247461 | archive-date=2015-09-24 | url-status=dead }}</ref><ref>{{cite book | first1=H.-O. | last1=Peitgen | first2=H. | last2=Jürgens | first3=D. | last3=Saupe | title=Chaos and Fractals: New Frontiers of Science | url=https://archive.org/details/chaosfractals00hein | url-access=limited | edition=2nd | location=N.Y., N.Y. | publisher=Springer Verlag | date=2004 | page=[https://archive.org/details/chaosfractals00hein/page/n79 65] | isbn=978-1-4684-9396-2}}</ref>

Through consideration of this set, Cantor and others helped lay the foundations of modern [[point-set topology]]. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the '''Cantor ternary set''', built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor himself mentioned the ternary construction only in passing, as an example of a more general idea, that of a [[perfect set]] that is [[Nowhere dense set|nowhere dense]].

[[File:Cantor Zoom.gif|center|thumb|600px|Zoom in Cantor set. Each point in the set is represented here by a vertical line.]]

==Construction and formula of the ternary set==
The Cantor ternary set <math>\mathcal{C}</math> is created by iteratively deleting the [[open interval|''open'']] middle third from a set of line segments. One starts by deleting the open middle third <math display="inline">\left(\frac{1}{3}, \frac{2}{3}\right)</math> from the [[interval (mathematics)|interval]] <math>\textstyle\left[0, 1\right]</math>, leaving two line segments: <math display="inline">\left[0, \frac{1}{3}\right]\cup\left[\frac{2}{3}, 1\right]</math>. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: <math display="inline">\left[0, \frac{1}{9}\right]\cup\left[\frac{2}{9}, \frac{1}{3}\right]\cup\left[\frac{2}{3}, \frac{7}{9}\right]\cup\left[\frac{8}{9}, 1\right]</math>.
The Cantor ternary set contains all points in the interval <math>[0,1]</math> that are not deleted at any step in this [[ad infinitum|infinite process]]. The same facts can be described recursively by setting
: <math>C_0 := [0,1]</math>
and
: <math>C_n := \frac{C_{n-1}} 3 \cup \left(\frac 2 {3} +\frac{C_{n-1}} 3 \right) = \frac13 \bigl(C_{n-1} \cup \left(2 + C_{n-1} \right)\bigr) </math>
for <math>n \ge 1 </math>, so that
: <math> \mathcal{C} :=</math>[[Set-theoretic limit#Monotone sequences|<math>{\color{Blue}\lim_{n\to\infty}C_n}</math>]]<math> = \bigcap_{n=0}^\infty C_n = \bigcap_{n=m}^\infty C_n </math>   for any   <math> m \ge 0</math>.

The first six steps of this process are illustrated below.

[[Image:Cantor set in seven iterations.svg|729px|Cantor ternary set, in seven iterations]]

Using the idea of self-similar transformations, <math>T_L(x)=x/3, T_R(x)=(2+x)/3</math> and <math> C_n =T_L(C_{n-1})\cup T_R(C_{n-1}),</math> the explicit closed formulas for the Cantor set are<ref>{{cite journal | first=Mohsen | last=Soltanifar | title=A Different Description of A Family of Middle-a Cantor Sets | journal=American Journal of Undergraduate Research | volume=5 | issue=2 | pages=9–12 | date=2006 | doi=10.33697/ajur.2006.014| doi-access=free }}</ref>
: <math> \mathcal{C}=[0,1] \,\setminus\, \bigcup_{n=0}^\infty \bigcup_{k=0}^{3^n-1} \left(\frac{3k+1}{3^{n+1}},\frac{3k+2}{3^{n+1}}\right), </math>
where every middle third is removed as the open interval <math display="inline">\left(\frac{3k+1}{3^{n+1}},\frac{3k+2}{3^{n+1}}\right) </math> from the closed interval <math display="inline">\left[\frac{3k+0}{3^{n+1}},\frac{3k+3}{3^{n+1}}\right] = \left[\frac{k+0}{3^n},\frac{k+1}{3^n}\right] </math> surrounding it, or
: <math> \mathcal{C}=\bigcap_{n=1}^\infty \bigcup_{k=0}^{3^{n-1}-1} \left( \left[\frac{3k+0}{3^n},\frac{3k+1}{3^n}\right] \cup \left[\frac{3k+2}{3^n},\frac{3k+3}{3^n}\right] \right), </math>
where the middle third <math display="inline">\left(\frac{3k+1}{3^n},\frac{3k+2}{3^n}\right) </math> of the foregoing closed interval <math display="inline">\left[\frac{k+0}{3^{n-1}},\frac{k+1}{3^{n-1}}\right] = \left[\frac{3k+0}{3^n},\frac{3k+3}{3^n}\right] </math> is removed by intersecting with <math display="inline">\left[\frac{3k+0}{3^n},\frac{3k+1}{3^n}\right] \cup \left[\frac{3k+2}{3^n},\frac{3k+3}{3^n}\right] .</math>

This process of removing middle thirds is a simple example of a [[finite subdivision rule]]. The Cantor ternary set is an example of a [[fractal string]].

[[File:Cantor set binary tree.svg|400px]]

In arithmetical terms, the Cantor set consists of all real numbers of the [[unit interval]] <math>[0,1]</math> that do not require the digit 1 in order to be expressed as a [[Ternary numeral system|ternary]] (base 3) fraction. As the above diagram illustrates, each point in the Cantor set is uniquely located by a path through an infinitely deep binary tree, where the path turns left or right at each level according to which side of a deleted segment the point lies on. Representing each left turn with 0 and each right turn with 2 yields the ternary fraction for a point.

== Composition ==
Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., [[Lebesgue measure|measure]]) of the unit interval remaining can be found by total length removed. This total is the [[geometric progression]]

:<math>\sum_{n=0}^\infty \frac{2^n}{3^{n+1}} = \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{81} + \cdots = \frac{1}{3}\left(\frac{1}{1-\frac{2}{3}}\right) = 1.</math>

So that the proportion left is 1 − 1 = 0.

This calculation suggests that the Cantor set cannot contain any [[interval (mathematics)|interval]] of non-zero length. It may seem surprising that there should be anything left—after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing [[open set]]s (sets that do not include their endpoints). So removing the line segment ({{sfrac|1|3}}, {{sfrac|2|3}}) from the original interval [0, 1] leaves behind the points {{sfrac|1|3}} and {{sfrac|2|3}}. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points (as follows from the above description in terms of paths in an infinite binary tree).

It may appear that ''only'' the endpoints of the construction segments are left, but that is not the case either. The number {{sfrac|1|4}}, for example, has the unique ternary form 0.020202... = {{overline|0.|02}}. It is in the bottom third, and the top third of that third, and the bottom third of that top third, and so on. Since it is never in one of the middle segments, it is never removed. Yet it is also not an endpoint of any middle segment, because it is not a multiple of any power of 1/3.<ref name="College">{{citation
| last1 = Belcastro | first1 = Sarah-Marie
| last2 = Green | first2 = Michael
| date = January 2001
| doi = 10.2307/2687224
| issue = 1
| journal = The College Mathematics Journal
| page = 55
| title = The Cantor set contains <math>\tfrac{1}{4}</math>? Really?
| volume = 32| jstor = 2687224
}}</ref>
All endpoints of segments are ''terminating'' ternary fractions and are contained in the set
:<math> \left\{x \in [0,1] \mid \exists i \in \N_0: x \, 3^i \in \Z \right\} \qquad \Bigl(\subset \N_0 \, 3^{-\N_0} \Bigr) </math>
which is a [[countably infinite]] set.
As to [[cardinality]], [[almost all]] elements of the Cantor set are not endpoints of intervals, nor rational points like 1/4. The whole Cantor set is in fact not countable.

== Properties ==

=== Cardinality ===
It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is [[uncountable set|uncountable]]. To see this, we show that there is a function ''f'' from the Cantor set <math>\mathcal{C}</math> to the closed interval [0,1] that is [[Surjective function|surjective]] (i.e. ''f'' maps from <math>\mathcal{C}</math> onto [0,1]) so that the [[cardinality]] of <math>\mathcal{C}</math> is no less than that of [0,1]. Since <math>\mathcal{C}</math> is a subset of [0,1], its cardinality is also no greater, so the two cardinalities must in fact be equal, by the [[Cantor–Bernstein–Schröder theorem]].

To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or [[ternary numeral system|ternary]]) notation. Recall that the proper ternary fractions, more precisely: the elements of <math>\bigl(\Z \setminus \{0\}\bigr) \cdot 3^{-\N_0}</math>, admit more than one representation in this notation, as for example {{sfrac|1|3}}, that can be written as 0.13 = {{overline|0.1|0}}3, but also as 0.0222...3 = {{overline|0.0|2}}3, and {{sfrac|2|3}}, that can be written as 0.23 = {{overline|0.2|0}}3 but also as 0.1222...3 = {{overline|0.1|2}}3.<ref>This alternative recurring representation of a number with a terminating numeral occurs in any [[Numeral system#Positional systems in detail|positional system]] with [[Absolute value (algebra)#Types of absolute value|Archimedean absolute value]].</ref>
When we remove the middle third, this contains the numbers with ternary numerals of the form 0.1xxxxx...3 where xxxxx...3 is strictly between 00000...3 and 22222...3. So the numbers remaining after the first step consist of
* Numbers of the form 0.0xxxxx...3 (including 0.022222...3 = 1/3)
* Numbers of the form 0.2xxxxx...3 (including 0.222222...3 = 1)

This can be summarized by saying that those numbers with a ternary representation such that the first digit after the [[radix point]] is not 1 are the ones remaining after the first step.

The second step removes numbers of the form 0.01xxxx...3 and 0.21xxxx...3, and (with appropriate care for the endpoints) it can be concluded that the remaining numbers are those with a ternary numeral where neither of the first ''two'' digits is 1.

Continuing in this way, for a number not to be excluded at step ''n'', it must have a ternary representation whose ''n''th digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit a numeral representation consisting entirely of 0s and 2s.

It is worth emphasizing that numbers like 1, {{sfrac|1|3}} = 0.13 and {{sfrac|7|9}} = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...3 = {{overline|0.|2}}3, {{sfrac|1|3}} = 0.0222...3 = {{overline|0.0|2}}3 and {{sfrac|7|9}} = 0.20222...3 = {{overline|0.20|2}}3.
All the latter numbers are “endpoints”, and these examples are right [[limit point]]s of <math>\mathcal{C}</math>. The same is true for the left limit points of <math>\mathcal{C}</math>, e.g. {{sfrac|2|3}} = 0.1222...3 = {{overline|0.1|2}}3 = {{overline|0.2|0}}3 and {{sfrac|8|9}} = 0.21222...3 = {{overline|0.21|2}}3 = {{overline|0.22|0}}3. All these endpoints are ''proper ternary'' [[Rational number|fractions]] (elements of <math>\Z \cdot 3^{-\N_0}</math>) of the form {{sfrac|p|q}}, where denominator ''q'' is a power of 3 when the fraction is in its [[Irreducible fraction|irreducible]] form.<ref name="College"/> The ternary representation of these fractions terminates (i.e., is finite) or — recall from above that proper ternary fractions each have 2 representations — is infinite and “ends” in either infinitely many recurring 0s or infinitely many recurring 2s. Such a fraction is a left [[limit point]] of <math>\mathcal{C}</math> if its ternary representation contains no 1's and “ends” in infinitely many recurring 0s. Similarly, a proper ternary fraction is a right limit point of <math>\mathcal{C}</math> if it again its ternary expansion contains no 1's and “ends” in infinitely many recurring 2s.

This set of endpoints is [[Dense set|dense]] in <math>\mathcal{C}</math> (but not dense in [0, 1]) and makes up a [[countably infinite]] set. The numbers in <math>\mathcal{C}</math> which are ''not'' endpoints also have only 0s and 2s in their ternary representation, but they cannot end in an infinite repetition of the digit 0, nor of the digit 2, because then it would be an endpoint.

The function from <math>\mathcal{C}</math> to [0,1] is defined by taking the ternary numerals that do consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a [[Binary numeral system#Representing real numbers|binary]] representation of a real number. In a formula,

:<math>f \bigg( \sum_{k\in \N} a_k 3^{-k} \bigg) = \sum_{k\in \N} \frac{a_k}{2} 2^{-k}</math>   where   <math>\forall k\in \N : a_k \in \{0,2\} .</math>

For any number ''y'' in [0,1], its binary representation can be translated into a ternary representation of a number ''x'' in <math>\mathcal{C}</math> by replacing all the 1s by 2s. With this, ''f''(''x'') = ''y'' so that ''y'' is in the [[Range of a function|range]] of ''f''. For instance if ''y'' = {{sfrac|3|5}} = 0.100110011001...2 = {{overline|0.|1001}}, we write ''x'' = {{overline|0.|2002}} = 0.200220022002...3 = {{sfrac|7|10}}. Consequently, ''f'' is surjective. However, ''f'' is ''not'' [[injective function|injective]] — the values for which ''f''(''x'') coincides are those at opposing ends of one of the ''middle thirds'' removed. For instance, take
:{{sfrac|1|3}} = {{overline|0.0|2}}3 (which is a right limit point of <math>\mathcal{C}</math> and a left limit point of the middle third [{{sfrac|1|3}}, {{sfrac|2|3}}])   and
:{{sfrac|2|3}} = {{overline|0.2|0}}3 (which is a left limit point of <math>\mathcal{C}</math> and a right limit point of the middle third [{{sfrac|1|3}}, {{sfrac|2|3}}])
so
:<math>\begin{array}{lcl}
f\bigl({}^1\!\!/\!_3 \bigr) = f(0.0\overline{2}_3) = 0.0\overline{1}_2 = \!\! & \!\! 0.1_2 \!\! & \!\! = 0.1\overline{0}_2 = f(0.2\overline{0}_3) = f\bigl({}^2\!\!/\!_3 \bigr) . \\
& \parallel \\
& {}^1\!\!/\!_2
\end{array}</math>
Thus there are as many points in the Cantor set as there are in the interval [0, 1] (which has the [[Uncountable set|uncountable]] cardinality {{nowrap|<math>\mathfrak{c} = 2^{\aleph_0}</math>).}} However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints. As noted above, one example of such a number is {{sfrac|1|4}}, which can be written as 0.020202...3 = {{overline|0.|02}} in ternary notation. In fact, given any <math>a\in[-1,1]</math>, there exist <math>x,y\in\mathcal{C}</math> such that <math>a = y-x</math>. This was first demonstrated by [[Hugo Steinhaus|Steinhaus]] in 1917, who proved, via a geometric argument, the equivalent assertion that <math>\{(x,y)\in\mathbb{R}^2 \mid y=x+a\} \; \cap \; (\mathcal{C}\times\mathcal{C}) \neq\emptyset</math> for every <math>a\in[-1,1]</math>.<ref>{{Cite book|title=Real Analysis|url=https://archive.org/details/realanalysis00caro_315|url-access=limited|last=Carothers|first=N. L.|publisher=Cambridge University Press|year=2000|isbn=978-0-521-69624-1|location=Cambridge|pages=[https://archive.org/details/realanalysis00caro_315/page/n41 31]–32}}</ref> Since this construction provides an injection from <math>[-1,1]</math> to <math>\mathcal{C}\times\mathcal{C}</math>, we have <math>|\mathcal{C}\times\mathcal{C}|\geq|[-1,1]|=\mathfrak{c}</math> as an immediate corollary. Assuming that <math>|A\times A|=|A|</math> for any infinite set <math>A</math> (a statement shown to be equivalent to the [[axiom of choice]] by [[Alfred Tarski|Tarski]]), this provides another demonstration that <math>|\mathcal{C}|=\mathfrak{c}</math>.

The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. The irrational numbers have the same property, but the Cantor set has the additional property of being [[Closed set|closed]], so it is not even [[Dense set|dense]] in any interval, unlike the irrational numbers which are dense in every interval.

It has been conjectured that all [[algebraic number|algebraic]] irrational numbers are [[normal number|normal]]. Since members of the Cantor set are not normal, this would imply that all members of the Cantor set are either rational or [[transcendental number|transcendental]].

=== Self-similarity ===
The Cantor set is the prototype of a [[fractal]]. It is [[self-similar]], because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. More precisely, the Cantor set is equal to the union of two functions, the left and right self-similarity transformations of itself, <math>T_L(x)=x/3</math> and <math>T_R(x)=(2+x)/3</math>, which leave the Cantor set invariant up to [[homeomorphism]]: <math>T_L(\mathcal{C})\cong T_R(\mathcal{C})\cong \mathcal{C}=T_L(\mathcal{C})\cup T_R(\mathcal{C}).</math>

Repeated [[iterated function|iteration]] of <math>T_L</math> and <math>T_R</math> can be visualized as an infinite [[binary tree]]. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set <math>\{T_L, T_R\}</math> together with [[function composition]] forms a [[monoid]], the [[dyadic monoid]].

The [[automorphism]]s of the binary tree are its hyperbolic rotations, and are given by the [[modular group]]. Thus, the Cantor set is a [[homogeneous space]] in the sense that for any two points <math>x</math> and <math>y</math> in the Cantor set <math>\mathcal{C}</math>, there exists a homeomorphism <math>h:\mathcal{C}\to \mathcal{C}</math> with <math>h(x)=y</math>. An explicit construction of <math>h</math> can be described more easily if we see the Cantor set [[Cantor set#Topological and analytical properties|as a product space]] of countably many copies of the discrete space <math>\{0,1\}</math>. Then the map <math>h:\{0,1\}^\N\to\{0,1\}^\N </math> defined by <math>h_n(u):=u_n+x_n+y_n \mod 2</math> is an involutive homeomorphism exchanging <math>x</math> and <math>y</math>.

=== Conservation law===

It has been found that some form of conservation law is always responsible behind scaling and self-similarity. In the case of Cantor set it can be seen that the <math>d_f</math>th moment (where <math>d_f=\ln(2)/\ln(3)</math> is the fractal dimension) of all the surviving intervals at any stage of the construction process is equal to constant which is equal to one in the case of Cantor set.<ref name="KBN95">{{cite journal | first1=P. L. | last1=Krapivsky | first2=E. | last2=Ben-Naim | title=Multiscaling in Stochastic Fractals | journal=Physics Letters A | volume=196 | issue=3–4 | date=1994 | page=168 | doi=10.1016/0375-9601(94)91220-3| bibcode=1994PhLA..196..168K }}</ref><ref name="HR95">{{cite journal | first1=M. K. | last1=Hassan | first2=G. J. | last2=Rodgers | title=Models of fragmentation and stochastic fractals | journal=Physics Letters A | page=208 | volume=95 | issue=1 | date=1995| bibcode=1995PhLA..208...95H | doi=10.1016/0375-9601(95)00727-K }}</ref>
We know that there are <math>N=2^n</math> intervals of size <math>1/3^n</math> present in the system at the <math>n</math>th step of its construction. Then if we label the surviving intervals as <math>x_1, x_2, \ldots, x_{2^n}</math> then the <math>d_f</math>th moment is <math>x_1^{d_f}+x_2^{d_f}+\cdots+x_{2^n}^{d_f}=1</math> since <math>x_1=x_2= \cdots =x_{2^n}=1/3^n</math>.

The [[Hausdorff dimension]] of the Cantor set is equal to ln(2)/ln(3) ≈ 0.631.

=== Topological and analytical properties ===

Although "the" Cantor set typically refers to the original, middle-thirds Cantor described above, topologists often talk about "a" Cantor set, which means any topological space that is [[homeomorphic]] (topologically equivalent) to it.

As the above summation argument shows, the Cantor set is uncountable but has [[Lebesgue measure]] 0. Since the Cantor set is the complement of a [[union (set theory)|union]] of [[open set]]s, it itself is a [[closed set|closed]] subset of the reals, and therefore a [[complete space|complete]] [[metric space]]. Since it is also [[totally bounded]], the [[Heine–Borel theorem]] says that it must be [[compact space|compact]].

For any point in the Cantor set and any arbitrarily small neighborhood of the point, there is some other number with a ternary numeral of only 0s and 2s, as well as numbers whose ternary numerals contain 1s. Hence, every point in the Cantor set is an [[accumulation point]] (also called a cluster point or limit point) of the Cantor set, but none is an [[interior point]]. A closed set in which every point is an accumulation point is also called a [[perfect set]] in [[topology]], while a closed subset of the interval with no interior points is [[Nowhere dense set|nowhere dense]] in the interval.

Every point of the Cantor set is also an accumulation point of the [[complement (set theory)|complement]] of the Cantor set.

For any two points in the Cantor set, there will be some ternary digit where they differ — one will have 0 and the other 2. By splitting the Cantor set into "halves" depending on the value of this digit, one obtains a partition of the Cantor set into two closed sets that separate the original two points. In the [[relative topology]] on the Cantor set, the points have been separated by a [[clopen set]]. Consequently, the Cantor set is [[totally disconnected]]. As a compact totally disconnected [[Hausdorff space]], the Cantor set is an example of a [[Stone space]].

As a [[topological space]], the Cantor set is naturally [[homeomorphism|homeomorphic]] to the [[product topology|product]] of [[countable|countably many]] copies of the space <math>\{0, 1\}</math>, where each copy carries the [[discrete space|discrete topology]]. This is the space of all [[sequence]]s in two digits
:<math>2^\mathbb{N}=\{(x_n)\mid x_n\in \{0,1\} \text{ for } n\in \mathbb{N}\},</math>

which can also be identified with the set of [[p-adic numbers|2-adic integers]]. The [[basis (topology)|basis]] for the open sets of the product topology are [[cylinder set]]s; the homeomorphism maps these to the [[subspace topology]] that the Cantor set inherits from the natural topology on the real number line. This characterization of the [[Cantor space]] as a product of compact spaces gives a second proof that Cantor space is compact, via [[Tychonoff's theorem]].

From the above characterization, the Cantor set is homeomorphic to the [[p-adic numbers|p-adic integers]], and, if one point is removed from it, to the [[p-adic numbers]].

The Cantor set is a subset of the reals, which are a [[metric space]] with respect to the [[absolute difference|ordinary distance metric]]; therefore the Cantor set itself is a metric space, by using that same metric. Alternatively, one can use the [[p-adic metric]] on <math>2^\mathbb{N}</math>: given two sequences <math>(x_n),(y_n)\in 2^\mathbb{N}</math>, the distance between them is <math>d((x_n),(y_n)) = 2^{-k}</math>, where <math>k</math> is the smallest index such that <math>x_k \ne y_k</math>; if there is no such index, then the two sequences are the same, and one defines the distance to be zero. These two metrics generate the same [[topological space|topology]] on the Cantor set.

We have seen above that the Cantor set is a totally disconnected perfect compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set. See [[Cantor space]] for more on spaces homeomorphic to the Cantor set.

The Cantor set is sometimes regarded as "universal" in the [[Category theory|category]] of [[Compact space|compact]] [[metric space]]s, since any compact metric space is a continuous image of the Cantor set; however this construction is not unique and so the Cantor set is not [[universal property|universal]] in the precise categorical sense. The "universal" property has important applications in [[functional analysis]], where it is sometimes known as the ''representation theorem for compact metric spaces''.<ref>{{cite book | first=Stephen | last=Willard | title=General Topology | publisher=Addison-Wesley | date=1968 | asin=B0000EG7Q0}}</ref>

For any integer ''q'' ≥ 2, the topology on the group G='''Z'''''q''ω (the countable direct sum) is discrete. Although the [[Pontrjagin dual]] Γ is also '''Z'''''q''ω, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it is homeomorphic to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case ''q''=2. (See Rudin 1962 p 40.)

The [[geometric mean]] of the Cantor set is approximately 0.274974.<ref>{{cite web| url = https://math.stackexchange.com/q/1889476| title = Cantor Set Geometric Mean}}</ref>{{unreliable source?|date=December 2017}}

===Measure and probability===
The Cantor set can be seen as the [[compact group]] of binary sequences, and as such, it is endowed with a natural [[Haar measure]]. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses. Furthermore, one can show that the usual [[Lebesgue measure]] on the interval is an image of the Haar measure on the Cantor set, while the natural injection into the ternary set is a canonical example of a [[singular measure]]. It can also be shown that the Haar measure is an image of any [[probability]], making the Cantor set a universal probability space in some ways.

In [[Lebesgue measure]] theory, the Cantor set is an example of a set which is uncountable and has zero measure.<ref>{{cite web | url=http://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/ | title=Theorem 36: the Cantor set is an uncountable set with zero measure | first=Laura | last=Irvine | website=Theorem of the week | access-date=2012-09-27 | archive-url=https://web.archive.org/web/20160315212203/https://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/ | archive-date=2016-03-15 | url-status=dead }}</ref> In contrast, the set has a [[Hausdorff measure]] of 1 in its dimension of log 2 / log 3.<ref>
{{cite book |last=Falconer |first=K. J. |date=July 24, 1986 |title=The Geometry of Fractal Sets |url=http://mate.dm.uba.ar/~umolter/materias/referencias/1.pdf |pages=14–15 |publisher=Cambridge University Press |isbn=9780521337052}}
</ref>

===Cantor numbers===
If we define a Cantor number as a member of the Cantor set, then<ref>{{cite book | title=Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise | first=Manfred | last=Schroeder | publisher=Dover | date=1991 | pages=164–165 | isbn=0486472043}}</ref>
# Every real number in [0, 2] is the sum of two Cantor numbers.
# Between any two Cantor numbers there is a number that is not a Cantor number.

=== Descriptive set theory ===
The Cantor set is a [[Meagre set|meagre set]] (or a set of first category) as a subset of [0,1] (although not as a subset of itself, since it is a [[Baire space]]). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set <math>\mathbb{Q}\cap[0,1]</math>, the Cantor set <math>\mathcal{C}</math> is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of [0,1]. However, unlike <math>\mathbb{Q}\cap[0,1]</math>, which is countable and has a "small" cardinality, <math>\aleph_0</math>, the cardinality of <math>\mathcal{C}</math> is the same as that of [0,1], the continuum <math>\mathfrak{c}</math>, and is "large" in the sense of cardinality. In fact, it is also possible to construct a subset of [0,1] that is meagre but of positive measure and a subset that is non-meagre but of measure zero:<ref>{{Cite book|title=Counterexamples in analysis|last=Gelbaum, Bernard R.|date=1964|publisher=Holden-Day|others=Olmsted, John M. H. (John Meigs Hubbell), 1911-1997|isbn=0486428753|location=San Francisco|oclc=527671}}</ref> By taking the countable union of "fat" Cantor sets <math>\mathcal{C}^{(n)}</math> of measure <math>\lambda = (n-1)/n</math> (see Smith–Volterra–Cantor set below for the construction), we obtain a set <math display="inline">\mathcal{A} := \bigcup_{n=1}^{\infty}\mathcal{C}^{(n)}</math>which has a positive measure (equal to 1) but is meagre in [0,1], since each <math>\mathcal{C}^{(n)}</math> is nowhere dense. Then consider the set <math display="inline">\mathcal{A}^{\mathrm{c}} = [0,1] \setminus\bigcup_{n=1}^\infty \mathcal{C}^{(n)}</math>. Since <math>\mathcal{A}\cup\mathcal{A}^{\mathrm{c}} = [0,1]</math>, <math>\mathcal{A}^{\mathrm{c}}</math> cannot be meagre, but since <math>\mu(\mathcal{A})=1</math>, <math>\mathcal{A}^{\mathrm{c}}</math> must have measure zero.

== Variants ==

===Smith–Volterra–Cantor set===
{{main|Smith–Volterra–Cantor set}}

Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle. In the case where the middle {{sfrac|8|10}} of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0s and 9s. If a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder <math>(1-f)^n\to 0</math> as <math>n\to\infty</math> for any ''f'' such that <math>0<f\leq 1</math>.

On the other hand, "fat Cantor sets" of positive measure can be generated by removal of smaller fractions of the middle of the segment in each iteration. Thus, one can construct sets homeomorphic to the Cantor set that have positive Lebesgue measure while still being nowhere dense. If an interval of length <math>r^n</math> (<math>r\leq 1/3</math>) is removed from the middle of each segment at the ''n''th iteration, then the total length removed is <math display="inline">\sum_{n=1}^\infty 2^{n-1}r^n=r/(1-2r)</math>, and the limiting set will have a [[Lebesgue measure]] of <math>\lambda=(1-3r)/(1-2r)</math>. Thus, in a sense, the middle-thirds Cantor set is a limiting case with <math>r=1/3</math>. If <math>0<r<1/3</math>, then the remainder will have positive measure with <math>0<\lambda<1</math>. The case <math>r=1/4</math> is known as the [[Smith–Volterra–Cantor set]], which has a Lebesgue measure of <math>1/2</math>.

===Stochastic Cantor set===

One can modify the construction of the Cantor set by dividing randomly instead of equally. Besides, to incorporate time we can divide only one of the available intervals at each step instead of dividing all the available intervals. In the case of stochastic triadic Cantor set the resulting process can be described by the following rate equation<ref name="KBN95"/><ref name="HR95"/>

:<math>\frac{\partial c(x,t)}{\partial t} =-\frac{x^2}{2} c(x,t) + 2\int_x^\infty (y-x)c(y,t) \, dy,</math>

and for the stochastic dyadic Cantor set<ref>{{cite journal | first1=M. K. | last1=Hassan | first2=N. I. | last2=Pavel | first3=R. K. | last3=Pandit | first4=J. | last4=Kurths | title=Dyadic Cantor set and its kinetic and stochastic counterpart | journal=Chaos, Solitons & Fractals | volume=60 | pages=31–39 | date=2014 | doi=10.1016/j.chaos.2013.12.010| bibcode=2014CSF....60...31H | arxiv=1401.0249 | s2cid=14494072 }}</ref>

:<math>{{\partial c(x,t)}\over{\partial t}}=-xc(x,t)+(1+p)\int_x^\infty c(y,t) \, dy,</math>

where <math>c(x,t)dx</math> is the number of intervals of size between <math>x</math> and <math>x+dx</math>. In the case of triadic Cantor set the fractal dimension is <math>0.5616</math> which is
less than its deterministic counterpart <math>0.6309</math>. In the case of stochastic dyadic Cantor set
the fractal dimension is <math>p</math> which is again less than that of its deterministic counterpart <math>\ln (1+p)/\ln 2</math>. In the case of stochastic dyadic Cantor set the solution for <math>c(x,t)</math> exhibits [[dynamic scaling]] as its solution in the long-time limit is <math>t^{-(1+d_f)}e^{-xt}</math> where the fractal dimension of the stochastic dyadic Cantor set <math>d_f=p</math>. In either case, like triadic Cantor set, the <math>d_f</math>th moment (<math display="inline">\int x^{d_f} c(x,t) \, dx = \text{constant}</math>) of stochastic triadic and dyadic Cantor set too are conserved quantities.

=== Cantor dust ===

'''Cantor dust''' is a multi-dimensional version of the Cantor set. It can be formed by taking a finite [[Cartesian product]] of the Cantor set with itself, making it a [[Cantor space]]. Like the Cantor set, Cantor dust has [[Measure zero|zero measure]].<ref>{{cite book|author=Helmberg, Gilbert|title=Getting Acquainted With Fractals|publisher=Walter de Gruyter|year=2007|isbn=978-3-11-019092-2|page=46|url=https://books.google.com/books?id=PbrlYO83Oq8C&pg=PA46}}</ref>
[[File:Cantorcubes.gif|thumb|right|250px|[[Cantor cube]]s recursion progression towards Cantor dust]]
{|
|[[Image:Cantor dust.svg|thumb|'''Cantor dust''' (2D)]]
|[[Image:Cantors cube.jpg|thumb|'''Cantor dust''' (3D)]]
|}

A different 2D analogue of the Cantor set is the [[Sierpinski carpet]], where a square is divided up into nine smaller squares, and the middle one removed. The remaining squares are then further divided into nine each and the middle removed, and so on ad infinitum.<ref>{{cite book|author=Helmberg, Gilbert|title=Getting Acquainted With Fractals|publisher=Walter de Gruyter|year=2007|isbn=978-3-11-019092-2|page=48|url=https://books.google.com/books?id=PbrlYO83Oq8C&pg=PA48}}</ref> One 3D analogue of this is the [[Menger sponge]].

==Historical remarks==
[[File:Cantor-like Column Capital Ile de Philae Description d'Egypte 1809.jpg|thumb|Column capital with pattern evocative of the Cantor set, but expressed in binary rather than ternary. Engraving of Île de Philae from ''Description d'Égypte'' by Jean-Baptiste Prosper Jollois and Édouard Devilliers, Imprimerie Impériale, Paris, 1809-1828]]

Cantor himself defined the set in a general, abstract way, and mentioned the ternary construction only in passing, as an example of a more general idea, that of a [[perfect set]] that is [[Nowhere dense set|nowhere dense]]. The original paper provides several different constructions of the abstract concept.

This set would have been considered abstract at the time when Cantor devised it. Cantor himself was led to it by practical concerns about the set of points where a [[Fourier series|trigonometric series]] might fail to converge. The discovery did much to set him on the course for developing an [[axiomatic set theory|abstract, general theory of infinite sets]].

== See also ==
*[[Classification of discontinuities#The set of discontinuities of a function|The indicator function of the Cantor set]]
*[[Smith–Volterra–Cantor set]]
*[[Hexagrams (I Ching)]]
*[[Cantor function]]
*[[Cantor cube]]
*[[Antoine's necklace]]
*[[Koch snowflake]]
*[[Knaster–Kuratowski fan]]
*[[List of fractals by Hausdorff dimension]]
*[[Moser–de Bruijn sequence]]

==Notes==
{{Reflist|30em}}

==References==
{{refbegin}}
* {{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach Jr. | title=Counterexamples in Topology | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995 | at=Example 29| title-link=Counterexamples in Topology }}
* {{cite book | first1=Gary L. | last1=Wise | first2=Eric B. | last2=Hall | title=Counterexamples in Probability and Real Analysis | url=https://archive.org/details/counterexamplesi0000wise | url-access=registration | publisher=[[Oxford University Press]] | location=New York | date=1993 | isbn=0-19-507068-2 | at=Chapter 1}}
* {{cite book | author-link=K. J. Falconer | first=K. J. | last=Falconer | title=Geometry of Fractal Sets | url=https://archive.org/details/geometryoffracta00falc | url-access=registration | publisher=[[Cambridge University Press]] | date=24 July 1986 | series=Cambridge Tracts in Mathematics | issue=85 | isbn=0521337054}}
* {{cite book | first=Pertti | last=Mattila | title=Geometry of Sets and Measures in Euclidean Space: Fractals and rectifiability | publisher=Cambridge University Press | date=25 February 1999 | series=Cambridge studies in advanced mathematics | issue=44 | isbn=0521655951}}
* {{cite book | first=Pertti | last=Mattila | title=Fourier Analysis and Hausdorff Dimension | publisher=Cambridge University Press | date=2015 | series=Cambridge studies in advanced mathematics | issue=150 | isbn=9781316227619}}.
* {{cite book | first=A. | last=Zygmund | author-link=A. Zygmund | title=Trigonometric Series, Vols. I and II |title-link = Trigonometric Series | publisher=Cambridge University Press | date=1958}}
{{refend}}

==External links==
* {{springer|title=Cantor set|id=p/c020250}}
* [http://www.cut-the-knot.org/do_you_know/Cantor2.shtml Cantor Sets] and [http://www.cut-the-knot.org/do_you_know/cantor.shtml Cantor Set and Function] at [[cut-the-knot]]
* [https://platonicrealms.com/encyclopedia/Cantorn-set Cantor Set] at Platonic Realms

{{Fractals}}
{{Authority control}}

{{DEFAULTSORT:Cantor Set}}
[[Category:Measure theory]]
[[Category:Topological spaces]]
[[Category:Sets of real numbers]]
[[Category:Georg Cantor]]
[[Category:L-systems]]

Open set condition

2022-02-07T02:21:25Z

IntegralPython: box counting dimension

{{Short description|Condition for fractals in math}}
[[File:Open set condition.png|thumb|an open set covering of the [[sierpinski triangle]] along with one of its mappings ψ''i''.]]
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

==Computing Hausdorff dimension==
When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose [[Hausdorff dimension]] is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Strong open set condition==
The strong open set condition (SOSC) is an extension of the open set condition. A fractal F satisfies the SOSC if, in addition to satisfying the OSC, the intersection between F and the open set V is nonempty.<ref>{{Cite web | url=http://www.stat.uchicago.edu/~lalley/Papers/packing.pdf| title=The Packing and Covering Functions for Some Self-similar Fractals|last=Lalley|first=Steven|publisher=Purdue University|date=21 January 1988|access-date=2 February 2022}}</ref> The two conditions are equivalent for self-similar and self-conformal sets, but not for certain classes of other sets, such as function systems with infinite mappings and in non-euclidean metric spaces.<ref>{{Cite web| url=http://users.jyu.fi/~antakae/publications/preprints/009-controlled_moran.pdf| title=Separation Conditions on Controlled Moran Constructions| last1=Käenmäki| first1=Antti| last2=Vilppolainen| first2=Markku| access-date = 2 February 2022}}</ref><ref>{{Cite journal| last=Schief| first=Andreas| title=Self-similar Sets in Complete Metric Spaces| journal=Proceedings of the American Mathematical Society| volume=124| issue=2| year=1996| url=https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03158-9/S0002-9939-96-03158-9.pdf}}</ref> In these cases, SOCS is indeed a stronger condition.

==See also==
*[[Cantor set]]
*[[List of fractals by Hausdorff dimension]]
*[[Minkowski–Bouligand dimension]]
*[[Packing dimension]]

==References==
{{reflist}}

[[Category:Fractals]]
[[Category:Iterated function system fractals]]

List of fractals by Hausdorff dimension

2022-02-07T02:19:27Z

IntegralPython: bluelink

{{short description|Wikipedia list article}}
{{Use dmy dates|date=October 2019}}
[[Benoit Mandelbrot]] has stated that "A [[fractal]] is by definition a set for which the [[Hausdorff dimension|Hausdorff-Besicovitch dimension]] strictly exceeds the [[topological dimension]]."<ref>{{harvnb|Mandelbrot|1982|p=15}}</ref>
Presented here is a list of fractals ordered by increasing Hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension.

==Deterministic fractals==
{| class="wikitable sortable"
|-
! Hausdorff dimension (exact value) || Hausdorff dimension (approx.) || Name || Illustration || width="40%" | Remarks
|-
| Calculated || align="right" | 0.538 || [[Logistic map|Feigenbaum attractor]] || align="center" |[[File:Feigenbaum attractor.png|150px]] || The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the [[logistic function]] for the critical parameter value <math>\scriptstyle{\lambda_\infty = 3.570}</math>, where the period doubling is infinite. This dimension is the same for any differentiable and [[unimodal]] function.<ref>{{cite journal |first=Erik |last=Aurell |title=On the metric properties of the Feigenbaum attractor |journal=Journal of Statistical Physics |volume=47 |issue=3–4 |pages=439–458 |date=May 1987 |doi=10.1007/BF01007519 |bibcode=1987JSP....47..439A |s2cid=122213380 }}</ref>
|-
| <math>\log_3(2)</math> || align="right" | 0.6309 || [[Cantor set]] || align="center" |[[File:Cantor set in seven iterations.svg|200px]] || Built by removing the central third at each iteration. [[Nowhere dense]] and not a [[countable set]].
|-
| <math>\log_2(\varphi)=\log_2(1+\sqrt{5})-1</math> || align="right" | 0.6942 || Asymmetric [[Cantor set]] || align="center" |[[File:AsymmCantor.png|200px]] || The dimension is not <math>\frac{\ln2}{\ln\frac83}</math>, which is the generalized Cantor set with γ=1/4, which has the same length at each stage.<ref>{{Cite journal|author=Tsang, K. Y. |title=Dimensionality of Strange Attractors Determined Analytically |journal=Phys. Rev. Lett. |volume=57|issue=12|pages=1390–1393 |year=1986|pmid=10033437 |doi=10.1103/PhysRevLett.57.1390|bibcode=1986PhRvL..57.1390T}}</ref>
Built by removing the second quarter at each iteration. [[Nowhere dense]] and not a [[countable set]].
<math>\scriptstyle\varphi = \frac{1+\sqrt5}2</math> ([[golden cut]]).
|-
| <math>\log_{10}(5)=1-\log_{10}(2)</math> || align="right" | 0.69897 || [[Real number]]s whose base 10 digits are even || align="center" |[[File:Even digits.png|200px]] || Similar to the [[Cantor set]].<ref name="Falconer">{{Cite book
| last = Falconer | first = Kenneth | author-link=Kenneth Falconer (mathematician)
| title = Fractal Geometry: Mathematical Foundations and Applications
| publisher = John Wiley & Sons, Ltd.
| year = 1990–2003
| isbn = 978-0-470-84862-3
| no-pp = true
| page = xxv}}</ref>
|-
| <math> \log(1+\sqrt{2})</math> || align="right" | 0.88137 || Spectrum of Fibonacci Hamiltonian|| align="center" | || The study of the spectrum of the Fibonacci Hamiltonian proves upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that the spectrum converges to an explicit constant.<ref>{{cite journal |last1=Damanik |first1=D. |last2=Embree |first2=M. |last3=Gorodetski |first3=A. |first4=S. |last4=Tcheremchantse |title=The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian |journal=Commun. Math. Phys. |volume=280 |issue=2 |pages=499–516 |year=2008 |doi=10.1007/s00220-008-0451-3 |arxiv=0705.0338|bibcode=2008CMaPh.280..499D |s2cid=12245755 }}</ref>{{page needed|date=October 2018}}
|-
| <math>\frac{-\log(2)}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}</math> || align="right" | 0<D<1 || Generalized Cantor set || align="center" |[[File:generalized cantor set.png|200px]] || Built by removing at the <math>m</math>th iteration the central interval of length <math>\gamma\,l_{m-1}</math> from each remaining segment (of length <math>l_{m-1}=(1-\gamma)^{m-1}/2^{m-1}</math>). At <math>\scriptstyle\gamma=1/3</math> one obtains the usual [[Cantor set]]. Varying <math>\scriptstyle\gamma</math> between 0 and 1 yields any fractal dimension <math>\scriptstyle 0\,<\,D\,<\,1</math>.<ref>{{cite journal |first1=A. Yu |last1=Cherny |first2=E.M. |last2=Anitas |first3=A.I. |last3=Kuklin |first4=M. |last4=Balasoiu |first5=V.A. |last5=Osipov |title=The scattering from generalized Cantor fractals |journal=J. Appl. Crystallogr. |volume=43 |issue= 4|pages=790–7 |year=2010 |doi=10.1107/S0021889810014184 |arxiv=0911.2497 |s2cid=94779870 }}</ref>
|-
| <math>1</math> || align="right" | 1 || [[Smith–Volterra–Cantor set]] || align="center" |[[File:Smith-Volterra-Cantor set.svg|200px]] || Built by removing a central interval of length <math>2^{-2n}</math> of each remaining interval at the ''n''th iteration. Nowhere dense but has a [[Lebesgue measure]] of {{sfrac|1|2}}.
|-
| <math>2+\log_2\left(\frac{1}{2}\right)=1</math> || align="right" | 1 || [[Takagi curve|Takagi or Blancmange curve]] || align="center" |[[File:Takagi curve.png|150px]] || Defined on the unit interval by <math>f(x) = \sum_{n=0}^\infty 2^{-n}s(2^{n}x)</math>, where <math>s(x)</math> is the [[Triangle wave|triangle wave function]]. Special case of the Takahi-Landsberg curve: <math>f(x) = \sum_{n=0}^\infty w^n s(2^n x)</math> with <math>w = 1/2</math>. The Hausdorff dimension equals <math>2+\log_2(w)</math> for <math>w</math> in <math>\left[1/2,1\right]</math>. (Hunt cited by Mandelbrot<ref>{{Cite book| last = Mandelbrot| first = Benoit | title = Gaussian self-affinity and Fractals | isbn = 978-0-387-98993-8 | year = 2002 }}</ref>).
|-
| Calculated|| align="right" | 1.0812 || [[Julia set]] z² + 1/4 || align="center" |[[File:Julia z2+0,25.png|100px]] || Julia set for ''c'' = 1/4.<ref name="McMullen"/>
|-
| Solution ''s'' of <math>2|\alpha|^{3s}+|\alpha|^{4s}=1</math>|| align="right" | 1.0933 || Boundary of the [[Rauzy fractal]]|| align="center" |[[File:Rauzy fractal.png|150px]] || Fractal representation introduced by G.Rauzy of the dynamics associated to the Tribonacci morphism: <math>1\mapsto12</math>, <math>2\mapsto13</math> and <math>3\mapsto1</math>.<ref>Messaoudi, Ali. [http://matwbn.icm.edu.pl/ksiazki/aa/aa95/aa9531.pdf Frontième de numération complexe]", ''matwbn.icm.edu.pl''. {{in lang|fr}} Accessed: 27 October 2018.</ref>{{page needed|date=October 2018}}<ref>{{Citation | last1=Lothaire | first1=M. | author-link=M. Lothaire | title=Applied combinatorics on words | url=https://archive.org/details/appliedcombinato0000loth/page/525 | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-84802-2 | mr=2165687 | zbl=1133.68067 | year=2005 | volume=105 | page=[https://archive.org/details/appliedcombinato0000loth/page/525 525] }}</ref> <math>\alpha</math> is one of the conjugated roots of <math>z^3-z^2-z-1=0</math>.
|-
| <math>2\log_7(3)</math> || align="right" | 1.12915 || contour of the [[Gosper island]] || align="center" |[[File:Gosper Island 4.svg|100px]] || Term used by Mandelbrot (1977).<ref>{{MathWorld |id=GosperIsland |title=Gosper Island |access-date=27 October 2018}}</ref> The Gosper island is the limit of the [[Gosper curve]].
|-
| Measured (box counting) || align="right" | 1.2 || Dendrite [[Julia set]] || align="center" |[[File:Dendrite julia.png|150px]] || Julia set for parameters: Real = 0 and Imaginary = 1.
|-
| <math>3\frac{\log(\varphi)}{\log\left(\displaystyle\frac{3+\sqrt{13}}{2}\right)}</math> || align="right" | 1.2083 || [[Fibonacci word fractal|Fibonacci word fractal 60°]] || align="center" | [[File:Fibo 60deg F18.png|200px]] || Build from the [[Fibonacci word]]. See also the standard Fibonacci word fractal.
<math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| |<math>\begin{align}&2\log_2\left(\displaystyle\frac{\sqrt[3]{27-3\sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right),\\ &\text{or root of }2^x-1=2^{(2-x)/2}\end{align}</math>|| align="right" | 1.2108 || Boundary of the tame twindragon || align="center" |[[File:TameTwindragontile.png|150px]] || One of the six 2-[[rep-tile]]s in the plane (can be tiled by two copies of itself, of equal size).<ref name="2-reptiles">Ngai, Sirvent, Veerman, and Wang (October 2000). "[https://doi.org/10.1023%2FA%3A1005206301454 On 2-Reptiles in the Plane 1999]", ''Geometriae Dedicata'', Volume 82. Accessed: 29 October 2018.</ref><ref name="Boundary">Duda, Jarek (March 2011). "[http://demonstrations.wolfram.com/TheBoundaryOfPeriodicIteratedFunctionSystems/ The Boundary of Periodic Iterated Function Systems]", ''Wolfram.com''.</ref>
|-
| || align="right" | 1.26 || [[Hénon map]] || align="center" |[[File:Henon.jpg|100px]] || The canonical [[Hénon map]] (with parameters ''a'' = 1.4 and ''b'' = 0.3) has Hausdorff dimension 1.261 ± 0.003. Different parameters yield different dimension values.
|-
| <math>\log_3(4)</math> || align="right" | 1.2619 || [[Triflake]] || align="center" | [[File:Triflake.png|150px]] || Three anti-snowflakes arranged in a way that a koch-snowflake forms in between the anti-snowflakes.
|-
| <math>\log_3(4)</math> || align="right" | 1.2619 || [[Koch curve]] || align="center" | [[File:Koch curve.svg|200px]] || 3 Koch curves form the Koch snowflake or the anti-snowflake.
|-
| <math>\log_3(4)</math> || align="right" | 1.2619 || boundary of [[Dragon curve|Terdragon curve]] || align="center" |[[File:Terdragon boundary.png|150px]] || L-system: same as dragon curve with angle = 30°. The Fudgeflake is based on 3 initial segments placed in a triangle.
|-
| <math>\log_3(4)</math> || align="right" | 1.2619 || 2D [[Cantor dust]] || align="center" |[[File:Carre cantor.gif|100px]] || Cantor set in 2 dimensions.
|-
| <math>\log_3(4)</math> || align="right" | 1.2619 || 2D [[L-system]] branch || align="center" |[[File:Onetwosix.png|200px]] || L-Systems branching pattern having 4 new pieces scaled by 1/3. Generating the pattern using statistical instead of exact self-similarity yields the same fractal dimension.
|-
| Calculated|| align="right" | 1.2683 || [[Julia set]] z2 − 1 || align="center" |[[File:Julia z2-1.png|200px]] || Julia set for ''c'' = −1.<ref name="McMullen"/>
|-
| || align="right" | 1.3057 || [[Apollonian gasket]] || align="center" |[[File:Apollonian gasket.svg|100px]] || Starting with 3 tangent circles, repeatedly packing new circles into the complementary interstices. Also the limit set generated by reflections in 4 mutually tangent circles. See<ref name="McMullen">McMullen, Curtis T. (3 October 1997). "[http://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf Hausdorff dimension and conformal dynamics III: Computation of dimension]", ''Abel.Math.Harvard.edu''. Accessed: 27 October 2018.</ref>
|-
| || align="right" | 1.328 || 5 [[Circle inversion|circles inversion]] fractal || align="center" |[[File:Cicle inversion.svg|100px]] || The limit set generated by iterated inversions with respect to 5 mutually tangent circles (in red). Also an Apollonian packing. See<ref>Chang, Angel and Zhang, Tianrong. {{Cite web |url=http://classes.yale.edu/fractals/CircInvFrac/CircDim/CircDim2.html |title=On the Fractal Structure of the Boundary of Dragon Curve |access-date=9 February 2019 |archive-url=https://web.archive.org/web/20110614063904/http://classes.yale.edu/Fractals/CircInvFrac/CircDim/CircDim2.html |archive-date=14 June 2011 |url-status=bot: unknown }} [https://stanford.edu/~angelx/pubs/dragonbound.pdf pdf]</ref>
|-
| <math>\log_5(9)</math>|| align="right" | 1.36521<ref>Mandelbrot, B. B. (1983). ''The Fractal Geometry of Nature'', p.48. New York: W. H. Freeman. {{ISBN|9780716711865}}. Cited in: {{MathWorld |id=MinkowskiSausage |title=Minkowski Sausage |access-date=22 September 2019}}</ref> || [[Koch island|Quadratic von Koch island]] using the type 1 curve as generator|| align="center" |[[File:Karperienflakeani2.gif|150px]] || Also known as the [[Minkowski Sausage]]
|-
| Calculated|| align="right" | 1.3934 || [[Douady rabbit]] || align="center" |[[File:Douady rabbit.png|150px]] || Julia set for ''c'' = −0,123 + 0.745i.<ref name="McMullen"/>
|-
| <math>\log_3(5)</math>|| align="right" | 1.4649 || [[Vicsek fractal]] || align="center" |[[File:Box fractal.svg|100px]] || Built by exchanging iteratively each square by a cross of 5 squares.
|-
| <math>\log_3(5)</math>|| align="right" | 1.4649 || [[Koch curve|Quadratic von Koch curve (type 1)]]|| align="center" |[[File:Quadratic Koch 2.svg|150px]] || One can recognize the pattern of the Vicsek fractal (above).
|-
| <math>\log_{\sqrt{5}}\left(\frac{10}{3}\right)</math>|| align="right" | 1.4961 ||Quadric cross || align="center" |{{anchor|cross}}[[File:Quadriccross.gif|150px]] ||[[File:Q Cross Fractal Generator.jpg|thumb|The quadric cross is made by scaling the 3-segment generator unit by 51/2 then adding 3 full scaled units, one to each original segment, plus a third of a scaled unit (blue) to increase the length of the pedestal of the starting 3-segment unit (purple).]]Built by replacing each end segment with a cross segment scaled by a factor of 51/2, consisting of 3 1/3 new segments, as illustrated in the inset.

''Images generated with Fractal Generator for ImageJ.''
|-
|<math>2-\log_2(\sqrt{2})=\frac{3}{2}</math>|| align="right" | 1.5000 || a [[Weierstrass function]]: <math>\displaystyle f(x)=\sum_{k=1}^\infty \frac{\sin(2^k x)}{\sqrt{2}^k}</math> || align="center" |[[File:Weierstrass functionAMD.png|150px]] || The Hausdorff dimension of the Weierstrass function <math>f:[0,1]\to\mathbb{R}</math> defined by <math>f(x)=\sum_{k=1}^\infty a^{-k}\sin(b^k x)</math> with <math>\frac{1}{b}<a<1</math> and <math>b>1</math> is <math>2 +\log_b(a)</math>.<ref>{{Cite journal|last=Shen|first=Weixiao|date=2018|title=Hausdorff dimension of the graphs of the classical Weierstrass functions|journal=Mathematische Zeitschrift|language=en|volume=289|issue=1–2|pages=223–266|doi=10.1007/s00209-017-1949-1|issn=0025-5874|arxiv=1505.03986|s2cid=118844077}}</ref><ref>N. Zhang. The Hausdorff dimension of the graphs of fractal functions. (In Chinese). Master Thesis. Zhejiang University, 2018.</ref>
|-
|<math>\log_4(8)=\frac{3}{2}</math>|| align="right" | 1.5000 || [[Minkowski Sausage|Quadratic von Koch curve (type 2)]] || align="center" |[[File:Quadratic Koch.svg|150px]] || Also called "Minkowski sausage".
|-
|<math>\log_2\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)</math> || align="right" | 1.5236 || Boundary of the [[Dragon curve]] || align="center" | [[File:Boundary dragon curve.png|150px]]|| cf. Chang & Zhang.<ref>[http://poignance.coiraweb.com/math/Fractals/Dragon/Bound.html Fractal dimension of the boundary of the dragon fractal]</ref><ref name="Boundary"/>
|-
|<math>\log_2\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)</math> || align="right" | 1.5236 || Boundary of the [[Dragon curve|twindragon curve]]|| align="center" |[[File:Twindragontile.png|150px]] || Can be built with two dragon curves. One of the six 2-[[rep-tile]]s in the plane (can be tiled by two copies of itself, of equal size).<ref name="2-reptiles"/>
|-
| <math>\log_2(3)</math> || align="right" | 1.5850 || 3-branches tree || align="center" | [[File:Arbre 3 branches.png|110px]] [[File:Arbre 3 branches2.png|110px]] || Each branch carries 3 branches (here 90° and 60°). The fractal dimension of the entire tree is the fractal dimension of the terminal branches. NB: the 2-branches tree has a fractal dimension of only 1.
|-
| <math>\log_2(3)</math> || align="right" | 1.5850 || [[Sierpinski triangle]] || align="center" | [[File:Sierpinski8.svg|100px]] || Also the triangle of Pascal modulo 2.
|-
| <math>\log_2(3)</math> || align="right" | 1.5850 || [[Sierpiński arrowhead curve]] || align="center" | [[File:PfeilspitzenFraktal.PNG|100px]] || Same limit as the triangle (above) but built with a one-dimensional curve.
|-
| <math>\log_2(3)</math> || align="right" | 1.5850 || Boundary of the [[T-square (fractal)|T-square]] fractal || align="center" | [[File:T-Square fractal (evolution).png|200px]] || The dimension of the fractal itself (not the boundary) is <math>\log_2(4)=2</math>
|-
| <math>\log_{\sqrt[\varphi]{\varphi}}(\varphi)=\varphi</math> || align="right" | 1.61803 || a golden [[dragon curve|dragon]] || align="center" | [[File:Phi glito.png|150px]] || Built from two similarities of ratios <math>r</math> and <math>r^2</math>, with <math>r = 1 / \varphi^{1/\varphi}</math>. Its dimension equals <math>\varphi</math> because <math>({r^2})^\varphi+r^\varphi = 1</math>. With <math>\varphi = (1+\sqrt{5})/2</math> ([[Golden ratio|Golden number]]).
|-
| <math>1+\log_3(2)</math> || align="right" | 1.6309 || [[Pascal triangle]] modulo 3 || align="center" | [[File:Pascal triangle modulo 3.png|160px]] || For a triangle modulo ''k'', if ''k'' is prime, the fractal dimension is <math>\scriptstyle{1 + \log_k\left(\frac{k+1}{2}\right)}</math> (cf. [[Stephen Wolfram]]<ref name="stephenwolfram.com">[http://www.stephenwolfram.com/publications/articles/ca/84-geometry/1/text.html Fractal dimension of the Pascal triangle modulo k]</ref>).
|-
| <math>1+\log_3(2)</math> || align="right" | 1.6309 || [[N-flake#Hexaflake|Sierpinski Hexagon]] || align="center" | [[File:Sierpinski hexagon 4th Iteration.svg|150px]] || Built in the manner of the [[Sierpinski carpet]], on an hexagonal grid, with 6 similitudes of ratio 1/3. The [[Koch snowflake]] is present at all scales.
|-
| <math>3\frac{\log(\varphi)}{\log (1+\sqrt{2})}</math> || align="right" | 1.6379 || [[Fibonacci word fractal]] || align="center" | [[File:Fibonacci fractal F23 steps.png|150px]] || Fractal based on the [[Fibonacci word]] (or Rabbit sequence) Sloane A005614. Illustration : [[Fractal curve]] after 23 steps (''F''23 = 28657 segments).<ref name="AMD">[http://hal.archives-ouvertes.fr/hal-00367972/en/ The Fibonacci word fractal]</ref> <math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| Solution of <math>(1/3)^s + (1/2)^s + (2/3)^s = 1</math> || align="right" | 1.6402 || Attractor of [[Iterated function system|IFS]] with 3 [[Similarity (geometry)|similarities]] of ratios 1/3, 1/2 and 2/3 || align="center" | [[File:IFS3sim3ratios.png|200px]] || Generalization : Providing the [[open set condition]] holds, the attractor of an [[iterated function system]] consisting of <math>n</math> similarities of ratios <math>c_n</math>, has Hausdorff dimension <math>s</math>, solution of the equation coinciding with the iteration function of the Euclidean contraction factor: <math>\sum_{k=1}^n c_k^s = 1</math>.<ref name="Falconer"/>
|-
|<math>\log_{8}(32)=\frac{5}{3}</math> || align="right" | 1.6667 || 32-segment quadric fractal (1/8 scaling rule)
||[[File:8 scale fractal.png|frameless]] see also: [[:File:32 Segment One Eighth Scale Quadric Fractal.jpg]]
||[[File:32SegmentSmall.jpg|thumb|Generator for 32 segment 1/8 scale quadric fractal.]]Built by scaling the 32 segment generator (see inset) by 1/8 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 32/log 8 = 1.6667. ''Images generated with Fractal Generator for ImageJ.''
|-
| <math>1+\log_5(3)</math> || align="right" | 1.6826 || [[Pascal triangle]] modulo 5 || align="center" | [[File:Pascal triangle modulo 5.png|160px]] || For a triangle modulo ''k'', if ''k'' is prime, the fractal dimension is <math>\scriptstyle{1 + \log_k\left(\frac{k+1}{2}\right)}</math> (cf. [[Stephen Wolfram]]<ref name="stephenwolfram.com"/>).
|-
| Measured (box-counting) || align="right" | 1.7 || [[Ikeda map]] attractor || align="center" | [[File:Ikeda map a=1 b=0.9 k=0.4 p=6.jpg|100px]] || For parameters a=1, b=0.9, k=0.4 and p=6 in the Ikeda map <math>z_{n+1} = a + bz_n \exp\left[i\left[k - p/\left(1 + \lfloor z_n \rfloor^2\right)\right]\right]</math>. It derives from a model of the plane-wave interactivity field in an optical ring laser. Different parameters yield different values.<ref>{{cite journal |first=James |last=Theiler |title=Estimating fractal dimension |journal=J. Opt. Soc. Am. A |volume=7 |issue=6 |pages=1055–73 |year=1990 |doi=10.1364/JOSAA.7.001055 |bibcode=1990JOSAA...7.1055T |url=http://public.lanl.gov/jt/Papers/est-fractal-dim.pdf }}</ref>
|-
| <math>1+\log_{10}(5)</math> || align="right" | 1.6990 || 50 segment quadric fractal (1/10 scaling rule) || align="center" | [[File:50seg.tif|150px]] || Built by scaling the 50 segment generator (see inset) by 1/10 for each iteration, and replacing each segment of the previous structure with a scaled copy of the entire generator. The structure shown is made of 4 generator units and is iterated 3 times. The fractal dimension for the theoretical structure is log 50/log 10 = 1.6990. ''Images generated with Fractal Generator for ImageJ''<ref>[http://rsb.info.nih.gov/ij/plugins/fractal-generator.html Fractal Generator for ImageJ] {{webarchive|url=https://web.archive.org/web/20120320124725/http://rsb.info.nih.gov/ij/plugins/fractal-generator.html |date=20 March 2012 }}.</ref>''.''[[File:50SegmentSmall.jpg|thumb|Generator for 50 Segment Fractal.]]
|-
| <math>4\log_5(2)</math> || align="right" | 1.7227 || [[Pinwheel tiling|Pinwheel fractal]] || align="center" | [[File:Pinwheel fractal.png|150px]] || Built with Conway's Pinwheel tile.
|-
| <math>\log_3(7)</math> || align="right" | 1.7712 || [[Sphinx tiling|Sphinx fractal]] || align="center" | [[File:Sphinx rep-tile fractal.gif|150px]] || Built with the Sphinx hexiamond tiling, removing two of the nine sub-sphinxes.<ref>W. Trump, G. Huber, C. Knecht, R. Ziff, to be published</ref>
|-
| <math>\log_3(7)</math> || align="right" | 1.7712 || [[Hexaflake]] || align="center" | [[File:HexaFlake_5th_Iteration_Center.svg|100px]] || Built by exchanging iteratively each hexagon by a flake of 7 hexagons. Its boundary is the von Koch flake and contains an infinity of Koch snowflakes (black or white).
|-
| <math>\log_3(7)</math> || align="right" | 1.7712 || Fractal H-I de Rivera || align="center" | [[File:Fractal H-I de Rivera.jpg|100px]] || Starting from a unit square dividing its dimensions into three equal parts to form nine self-similar squares with the first square, two middle squares (the one that is above and the one below the central square) are removed in each of the seven squares not eliminated the process is repeated, so it continues indefinitely.
|-
| <math>\frac{\log(4)}{\log(2+2\cos(85^\circ))}</math> || align="right" | 1.7848 || [[Koch curve|Von Koch curve 85°]] || align="center" | [[File:Koch Curve 85degrees.png|150px]] || Generalizing the von Koch curve with an angle ''a'' chosen between 0 and 90°. The fractal dimension is then <math>\frac{\log(4)}{\log(2+2\cos(a))} \in [1,2]</math>.
|-
| <math>\log_2\left(3^{0.63}+2^{0.63}\right)</math> || align="right" | 1.8272 || A self-[[affine transformation|affine]] fractal set || align="center" | [[File:Self-affine set.png|200px]] || Build iteratively from a <math>p \times q</math> array on a square, with <math>p \le q</math>. Its Hausdorff dimension equals <math>\log_p\left(\sum_{k=1}^p n_k^a\right)</math><ref name="Falconer"/> with <math>a=\log_q(p)</math> and <math>n_k</math> is the number of elements in the <math>k</math>th column. The [[Minkowski–Bouligand dimension|box-counting dimension]] yields a different formula, therefore, a different value. Unlike self-similar sets, the Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case.
|-
| <math>\frac{\log(6)}{\log(1+\varphi)}</math> || align="right" | 1.8617 || [[N-flake#Pentaflake|Pentaflake]] || align="center" | [[File:Pentaflake-C 3rd Iteration Blue.svg|100px]] || Built by exchanging iteratively each pentagon by a flake of 6 pentagons. <math>\varphi=(1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| solution of <math>6(1/3)^s+5{(1/3\sqrt{3})}^s=1</math> || align="right" | 1.8687 || Monkeys tree || align="center" | [[File:Monkeytree.svg|100px]] || This curve appeared in [[Benoit Mandelbrot]]'s "Fractal geometry of Nature" (1983). It is based on 6 similarities of ratio <math>1/3</math> and 5 similarities of ratio <math>1/3\sqrt{3}</math>.<ref>[http://www.coaauw.org/boulder-eyh/eyh_fractal.html Monkeys tree fractal curve] {{webarchive|url=https://archive.today/20020921135308/http://www.coaauw.org/boulder-eyh/eyh_fractal.html |date=21 September 2002 }}</ref>
|-
| <math>\log_3(8)</math> || align="right" | 1.8928 || [[Sierpinski carpet]] || align="center" | [[File:Sierpinski carpet 6.png|100px]] || Each face of the Menger sponge is a Sierpinski carpet, as is the bottom surface of the 3D quadratic Koch surface (type 1).
|-
| <math>\log_3(8)</math> || align="right" | 1.8928 || 3D [[Cantor dust]] || align="center" | [[File:Cantor3D3.png|200px]]|| Cantor set in 3 dimensions.
|-
| <math>\log_3(4)+\log_3(2)=\frac {\log(4)} {\log(3)}+\frac {\log(2)} {\log(3)}=\frac {\log(8)} {\log(3)}</math> || align="right" | {{formatnum:1.8928}} || Cartesian product of the [[von Koch curve]] and the [[Cantor set]] || align="center" | [[File:Koch Cantor cartesian product.png|150px]]|| Generalization : Let F×G be the cartesian product of two fractals sets F and G. Then <math>\dim_H(F \times G) = \dim_H(F) + \dim_H(G)</math>.<ref name="Falconer"/> See also the 2D [[Cantor dust]] and the [[Cantor cube]].
|-
|<math>2\log_2(x)</math> where <math> x^9-3x^8+3x^7-3x^6+2x^5+4x^4-8x^3+</math><math>8x^2-16x+8=0</math>|| align="right" | 1.9340 || Boundary of the [[Lévy C curve]] || align="center" | [[File:LevyFractal.png|100px]] || Estimated by Duvall and Keesling (1999). The curve itself has a fractal dimension of 2.
|-
| || align="right" | 2 || [[Penrose tiling]] || align="center" |[[File:pen0305c.gif|100px]] || See Ramachandrarao, Sinha & Sanyal.<ref>[http://www.iisc.ernet.in/currsci/aug102000/rc80.pdf Fractal dimension of a Penrose tiling]</ref>
|-
| <math>2</math> || align="right" | 2 || Boundary of the [[Mandelbrot set]] || align="center" | [[File:Boundary mandelbrot set.png|100px]] || The boundary and the set itself have the same Hausdorff dimension.<ref name=Shishikura91>{{cite arXiv |first=Mitsuhiro |last=Shishikura |title=The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets |date=1991 |eprint=math/9201282}}</ref>
|-
| <math>2</math> || align="right" | 2 || [[Julia set]] || align="center" | [[File:Juliadim2.png|150px]] || For determined values of ''c'' (including ''c'' [[Misiurewicz point|belonging to the boundary]] of the Mandelbrot set), the Julia set has a dimension of 2.<ref name=Shishikura91/>
|-
| <math>2</math> || align="right" | 2 || [[Sierpiński curve]] || align="center" | [[File:Sierpinski-Curve-3.png|100px]] || Every [[Peano curve]] filling the plane has a Hausdorff dimension of 2.
|-
| <math>2</math> || align="right" | 2 || [[Hilbert curve]] || align="center" | [[File:Hilbert curve 3.svg|100px]]||
|-
| <math>2</math> || align="right" | 2 || [[Peano curve]] || align="center" | [[File:Peano curve.png|100px]]|| And a family of curves built in a similar way, such as the [[Wunderlich curves]].
|-
| <math>2</math> || align="right" | 2 || [[Moore curve]] || align="center" | [[File:Moore-curve-stages-1-through-4.svg|100px]]|| Can be extended in 3 dimensions.
|-
| || align="right" | 2 || [[z-order (curve)|Lebesgue curve or z-order curve]] || align="center" | [[File:z-order curve.png|100px]]|| Unlike the previous ones this space-filling curve is differentiable almost everywhere. Another type can be defined in 2D. Like the Hilbert Curve it can be extended in 3D.<ref>[http://www.mathcurve.com/fractals/lebesgue/lebesgue.shtml Lebesgue curve variants]</ref>
|-
| <math>\log_\sqrt{2}(2)=2</math> || align="right" | 2 || [[Dragon curve]] || align="center" | [[File:Courbe du dragon.png|150px]]|| And its boundary has a fractal dimension of 1.5236270862.<ref>{{cite arXiv |first=Jarek |last=Duda |title=Complex base numeral systems |date=2008 |class=math.DS |eprint=0712.1309v3}}</ref>
|-
| || align="right" | 2 || [[Dragon curve|Terdragon curve]] || align="center" | [[File:Terdragon curve.png|150px]]|| L-system: ''F'' → ''F'' + F – F, angle = 120°.
|-
| <math>\log_2(4)=2</math> || align="right" | 2 || [[Gosper curve]] || align="center" | [[File:Gosper curve 3.svg|100px]]|| Its boundary is the Gosper island.
|-
| Solution of <math>7({1/3})^s+6({1/3\sqrt{3}})^s=1</math> || align="right" | 2 || Curve filling the [[Koch snowflake]] || align="center" | [[File:Mandeltree.svg|100px]]|| Proposed by Mandelbrot in 1982,<ref>{{cite book |title=Penser les mathématiques |last=Seuil |isbn=2-02-006061-2 |year=1982}}</ref> it fills the [[Koch snowflake]]. It is based on 7 similarities of ratio 1/3 and 6 similarities of ratio <math>1/3\sqrt{3}</math>.
|-
| <math>\log_2(4)=2</math> || align="right" | 2 || [[Sierpiński triangle|Sierpiński tetrahedron]] || align="center" | [[File:Tetraedre Sierpinski.png|80px]]|| Each [[tetrahedron]] is replaced by 4 tetrahedra.
|-
| <math>\log_2(4)=2</math> || align="right" | 2 || [[H-fractal]] || align="center" |[[File:H fractal2.png|150px]]|| Also the [[Mandelbrot tree]] which has a similar pattern.
|-
| <math>\frac{\log(2)}{\log(2/\sqrt{2})}=2</math> || align="right" | {{formatnum:2}} || [[Pythagoras tree (fractal)]] || align="center" |[[File:PythagorasTree.png|150px]]|| Every square generates two squares with a reduction ratio of <math>1/\sqrt{2}</math>.
|-
| <math>\log_2(4)=2</math> || align="right" | 2 || [[Iterated function system#Example: Greek cross fractal|2D Greek cross fractal]] || align="center" |[[File:Greek cross fractal stage 4.svg|100px]] || Each segment is replaced by a cross formed by 4 segments.
|-
| Measured || align="right" | 2.01 ±0.01|| [[Rössler attractor]] || align="center" | [[File:Roessler attractor.png|100px]] || The fractal dimension of the Rössler attractor is slightly above 2. For a=0.1, b=0.1 and c=14 it has been estimated between 2.01 and 2.02.<ref>[http://www.ocf.berkeley.edu/~trose/rossler.html Fractals and the Rössler attractor]</ref>
|-
| Measured || align="right" | 2.06 ±0.01|| [[Lorenz attractor]] || align="center" |[[File:Lorenz attractor.png|100px]] || For parameters <math>\rho=40</math>,<math>\sigma</math>=16 and <math>\beta=4</math> . See McGuinness (1983)<ref>{{cite journal |first=M.J. |last=McGuinness |title=The fractal dimension of the Lorenz attractor |journal=Physics Letters |volume=99A |pages=5–9 |year=1983 |issue=1 |doi=10.1016/0375-9601(83)90052-X |bibcode=1983PhLA...99....5M }}</ref>
|-
|<math>4+c^D+d^D=(c+d)^D</math>|| align="right" |2<D<2.3 || Pyramid surface || align="center" |[[File:Pyramid surface fractal.png|200px]]|| Each triangle is replaced by 6 triangles, of which 4 identical triangles form a diamond based pyramid and the remaining two remain flat with lengths <math>c</math> and <math>d</math> relative to the pyramid triangles. The dimension is a parameter, self-intersection occurs for values greater than 2.3.<ref>{{Cite journal|last=Lowe|first=Thomas|date=24 October 2016|title=Three Variable Dimension Surfaces|url=https://www.researchgate.net/publication/309391846|journal=ResearchGate}}</ref>
|-
| <math>\log_2(5)</math>|| align="right" | 2.3219 || Fractal pyramid || align="center" |[[File:Fractal pyramid.jpg|100px]]|| Each [[square pyramid]] is replaced by 5 half-size square pyramids. (Different from the Sierpinski tetrahedron, which replaces each [[triangular pyramid]] with 4 half-size triangular pyramids).
|-
| <math>\frac{\log(20)}{\log(2+\varphi)}</math>|| align="right" | 2.3296 || [[N-flake#Dodecahedron flake|Dodecahedron fractal]]|| align="center" |[[File:Dodecaedron fractal.jpg|100px]]|| Each [[dodecahedron]] is replaced by 20 dodecahedra. <math>\varphi = (1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| <math>\log_3(13)</math> || align="right" | 2.3347 || [[Koch curve|3D quadratic Koch surface (type 1)]] || align="center" |[[Image:koch_quadratic_3d_fractal.svg|150px]]|| Extension in 3D of the quadratic Koch curve (type 1). The illustration shows the first (blue block), second (plus green blocks), third (plus yellow blocks) and fourth (plus clear blocks) iterations.
|-
| || align="right" | 2.4739 || [[Apollonian sphere packing]] || align="center" |[[File:Apollonian spheres2.png|100px]] || The interstice left by the Apollonian spheres. Apollonian gasket in 3D. Dimension calculated by M. Borkovec, W. De Paris, and R. Peikert.<ref>[http://www.scivis.ethz.ch/publications/pdf/1994/borkovec1994fractal.pdf The Fractal dimension of the apollonian sphere packing] {{webarchive|url=https://web.archive.org/web/20160506190118/http://www.scivis.ethz.ch/publications/pdf/1994/borkovec1994fractal.pdf |date=6 May 2016 }}</ref>
|-
| <math>\log_4(32)=\frac{5}{2}</math> || align="right" | 2.50 || [[Koch curve|3D quadratic Koch surface (type 2)]] || align="center" |[[File:Quadratic Koch 3D (type2 stage2).png|150px]]|| Extension in 3D of the quadratic Koch curve (type 2). The illustration shows the second iteration.
|-
| <math>\frac{\log\left(\frac{\sqrt{7}}{6}-\frac{1}{3}\right)}{\log(\sqrt2-1)}</math> || align="right" | 2.529 || [[Jerusalem cube]] || align="center" | [[File:Jerusalem Cube.jpg|150px]] || The iteration n is built with 8 cubes of iteration n-1 (at the corners) and 12 cubes of iteration n-2 (linking the corners). The contraction ratio is <math>\sqrt{2}-1</math>.
|-
| <math>\frac{\log(12)}{\log(1+\varphi)}</math> || align="right" | 2.5819 || [[N-flake#Icosahedron flake|Icosahedron fractal]] || align="center" |[[File:Icosaedron fractal.jpg|100px]]|| Each [[icosahedron]] is replaced by 12 icosahedra. <math>\varphi=(1+\sqrt{5})/2</math> ([[golden ratio]]).
|-
| <math>1+\log_2(3)</math> || align="right" | 2.5849 || [[Iterated function system#Example: Greek cross fractal|3D Greek cross fractal]] || align="center" |[[File:Greek cross 3D 1 through 4.png|200px]]|| Each segment is replaced by a cross formed by 6 segments.
|-
| <math>1+\log_2(3)</math> || align="right" | 2.5849 || [[N-flake#Octahedron flake|Octahedron fractal]] || align="center" |[[File:Octaedron fractal.jpg|100px]]|| Each [[octahedron]] is replaced by 6 octahedra.
|-
| <math>1+\log_2(3)</math> || align="right" | 2.5849 || [[Koch curve|von Koch surface]] || align="center" |[[File:Koch surface 3.png|150px]]|| Each equilateral triangular face is cut into 4 equal triangles.
Using the central triangle as the base, form a tetrahedron. Replace the triangular base with the tetrahedral "tent".
|-
| <math>\frac{\log(3)}{\log(3/2)}</math> || align="right" | 2.7095 || [[Koch curve|Von Koch in 3D]] || align="center" | [[File:Koch Curve in Three Dimensions ("Delta" fractal).jpg|100px]] || Start with a 6-sided polyhedron whose faces are isosceles triangles with sides of ratio 2:2:3 . Replace each polyhedron with 3 copies of itself, 2/3 smaller.<ref>[https://www.researchgate.net/publication/262600735_The_Koch_curve_in_three_dimensions]</ref>
|-
| <math>\log_3(20)</math> || align="right" | 2.7268 || [[Menger sponge]] || align="center" | [[File:Menger.png|100px]] || And its surface has a fractal dimension of <math>\log_3(20)</math>, which is the same as that by volume.
|-
| <math>\log_2(8)=3</math> || align="right" | 3 || [[Hilbert curve|3D Hilbert curve]] || align="center" | [[File:Hilbert3d-step3.png|100px]]|| A Hilbert curve extended to 3 dimensions.
|-
| <math>\log_2(8)=3</math> || align="right" | 3 || [[z-order (curve)|3D Lebesgue curve]] || align="center" | [[File:Lebesgue-3d-step3.png|100px]]|| A Lebesgue curve extended to 3 dimensions.
|-
| <math>\log_2(8)=3</math> || align="right" | 3 || [[Moore curve|3D Moore curve]] || align="center" | [[File:Moore3d-step3.png|100px]]|| A Moore curve extended to 3 dimensions.
|-
| <math>\log_2(8)=3</math> || align="right" | 3 || 3D [[H-fractal]] || align="center" | [[File:3D H-fractal.png|120px]]|| A H-fractal extended to 3 dimensions.<ref>{{cite journal |first1=B. |last1=Hou |first2=H. |last2=Xie |first3=W. |last3=Wen |first4=P. |last4=Sheng | year = 2008
| title = Three-dimensional metallic fractals and their photonic crystal characteristics |journal= Phys. Rev. B |volume=77 |issue=12 |page=125113
|bibcode=2008PhRvB..77l5113H |doi=10.1103/PhysRevB.77.125113 |url=http://repository.ust.hk/ir/bitstream/1783.1-25969/1/PhysRevB.77.125113.pdf }}</ref>
|-
| <math>3</math> (conjectured) || align="right" | {{formatnum:3}} (to be confirmed) || [[Mandelbulb]] || align="center" |[[File:Mandelbulb 1,024GP Overview 20211110 002 ALT.png|100px]]|| Extension of the Mandelbrot set (power 9) in 3 dimensions<ref>[http://www.fractalforums.com/theory/hausdorff-dimension-of-the-mandelbulb/15/ Hausdorff dimension of the Mandelbulb]</ref>{{Unreliable source?|date=September 2011}}
|}

==Random and natural fractals==
{| class="wikitable"
|-
! Hausdorff dimension (exact value) || Hausdorff dimension (approx.) || Name || Illustration || width="40%" | Remarks
|-
|1/2 || align="right" | 0.5 || Zeros of a [[Wiener process]] || align="center" |[[File:Wiener process set of zeros.gif|150px]] || The zeros of a Wiener process (Brownian motion) are a [[nowhere dense set]] of [[Lebesgue measure]] 0 with a fractal structure.<ref name="Falconer"/><ref>Peter Mörters, Yuval Peres, Oded Schramm, "Brownian Motion", Cambridge University Press, 2010</ref>
|-
| Solution of <math>E(C_1^s + C_2^s)=1</math> where <math>E(C_1)=0.5</math> and <math>E(C_2)=0.3</math>|| align="right" | 0.7499 || a random [[Cantor set]] with 50% - 30% || align="center" |[[File:Random Cantor set.png|150px]] || Generalization: at each iteration, the length of the left interval is defined with a random variable <math>C_1</math>, a variable percentage of the length of the original interval. Same for the right interval, with a random variable <math>C_2</math>. Its Hausdorff Dimension <math>s</math> satisfies: <math>E(C_1^s + C_2^s)=1</math> (where <math>E(X)</math> is the [[expected value]] of <math>X</math>).<ref name="Falconer"/>
|-
|Solution of <math>s+1=12\cdot2^{-(s+1)}-6\cdot3^{-(s+1)}</math>||align="right"|1.144...||[[von Koch curve]] with random interval||align="center"| [[File:Random interval koch.png|200px]] || The length of the middle interval is a random variable with uniform distribution on the interval (0,1/3).<ref name="Falconer"/>
|-
|Measured||align="right"|1.22±0.02||Coastline of Ireland||align="center"| [[File:Ireland (MODIS).jpg|150px]] || Values for the fractal dimension of the entire coast of Ireland were determined by McCartney, Abernethy and Gault<ref>{{cite journal|last1=McCartney|first1=Mark|first2=Gavin |last2=Abernethya |first3=Lisa |last3=Gaulta|title=The Divider Dimension of the Irish Coast|journal=Irish Geography|date=24 June 2010|volume=43|issue=3|pages=277–284|doi=10.1080/00750778.2011.582632}}</ref> at the [[University of Ulster]] and [[Theoretical Physics]] students at [[Trinity College, Dublin]], under the supervision of S. Hutzler.<ref name="S.Hutzler">{{cite journal|last1=Hutzler |first1=S. |title=Fractal Ireland |journal=Science Spin |date=2013 |volume=58 |pages=19–20 |url=https://issuu.com/spin35/docs/spin_58_all |access-date=15 November 2016 }}
(See [https://web.archive.org/web/20130726164417/http://www.sciencespin.com/magazine/archive/2013/05/ contents page], archived 26 July 2013)</ref>

Note that there are marked differences between Ireland's ragged west coast (fractal dimension of about 1.26) and the much smoother east coast (fractal dimension 1.10)<ref name="S.Hutzler" />
|-
|Measured||align="right"|1.25||[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|Coastline of Great Britain]]||align="center"| [[File:Britain-fractal-coastline-combined.jpg|200px]] || Fractal dimension of the west coast of Great Britain, as measured by [[Lewis Fry Richardson]] and cited by [[Benoît Mandelbrot]].<ref>[http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf How long is the coast of Britain? Statistical self-similarity and fractional dimension], B. Mandelbrot</ref>
|-
| <math>\frac {\log(4)} {\log(3)}</math> || align="right" | 1.2619 || [[von Koch curve]] with random orientation || align="center" | [[File:Random orientation koch.png|200px]] || One introduces here an element of randomness which does not affect the dimension, by choosing, at each iteration, to place the equilateral triangle above or below the curve.<ref name="Falconer"/>
|-
|<math>\frac {4}{3}</math> || align="right" | 1.333 || Boundary of Brownian motion || align="center" |[[File:Front mouvt brownien.png|150px]] || (cf. Mandelbrot, [[Gregory Lawler|Lawler]], [[Oded Schramm|Schramm]], [[Wendelin Werner|Werner]]).<ref>{{cite journal |first1=Gregory F. |last1=Lawler |first2=Oded |last2=Schramm |first3=Wendelin |last3=Werner |title=The Dimension of the Planar Brownian Frontier is 4/3 |journal=Math. Res. Lett. |volume=8 |issue=4 |pages=401–411 |year=2001 |arxiv=math/0010165|bibcode=2000math.....10165L |doi=10.4310/MRL.2001.v8.n4.a1 |s2cid=5877745 }}</ref>
|-
|<math>\frac {4}{3}</math> || align="right" | 1.333 || [[Polymer]] in 2D || align="center" | || Similar to the Brownian motion in 2D with non-self-intersection.<ref name="sapoval">{{cite book |first=Bernard |last=Sapoval |title=Universalités et fractales |publisher=Flammarion-Champs |year=2001 |isbn=2-08-081466-4 }}</ref>
|-
|<math>\frac {4}{3}</math> || align="right" | 1.333 || [[Percolation]] front in 2D, [[Corrosion]] front in 2D || align="center" | [[File:Front de percolation.png|150px]] || Fractal dimension of the percolation-by-invasion front (accessible perimeter), at the [[percolation threshold]] (59.3%). It's also the fractal dimension of a stopped corrosion front.<ref name="sapoval" />
|-
| || align="right" | 1.40 || [[diffusion-limited aggregation|Clusters of clusters 2D]] || align="center" | || When limited by diffusion, clusters combine progressively to a unique cluster of dimension 1.4.<ref name="sapoval" />
|-
| <math>2-\frac{1}{2}</math>|| align="right" | 1.5|| Graph of a regular [[Fractional Brownian motion|Brownian]] function ([[Wiener process]]) || align="center" | [[File:Wiener process zoom.png|150px]] || Graph of a function <math>f</math> such that, for any two positive reals <math>x</math> and <math>x+h</math>, the difference of their images <math>f(x+h)-f(x)</math> has the centered gaussian distribution with variance <math>= h</math>. Generalization: the [[fractional Brownian motion]] of index <math>\alpha</math> follows the same definition but with a variance <math>= h^{2\alpha}</math>, in that case its Hausdorff dimension <math>=2-\alpha</math>.<ref name="Falconer" />
|-
| Measured|| align="right" | 1.52|| [[Fjord|Coastline of Norway]] || align="center" |[[File:Norway municipalities 2020 blank.svg|100px]] || See J. Feder.<ref>Feder, J., "Fractals,", Plenum Press, New York, (1988).</ref>
|-
| Measured|| align="right" | 1.55 || [[Self-avoiding walk]] || align="center" | [[File:Polymer 2D.png|150px]]|| Random walk in a square lattice that avoids visiting the same place twice, with a "go-back" routine for avoiding dead ends.
|-
| <math>\frac {5} {3}</math>|| align="right" | 1.66|| Polymer in 3D || align="center" | || Similar to the Brownian motion in a cubic lattice, but without self-intersection.<ref name="sapoval" />
|-
| || align="right" | 1.70 || [[Diffusion-limited aggregation|2D DLA Cluster]] || align="center" | [[File:Aggregation limitee par diffusion.png|150px]]|| In 2 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 1.70.<ref name="sapoval" />
|-
| <math>\frac {\log(9\cdot0.75)} {\log(3)}</math>|| align="right" | 1.7381|| Fractal percolation with 75% probability|| align="center" |[[File:Fractal percolation 75.png|150px]] || The fractal percolation model is constructed by the progressive replacement of each square by a <math>3\times3</math> grid in which is placed a random collection of sub-squares, each sub-square being retained with probability ''p''. The "almost sure" Hausdorff dimension equals <math>\textstyle{\frac {\log(9p)} {\log(3)}}</math>.<ref name="Falconer" />
|-
| 7/4 || align="right" | 1.75 || 2D percolation cluster hull || align="center" | [[File:PercolationHull.png|150px]]|| The hull or boundary of a percolation cluster. Can also be generated by a hull-generating walk,<ref>[http://deepblue.lib.umich.edu/handle/2027.42/27787 Hull-generating walks]</ref> or by Schramm-Loewner Evolution.
|-
| <math>\frac {91} {48}</math> || align="right" | 1.8958 || [[2D percolation cluster]] || align="center" | [[File:Amas de percolation.png|150px]] || In a square lattice, under the site [[percolation threshold]] (59.3%) the percolation-by-invasion cluster has a fractal dimension of 91/48.<ref name="sapoval" /><ref name="Sahimi">{{cite book|author1=M Sahini|author2=M Sahimi|title=Applications Of Percolation Theory|url=https://books.google.com/books?id=MJwqsbWBc-YC|year=2003|publisher=CRC Press|isbn=978-0-203-22153-2}}</ref> Beyond that threshold, the cluster is infinite and 91/48 becomes the fractal dimension of the "clearings".
|-
| <math>\frac {\log(2)} {\log(\sqrt{2})} = 2</math> || align="right" | 2 || [[Brownian motion]] || align="center" | [[File:Mouvt brownien2.png|150px]]|| Or random walk. The Hausdorff dimensions equals 2 in 2D, in 3D and in all greater dimensions (K.Falconer "The geometry of fractal sets").
|-
| Measured || align="right" | Around 2 || Distribution of [[galaxy cluster]]s || align="center" | [[File:Abell 1835 Hubble.jpg|100px]]|| From the 2005 results of the Sloan Digital Sky Survey.<ref>[https://arxiv.org/abs/astro-ph/0501583v2 Basic properties of galaxy clustering in the light of recent results from the Sloan Digital Sky Survey]</ref>
|-
| || align="right" | 2.5 || Balls of crumpled paper || align="center" | [[File:Paperball.png|100px]] || When crumpling sheets of different sizes but made of the same type of paper and with the same aspect ratio (for example, different sizes in the [[ISO 216]] A series), then the diameter of the balls so obtained elevated to a non-integer exponent between 2 and 3 will be approximately proportional to the area of the sheets from which the balls have been made.<ref>{{Cite journal | publisher=Yale | url=http://classes.yale.edu/fractals/FracAndDim/BoxDim/PowerLaw/CrumpledPaper.html | title=Power Law Relations | access-date=29 July 2010 | url-status=dead | archive-url=https://web.archive.org/web/20100628020140/http://classes.yale.edu/fractals/FracAndDim/BoxDim/PowerLaw/CrumpledPaper.html | archive-date=28 June 2010 | df=dmy-all }}</ref> Creases will form at all size scales (see [[Universality (dynamical systems)]]).
|-
| || align="right" | 2.50 || [[diffusion-limited aggregation|3D DLA Cluster]] || align="center" | [[File:3D DLA.jpg|150px]] || In 3 dimensions, clusters formed by diffusion-limited aggregation, have a fractal dimension of around 2.50.<ref name="sapoval" />
|-
| || align="right" | 2.50 || [[Lichtenberg figure]] || align="center" | [[File:PlanePair2.jpg|100px]] || Their appearance and growth appear to be related to the process of diffusion-limited aggregation or DLA.<ref name="sapoval" />
|-
| <math>3-\frac{1}{2}</math>|| align="right" | 2.5|| regular [[Brownian surface]]|| align="center" | [[File:Brownian surface.png|150px]] || A function <math>f:\mathbb{R}^2 \to \mathbb{R}</math>, gives the height of a point <math>(x,y)</math> such that, for two given positive increments <math>h</math> and <math>k</math>, then <math>f(x+h,y+k)-f(x,y)</math> has a centered Gaussian distribution with variance = <math>\sqrt{h^2+k^2}</math>. Generalization: the [[fractional Brownian motion|fractional Brownian]] surface of index <math>\alpha</math> follows the same definition but with a variance <math>=(h^2+k^2)^\alpha</math>, in that case its Hausdorff dimension <math>=3-\alpha</math>.<ref name="Falconer" />
|-
| Measured || align="right" | 2.52 || 3D [[Percolation theory|percolation]] cluster || align="center" |[[File:3Dpercolation.png|225px]] || In a cubic lattice, at the site [[percolation threshold]] (31.1%), the 3D percolation-by-invasion cluster has a fractal dimension of around 2.52.<ref name="Sahimi"/> Beyond that threshold, the cluster is infinite.
|-
|Measured and calculated|| align="right" |~2.7|| The surface of [[Broccoli]] || align="center" | [[File:Broccoli DSC00862.png|100px]] ||San-Hoon Kim used a direct scanning method and a cross section analysis of a broccoli to conclude that the fractal dimension of it is ~2.7.<ref name=":0">{{Cite arXiv |eprint=cond-mat/0411597|title=Fractal dimensions of a green broccoli and a white cauliflower|last=Kim|first=Sang-Hoon|date=2 February 2008}}</ref>
|-
| || align="right" | 2.79 || Surface of [[Cerebral cortex|human brain]] || align="center" | [[File:Cerebellum NIH.png|100px]] ||<ref>[http://www.heise.de/tr/artikel/54311/2/0 Fractal dimension of the surface of the human brain]</ref>{{Failed verification|date=November 2018}}
|-
|Measured and calculated|| align="right" |~2.8|| [[Cauliflower]] || align="center" | [[File:Blumenkohl-1.jpg|100px]]||San-Hoon Kim used a direct scanning method and a mathematical analysis of the cross section of a cauliflower to conclude that the fractal dimension of it is ~2.8.<ref name=":0" />
|-
| || align="right" | 2.97 || Lung surface || align="center" |[[File:Thorax Lung 3d (2).jpg|100px]] || The alveoli of a lung form a fractal surface close to 3.<ref name="sapoval" />
|-
| Calculated || align="right" | <math>\in(0,2)</math> || [[Multiplicative cascade]] || align="center" | [[File:3fractals2.jpg|150px]] || This is an example of a [[multifractal]] distribution. However, by choosing its parameters in a particular way we can force the distribution to become a monofractal.<ref>[Meakin (1987)]</ref>{{Full citation needed|date=October 2018}}
|}

==See also==
{{Commons|Fractal|fractals}}
* [[Fractal dimension]]
* [[Hausdorff dimension]]
* [[Scale invariance]]

==Notes and references==
{{Reflist|30em}}

==Further reading==
*{{cite book |first=Benoît |last=Mandelbrot |title=The Fractal Geometry of Nature |publisher=W.H. Freeman |year=1982 |isbn=0-7167-1186-9 |url-access=registration |url=https://archive.org/details/fractalgeometryo00beno }}
*{{cite book |first=Heinz-Otto |last=Peitgen |editor-first=Dietmar |editor-last=Saupe |title=The Science of Fractal Images |publisher=Springer Verlag |year=1988 |isbn=0-387-96608-0 |url-access=registration |url=https://archive.org/details/scienceoffractal0000unse }}
*{{cite book |first=Michael F. |last=Barnsley |title=Fractals Everywhere |publisher=Morgan Kaufmann |isbn=0-12-079061-0 |date=1 January 1993 }}
*{{cite book |first1=Bernard |last1=Sapoval |first2=Benoît B. |last2=Mandelbrot |title=Universalités et fractales: jeux d'enfant ou délits d'initié? |publisher=Flammarion-Champs |year=2001 |isbn=2-08-081466-4 }}

==External links==
* [http://mathworld.wolfram.com/search/?query=fractal The fractals on Mathworld]
* [https://web.archive.org/web/20060905203033/http://local.wasp.uwa.edu.au/~pbourke/fractals/ Other fractals on Paul Bourke's website]
* [http://soler7.com/Fractals/FractalsSite.html Soler's Gallery]
* [http://www.mathcurve.com/fractals/fractals.shtml Fractals on mathcurve.com]
* [http://1000fractales.free.fr/index.htm 1000fractales.free.fr - Project gathering fractals created with various software]
* [https://web.archive.org/web/20060923100014/http://library.thinkquest.org/26242/full/index.html Fractals unleashed]
* [https://ifstile.com IFStile - software that computes the dimension of the boundary of self-affine tiles]

{{Fractals}}

{{DEFAULTSORT:List of Fractals By Hausdorff Dimension}}
[[Category:Fractals]]
[[Category:Fractal curves]]
[[Category:Mathematics-related lists|Fractals by Hausdorff dimension]]

Minkowski–Bouligand dimension

2022-02-07T02:18:38Z

IntegralPython: change blue link to new article

{{short description|Way of determining the dimension of a fractal set in a Euclidean space by counting the number of fixed-size boxes needed to cover the set as a function of the box size}}
[[image:Great Britain Box.svg|thumb|450px|Estimating the box-counting dimension of the coast of Great Britain]]
In [[fractal geometry]], the '''Minkowski–Bouligand dimension''', also known as '''Minkowski dimension''' or '''box-counting dimension''', is a way of determining the [[fractal dimension]] of a [[Set (mathematics)|set]] ''S'' in a [[Euclidean space]] '''R'''''n'', or more generally in a [[metric space]] (''X'', ''d''). It is named after the [[Poland|Polish]] [[mathematician]] [[Hermann Minkowski]] and the [[France|French]] mathematician [[Georges Bouligand]].

To calculate this dimension for a fractal ''S'', imagine this fractal lying on an evenly spaced grid, and count how many boxes are required to [[cover (topology)|cover]] the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a [[box counting|box-counting]] algorithm.

Suppose that ''N''(''ε'') is the number of boxes of side length ε required to cover the set. Then the box-counting dimension is defined as:

:<math>\dim_{\rm box}(S) := \lim_{\varepsilon \to 0} \frac {\log N(\varepsilon)}{\log (1/\varepsilon)}.</math>

Roughly speaking, this means the dimension is the exponent ''d'' such that ''N''(1/''n'') ≈ ''C nd'', which is what one would expect in the trivial case where ''S'' is a smooth space (a manifold) of integer dimension d.

If the above [[limit of a function|limit]] does not exist, one may still take the [[limit superior and limit inferior]], which respectively define the '''upper box dimension''' and '''lower box dimension'''. The upper box dimension is sometimes called the '''entropy dimension''', '''Kolmogorov dimension''', '''Kolmogorov capacity''', '''limit capacity''' or '''upper Minkowski dimension''', while the lower box dimension is also called the '''lower Minkowski dimension'''.

The upper and lower box dimensions are strongly related to the more popular [[Hausdorff dimension]]. Only in very special applications is it important to distinguish between the three (see [[#Relations to the Hausdorff dimension|below]]). Yet another measure of fractal dimension is the [[correlation dimension]].

== Alternative definitions ==
[[image:Great Britain coverings.svg|thumb|350px|Examples of ball packing, ball covering, and box covering.]]
It is possible to define the box dimensions using balls, with either the [[covering number]] or the packing number. The covering number <math>N_{\rm covering}(\varepsilon)</math> is the ''minimal'' number of [[open ball]]s of radius ε required to [[cover (topology)|cover]] the fractal, or in other words, such that their union contains the fractal. We can also
consider the intrinsic covering number <math>N'_{\rm covering}(\varepsilon)</math>, which is defined the same way but with the additional requirement that the centers of the open balls lie inside the set ''S''. The packing number <math>N_{\rm packing}(\varepsilon)</math> is the ''maximal'' number of [[Disjoint sets|disjoint]] open balls of radius ε one can situate such that their centers would be inside the fractal. While ''N'', ''N''covering, ''N'''covering and ''N''packing are not exactly identical, they are closely related, and give rise to identical definitions of the upper and lower box dimensions. This is easy to prove once the following inequalities are proven:

:<math> N_\text{packing}(\varepsilon) \leq N'_\text{covering}(\varepsilon) \leq N_\text{covering}(\varepsilon/2). \, </math>

These, in turn, follow with a little effort from the [[triangle inequality]].

The advantage of using balls rather than squares is that this definition generalizes to any [[metric space]]. In other words, the box definition is [[Differential_geometry#Intrinsic_versus_extrinsic|extrinsic]] — one assumes the fractal space ''S'' is contained in a [[Euclidean space]], and defines boxes according to the external geometry of the containing space. However, the dimension of ''S'' should be [[Differential_geometry#Intrinsic_versus_extrinsic|intrinsic]], independent of the environment into which ''S'' is placed, and the ball definition can be formulated intrinsically. One defines an internal ball as all points of ''S'' within a certain distance of a chosen center, and one counts such balls to get the dimension. (More precisely, the ''N''covering definition is extrinsic, but the other two are intrinsic.)

The advantage of using boxes is that in many cases ''N''(''ε'') may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal.

The [[logarithm]] of the packing and covering numbers are sometimes referred to as ''entropy numbers'', and are somewhat analogous to the concepts of [[entropy|thermodynamic entropy]] and [[entropy (information theory)|information-theoretic entropy]], in that they measure the amount of "disorder" in the metric space or fractal at scale ''ε'', and also measure how many bits or digits one would need to specify a point of the space to accuracy ''ε''.

Another equivalent (extrinsic) definition for the box-counting dimension, is given by the formula:

:<math>\dim_\text{box}(S) = n - \lim_{r \to 0} \frac{\log \text{vol}(S_r)}{\log r},</math>

where for each ''r'' > 0, the set <math>S_r</math> is defined to be the ''r''-neighborhood of ''S'', i.e. the set of all points in <math>R^n</math> which are at distance less than ''r'' from ''S'' (or equivalently, <math>S_r</math> is the union of all the open balls of radius ''r'' which are centered at a point in ''S'').

== Properties ==
Both box dimensions are finitely additive, i.e. if { ''A''1, .... ''A''''n'' } is a finite collection of sets then

:<math>\dim (A_1 \cup \dotsb \cup A_n) = \max \{ \dim A_1 ,\dots, \dim A_n \}. \, </math>

However, they are not [[countable set|countably]] additive, i.e. this equality does not hold for an ''infinite'' sequence of sets. For example, the box dimension of a single point is 0, but the box dimension of the collection of [[rational number]]s in the interval [0, 1] has dimension 1. The [[Hausdorff measure]] by comparison, is countably additive.

An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. If ''A'' and ''B'' are two sets in a Euclidean space then ''A'' + ''B'' is formed by taking all the pairs of points ''a,b'' where ''a'' is from ''A'' and ''b'' is from ''B'' and adding ''a+b''. One has

:<math>\dim_\text{upper box}(A+B)\leq \dim_\text{upper box}(A)+\dim_\text{upper box}(B).</math>

== Relations to the Hausdorff dimension ==
The box-counting dimension is one of a number of definitions for dimension that can be applied to fractals. For many well behaved fractals all these dimensions are equal; in particular, these dimensions coincide whenever the fractal satisfies the [[open set condition|open set condition (OSC)]].<ref name=Wagon214>{{cite book | title=Mathematica® in Action: Problem Solving Through Visualization and Computation | first=Stan | last=Wagon | publisher=[[Springer-Verlag]] | year=2010 | isbn=0-387-75477-6 | page=214 }}</ref> For example, the [[Hausdorff dimension]], lower box dimension, and upper box dimension of the [[Cantor set]] are all equal to log(2)/log(3). However, the definitions are not equivalent.

The box dimensions and the Hausdorff dimension are related by the inequality

:<math>\dim_{\operatorname{Haus}} \leq \dim_{\operatorname{lower box}} \leq \dim_{\operatorname{upper box}}.</math>

In general both inequalities may be [[strict inequality|strict]]. The upper box dimension may be bigger than the lower box dimension if the fractal has different behaviour in different scales. For example, examine the set of numbers in the interval [0,1] satisfying the condition

:for any ''n'', all the digits between the 22''n''-th digit and the (22''n''+1 − 1)th digit are zero

The digits in the "odd place-intervals", i.e. between digits 22''n''+1 and 22''n''+2 − 1 are not restricted and may take any value. This fractal has upper box dimension 2/3 and lower box dimension 1/3, a fact which may be easily verified by calculating ''N''(''ε'') for <math>\varepsilon=10^{-2^n}</math> and noting that their values behave differently for ''n'' even and odd.

More examples: The set of rational numbers <math>\mathbb{Q}</math>, a countable set with <math>\dim_{\operatorname{Haus}} = 0</math>, has <math>\dim_{\operatorname{box}} = 1</math> because its closure, <math>\mathbb{R}</math>, has dimension 1. In fact,

:<math> \dim_{\operatorname{box}} \left\{0,1,\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\right\} = \frac{1}{2}. </math>

These examples show that adding a countable set can change box-dimension, showing a kind of instability of this dimension.

== See also ==
*[[Correlation dimension]]
*[[Packing dimension]]
*[[Uncertainty exponent]]
*[[Weyl–Berry conjecture]]
*[[Lacunarity]]

==References==
{{reflist}}
* {{cite book | zbl=0689.28003 | last=Falconer | first=Kenneth | author-link=Kenneth Falconer (mathematician) | title=Fractal geometry: mathematical foundations and applications | url=https://archive.org/details/fractalgeometrym0000falc | url-access=registration | location=Chichester | publisher=John Wiley | year=1990 | isbn=0-471-92287-0 | pages=[https://archive.org/details/fractalgeometrym0000falc/page/38 38–47] }}
* {{mathworld| urlname=Minkowski-BouligandDimension|title=Minkowski-Bouligand Dimension}}

== External links ==
* [http://code.google.com/p/frakout/ FrakOut!: an OSS application for calculating the fractal dimension of a shape using the box counting method] (Does not automatically place the boxes for you).
* FracLac: online user guide and software [http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm ImageJ and FracLac box counting plugin; free user-friendly open source software for digital image analysis in biology]

{{Fractals}}

{{DEFAULTSORT:Minkowski-Bouligand dimension}}
[[Category:Fractals]]
[[Category:Dimension theory]]
[[Category:Hermann Minkowski]]

Hausdorff dimension

2022-02-04T01:19:57Z

IntegralPython: /* Formal definition */ typo

{{short description|Invariant}}
[[File:KochFlake.svg|thumb|280px|Example of non-integer dimensions. The first four [[iteration]]s of the [[Koch snowflake|Koch curve]], where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at ''Annenberg Learner:MATHematics illuminated'', see [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref>]]

In [[mathematics]], '''Hausdorff dimension''' is a measure of ''roughness'', or more specifically, [[fractal dimension]], that was first introduced in 1918 by [[mathematician]] [[Felix Hausdorff]].<ref>{{Cite journal |arxiv = 1101.1444|doi = 10.1214/11-STS370|title = Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data|journal = Statistical Science|volume = 27|issue = 2|pages = 247–277|year = 2012|last1 = Gneiting|first1 = Tilmann|last2 = Ševčíková|first2 = Hana|last3 = Percival|first3 = Donald B.|s2cid = 88512325}}</ref> For instance, the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an [[integer]] agreeing with the usual sense of dimension, also known as the [[Inductive dimension|topological dimension]]. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of [[scaling (geometry)|scaling]] and [[self-similarity]], one is led to the conclusion that particular objects—including [[fractal]]s—have non-integer Hausdorff dimensions. Because of the significant technical advances made by [[Abram Samoilovitch Besicovitch]] allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the ''Hausdorff–Besicovitch dimension.''

More specifically, the Hausdorff dimension is a dimensional number associated with a [[metric space]], i.e. a set where the distances between all members are defined. The dimension is drawn from the [[Extended real number line|extended real numbers]], <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real [[vector space]]. That is, the Hausdorff dimension of an ''n''-dimensional [[inner product space]] equals ''n''. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that [[fractal|irregular sets]] can have noninteger Hausdorff dimensions. For instance, the [[Koch snowflake]] shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new [[equilateral]] triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.<ref>Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see [http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm], accessed 5 March 2015.</ref> That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.<ref name=CampbellAnnenberg15/> Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD.<ref name=ClaytonSCTPLS96>Keith Clayton, 1996, "Fractals and the Fractal Dimension," ''Basic Concepts in Nonlinear Dynamics and Chaos'' (workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see [http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html], accessed 5 March 2015.</ref> This equation is easily solved for D, yielding the ratio of logarithms (or [[natural logarithm]]s) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or [[Minkowski–Bouligand dimension]].

==Intuition==
{{refimprove section|date=March 2015}}
The intuitive concept of dimension of a geometric object ''X'' is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the [[cardinality]] of the [[real plane]] is equal to the cardinality of the [[real line]] (this can be seen by an [[Cantor's diagonal argument|argument]] involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a [[space-filling curve]] shows that one can even map the real line to the real plane [[Surjective function|surjectively]] (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and ''continuously'', so that a one-dimensional object completely fills up a higher-dimensional object.

Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called [[Lebesgue covering dimension]], explains why. This dimension is ''n'' if, in every covering of ''X'' by small open balls, there is at least one point where ''n'' + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension ''n'' = 1.

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A [[fractal]] has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the [[metric space|metric]]. Consider the number ''N''(''r'') of [[ball (mathematics)|balls]] of radius at most ''r'' required to cover ''X'' completely. When ''r'' is very small, ''N''(''r'') grows polynomially with 1/''r''. For a sufficiently well-behaved ''X'', the Hausdorff dimension is the unique number ''d'' such that N(''r'') grows as 1/''rd'' as ''r'' approaches zero. More precisely, this defines the [[Minkowski–Bouligand dimension|box-counting dimension]], which equals the Hausdorff dimension when the value ''d'' is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But [[Benoit Mandelbrot]] observed that [[fractal]]s, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:

<blockquote>Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.<ref name="mandelbrot">{{cite book | last = Mandelbrot | first = Benoît | author-link = Benoit Mandelbrot | title = The Fractal Geometry of Nature | publisher = W. H. Freeman | series = Lecture notes in mathematics 1358 | year = 1982 | isbn = 0-7167-1186-9 | url-access = registration | url = https://archive.org/details/fractalgeometryo00beno }}</ref></blockquote>

For fractals that occur in nature, the Hausdorff and [[Minkowski–Bouligand dimension|box-counting dimension]] coincide. The [[packing dimension]] is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.{{Example needed|s|date=January 2022}}

==Formal definition==
{{main| Hausdorff measure}}
The formal definition of the Hausdorff dimension is arrived at by defining first the [[Hausdorff measure]], a fractional-dimension analogue of the [[Lebesgue measure]]. First, an [[outer measure]] is constructed:
Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞),

:<math>H^d_\delta(S)=\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S, \operatorname{diam} U_i<\delta\right \},</math>

where the [[infimum]] is taken over all countable covers ''Ui'' of ''S''. The Hausdorff outer measure is then defined as <math>\lim_{\delta\to 0}H^d_\delta(S).</math>, and the restriction of the mapping to [[non-measurable set| measurable set]]s justifies it as a measure, called the ''d''-dimensional Hausdorff Measure.<ref>{{cite web| last1=Briggs| first1=Jimmy| last2=Tyree|first2=Tim| title=Hausdorff Measure| url=https://sites.math.washington.edu/~farbod/teaching/cornell/math6210pdf/math6210Hausdorff.pdf| date=3 December 2016| access-date=3 February 2022| publisher=University of Washington}}</ref>

===Hausdorff dimension===
The '''Hausdorff dimension''' of ''X'' is defined by
:<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>

This is the same as the [[supremum]] of the set of ''d'' ∈ [0, ∞) such that the ''d''-dimensional Hausdorff measure of ''X'' is infinite (except that when this latter set of numbers ''d'' is empty the Hausdorff dimension is zero).

===Hausdorff content===
the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by
:<math>C_H^d(S):= H_\infty^d(S) = \inf\left \{ \sum_{i=1}^\infty (\operatorname{diam} U_k)^d: \bigcup_{i=1}^\infty U_k\supseteq S \right \}</math>

In other words, <math>C_H^d(S)</math> has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes (Here, we use the standard convention that [[infimum|inf Ø = ∞]]).<ref>{{cite web | last1=Farkas| first1=Abel| last2=Fraser| first2=Jonathan| title=On the equality of Hausdorff measure and Hausdorff content| date=30 July 2015| url=https://arxiv.org/pdf/1411.0867.pdf| access-date=3 February 2022}}</ref> The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree.

==Examples==
[[Image:Sierpinski deep.svg|thumb|250px|Dimension of a further [[fractal]] example. The [[Sierpinski triangle]], an object with Hausdorff dimension of log(3)/log(2)≈1.58.<ref name=ClaytonSCTPLS96/>]]
* [[Countable set]]s have Hausdorff dimension 0.<ref name="schleicher">{{cite journal |last1=Schleicher |first1=Dierk |title=Hausdorff Dimension, Its Properties, and Its Surprises |journal=The American Mathematical Monthly |date=June 2007 |volume=114 |issue=6 |pages=509–528 |doi=10.1080/00029890.2007.11920440 |language=en |issn=0002-9890|arxiv=math/0505099 |s2cid=9811750 }}</ref>
* The [[Euclidean space]] ℝ''n'' has Hausdorff dimension ''n'', and the circle '''S'''1 has Hausdorff dimension 1.<ref name="schleicher" />
* [[Fractal]]s often are spaces whose Hausdorff dimension strictly exceeds the [[topological dimension]].<ref name="mandelbrot" /> For example, the [[Cantor set]], a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63.<ref>{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}</ref> The [[Sierpinski triangle]] is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58.<ref name=CampbellAnnenberg15/> These Hausdorff dimensions are related to the "critical exponent" of the [[Master theorem (analysis of algorithms)|Master theorem]] for solving [[Recurrence relation|recurrence relations]] in the [[analysis of algorithms]].
* [[Space-filling curve]]s like the [[Peano curve]] have the same Hausdorff dimension as the space they fill.
* The trajectory of [[Brownian motion]] in dimension 2 and above is conjectured to be Hausdorff dimension 2.<ref>{{cite book | last=Morters | first=Peres | title= Brownian Motion | publisher=[[Cambridge University Press]] | year=2010 }}</ref>
[[image:Great Britain Hausdorff.svg|thumb|250px|Estimating the Hausdorff dimension of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coast of Great Britain]]]]
* [[Lewis Fry Richardson]] has performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of [[South Africa]] to 1.25 for the west coast of [[Great Britain]].<ref name="mandelbrot" />

==Properties of Hausdorff dimension==
{{refimprove section|date=March 2015}}

=== Hausdorff dimension and inductive dimension ===
Let ''X'' be an arbitrary [[Separable space|separable]] metric space. There is a [[topology|topological]] notion of [[inductive dimension]] for ''X'' which is defined recursively. It is always an integer (or +∞) and is denoted dimind(''X'').

'''Theorem'''. Suppose ''X'' is non-empty. Then
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
Moreover,
:<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
where ''Y'' ranges over metric spaces [[homeomorphic]] to ''X''. In other words, ''X'' and ''Y'' have the same underlying set of points and the metric ''d''''Y'' of ''Y'' is topologically equivalent to ''d''''X''.

These results were originally established by [[Edward Szpilrajn]] (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.{{full citation needed|date=March 2015}}

=== Hausdorff dimension and Minkowski dimension ===
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of [[rational number|rational]] points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.

=== Hausdorff dimensions and Frostman measures ===
If there is a [[measure (mathematics)|measure]] μ defined on [[Borel measure|Borel]] subsets of a metric space ''X'' such that ''μ''(''X'') > 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then dimHaus(''X'') ≥ ''s''. A partial converse is provided by [[Frostman's lemma]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>

=== Behaviour under unions and products ===
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then

:<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

This can be verified directly from the definition.

If ''X'' and ''Y'' are non-empty metric spaces, then the Hausdorff dimension of their product satisfies<ref>{{cite journal |author=Marstrand, J. M. |title=The dimension of Cartesian product sets |journal=Proc. Cambridge Philos. Soc. |volume=50 |issue=3 |pages=198–202 |year=1954 |doi=10.1017/S0305004100029236 |bibcode = 1954PCPS...50..198M }}</ref>

:<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.<ref>{{cite book | last = Falconer | first = Kenneth J. | title = Fractal geometry. Mathematical foundations and applications | publisher = John Wiley & Sons, Inc., Hoboken, New Jersey | year = 2003 }}</ref> In the opposite direction, it is known that when ''X'' and ''Y'' are Borel subsets of '''R'''''n'', the Hausdorff dimension of ''X'' × ''Y'' is bounded from above by the Hausdorff dimension of ''X'' plus the [[packing dimension|upper packing dimension]] of ''Y''. These facts are discussed in Mattila (1995).

==Self-similar sets==
{{refimprove section|date=March 2015}}
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set ''E'' is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(''E'') = ''E'', although the exact definition is given below.

<blockquote>'''Theorem'''. Suppose

:<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>

are [[Contraction mapping|contractive]] mappings on '''R'''''n'' with contraction constant ''rj'' < 1. Then there is a unique ''non-empty'' compact set ''A'' such that

:<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
</blockquote>

The theorem follows from [[Stefan Banach]]'s [[Contractive mapping theorem|contractive mapping fixed point theorem]] applied to the complete metric space of non-empty compact subsets of '''R'''''n'' with the [[Hausdorff distance]].<ref>{{cite book |author=Falconer, K. J. |title=The Geometry of Fractal Sets |publisher=Cambridge University Press |location=Cambridge, UK |year=1985 |isbn=0-521-25694-1 |chapter=Theorem 8.3}}</ref>

===The open set condition===
{{main|Open set condition}}
To determine the dimension of the self-similar set ''A'' (in certain cases), we need a technical condition called the ''open set condition'' (OSC) on the sequence of contractions ψ''i''.

There is a relatively compact open set ''V'' such that

:<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>

where the sets in union on the left are pairwise [[disjoint sets|disjoint]].

The open set condition is a separation condition that ensures the images ψ''i''(''V'') do not overlap "too much".

'''Theorem'''. Suppose the open set condition holds and each ψ''i'' is a similitude, that is a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is ''s'' where ''s'' is the unique solution of<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

The contraction coefficient of a similitude is the magnitude of the dilation.

In general, a set ''E'' which is a fixed point of a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections

:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''Hs'' denotes [[Hausdorff measure]]. This is clear in the case of the [[Sierpinski gasket]] (the intersections are just points), but is also true more generally:

'''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

==See also==
* [[List of fractals by Hausdorff dimension]] Examples of deterministic fractals, random and natural fractals.
* [[Assouad dimension]], another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
* [[Intrinsic dimension]]
* [[Packing dimension]]
* [[Fractal dimension]]

==References==
{{reflist}}

==Further reading==
* {{cite book |last1=Dodson |first1=M. Maurice |title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot |volume=72 |issue=1 |pages=305–347 |last2=Kristensen |first2=Simon |chapter=Hausdorff Dimension and Diophantine Approximation |date=June 12, 2003 |arxiv=math/0305399 |bibcode = 2003math......5399D |doi=10.1090/pspum/072.1/2112110|series=Proceedings of Symposia in Pure Mathematics |isbn=9780821836378 |s2cid=119613948 }}
* {{cite book |last1=Hurewicz |first1=Witold |author-link1=Witold Hurewicz |last2=Wallman |first2=Henry |author-link2=Henry Wallman |title=Dimension Theory |url=https://archive.org/details/in.ernet.dli.2015.84609 |publisher=Princeton University Press |year=1948 }}
* {{cite journal |author=E. Szpilrajn |author-link=Edward Marczewski |title=La dimension et la mesure |journal=Fundamenta Mathematicae |volume=28 |pages=81–9 |year=1937 }}
* {{cite journal
| last1=Marstrand
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
| volume=50
| issue=3
| pages=198–202
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
* {{Cite book
| last1=Mattila
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=[[Cambridge University Press]]
| isbn=978-0-521-65595-8 | year=1995}}
* {{cite journal |author=A. S. Besicovitch |author-link=A. S. Besicovitch |title=On Linear Sets of Points of Fractional Dimensions |journal=[[Mathematische Annalen]] |volume=101 |year=1929 | doi=10.1007/BF01454831| issue=1 |pages= 161–193|s2cid=125368661 }}
* {{cite journal |author1=A. S. Besicovitch |author-link1=A. S. Besicovitch |author2=H. D. Ursell |author-link2=H. D. Ursell |title=Sets of Fractional Dimensions |journal=Journal of the London Mathematical Society |volume=12 |year=1937 | issue=1 | doi=10.1112/jlms/s1-12.45.18 | pages=18–25 }} Several selections from this volume are reprinted in {{cite book |author=Edgar, Gerald A. |title=Classics on fractals |publisher=Addison-Wesley |location=Boston |year=1993 |isbn=0-201-58701-7}} See chapters 9,10,11
* {{cite journal |author=F. Hausdorff |author-link=F. Hausdorff |title=Dimension und äußeres Maß |journal=Mathematische Annalen |volume=79 |issue=1–2 |pages=157–179 |date=March 1919 |doi=10.1007/BF01457179|hdl=10338.dmlcz/100363 |s2cid=122001234 |url=http://dml.cz/bitstream/handle/10338.dmlcz/100363/CzechMathJ_09-1959-3_5.pdf }}
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}
*{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{Dimension topics}}

[[Category:Fractals]]
[[Category:Metric geometry]]
[[Category:Dimension theory]]

Hausdorff dimension

2022-02-04T01:18:52Z

IntegralPython: /* Formal definitions */ Most of this entire section seemed to be just wrong, for example using balls when arbitrary sets are required. I also moved the measure to be the main definition since that is the one that seems to be the standard

{{short description|Invariant}}
[[File:KochFlake.svg|thumb|280px|Example of non-integer dimensions. The first four [[iteration]]s of the [[Koch snowflake|Koch curve]], where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at ''Annenberg Learner:MATHematics illuminated'', see [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref>]]

In [[mathematics]], '''Hausdorff dimension''' is a measure of ''roughness'', or more specifically, [[fractal dimension]], that was first introduced in 1918 by [[mathematician]] [[Felix Hausdorff]].<ref>{{Cite journal |arxiv = 1101.1444|doi = 10.1214/11-STS370|title = Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data|journal = Statistical Science|volume = 27|issue = 2|pages = 247–277|year = 2012|last1 = Gneiting|first1 = Tilmann|last2 = Ševčíková|first2 = Hana|last3 = Percival|first3 = Donald B.|s2cid = 88512325}}</ref> For instance, the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an [[integer]] agreeing with the usual sense of dimension, also known as the [[Inductive dimension|topological dimension]]. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of [[scaling (geometry)|scaling]] and [[self-similarity]], one is led to the conclusion that particular objects—including [[fractal]]s—have non-integer Hausdorff dimensions. Because of the significant technical advances made by [[Abram Samoilovitch Besicovitch]] allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the ''Hausdorff–Besicovitch dimension.''

More specifically, the Hausdorff dimension is a dimensional number associated with a [[metric space]], i.e. a set where the distances between all members are defined. The dimension is drawn from the [[Extended real number line|extended real numbers]], <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real [[vector space]]. That is, the Hausdorff dimension of an ''n''-dimensional [[inner product space]] equals ''n''. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that [[fractal|irregular sets]] can have noninteger Hausdorff dimensions. For instance, the [[Koch snowflake]] shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new [[equilateral]] triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.<ref>Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see [http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm], accessed 5 March 2015.</ref> That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.<ref name=CampbellAnnenberg15/> Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD.<ref name=ClaytonSCTPLS96>Keith Clayton, 1996, "Fractals and the Fractal Dimension," ''Basic Concepts in Nonlinear Dynamics and Chaos'' (workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see [http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html], accessed 5 March 2015.</ref> This equation is easily solved for D, yielding the ratio of logarithms (or [[natural logarithm]]s) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or [[Minkowski–Bouligand dimension]].

==Intuition==
{{refimprove section|date=March 2015}}
The intuitive concept of dimension of a geometric object ''X'' is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the [[cardinality]] of the [[real plane]] is equal to the cardinality of the [[real line]] (this can be seen by an [[Cantor's diagonal argument|argument]] involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a [[space-filling curve]] shows that one can even map the real line to the real plane [[Surjective function|surjectively]] (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and ''continuously'', so that a one-dimensional object completely fills up a higher-dimensional object.

Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called [[Lebesgue covering dimension]], explains why. This dimension is ''n'' if, in every covering of ''X'' by small open balls, there is at least one point where ''n'' + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension ''n'' = 1.

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A [[fractal]] has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the [[metric space|metric]]. Consider the number ''N''(''r'') of [[ball (mathematics)|balls]] of radius at most ''r'' required to cover ''X'' completely. When ''r'' is very small, ''N''(''r'') grows polynomially with 1/''r''. For a sufficiently well-behaved ''X'', the Hausdorff dimension is the unique number ''d'' such that N(''r'') grows as 1/''rd'' as ''r'' approaches zero. More precisely, this defines the [[Minkowski–Bouligand dimension|box-counting dimension]], which equals the Hausdorff dimension when the value ''d'' is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But [[Benoit Mandelbrot]] observed that [[fractal]]s, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:

<blockquote>Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.<ref name="mandelbrot">{{cite book | last = Mandelbrot | first = Benoît | author-link = Benoit Mandelbrot | title = The Fractal Geometry of Nature | publisher = W. H. Freeman | series = Lecture notes in mathematics 1358 | year = 1982 | isbn = 0-7167-1186-9 | url-access = registration | url = https://archive.org/details/fractalgeometryo00beno }}</ref></blockquote>

For fractals that occur in nature, the Hausdorff and [[Minkowski–Bouligand dimension|box-counting dimension]] coincide. The [[packing dimension]] is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.{{Example needed|s|date=January 2022}}

==Formal definition==
{{main| Hausdorff measure}}
The formal definition of the Hausdorff dimension is arrived at by defining first the [[Hausdorff measure]], a fractional-dimension analogue of the [[Lebesgue measure]]. First, an [[outer measure]] is constructed:
Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞),

:<math>H^d_\delta(S)=\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S, \operatorname{diam} U_i<\delta\right \},</math>

where the [[infimum]] is taken over all countable covers ''Ui'' of ''S''. The Hausdorff outer measure is then defined as <math>\lim_{\delta\to 0}H^d_\delta(S).</math>, and the restriction of the mapping to [[non-measureable set| measureable set]]s justifies it as a measure, called the ''d''-dimensional Hausdorff Measure.<ref>{{cite web| last1=Briggs| first1=Jimmy| last2=Tyree|first2=Tim| title=Hausdorff Measure| url=https://sites.math.washington.edu/~farbod/teaching/cornell/math6210pdf/math6210Hausdorff.pdf| date=3 December 2016| access-date=3 February 2022| publisher=University of Washington}}</ref>

===Hausdorff dimension===
The '''Hausdorff dimension''' of ''X'' is defined by
:<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>

This is the same as the [[supremum]] of the set of ''d'' ∈ [0, ∞) such that the ''d''-dimensional Hausdorff measure of ''X'' is infinite (except that when this latter set of numbers ''d'' is empty the Hausdorff dimension is zero).

===Hausdorff content===
the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by
:<math>C_H^d(S):= H_\infty^d(S) = \inf\left \{ \sum_{i=1}^\infty (\operatorname{diam} U_k)^d: \bigcup_{i=1}^\infty U_k\supseteq S \right \}</math>

In other words, <math>C_H^d(S)</math> has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes (Here, we use the standard convention that [[infimum|inf Ø = ∞]]).<ref>{{cite web | last1=Farkas| first1=Abel| last2=Fraser| first2=Jonathan| title=On the equality of Hausdorff measure and Hausdorff content| date=30 July 2015| url=https://arxiv.org/pdf/1411.0867.pdf| access-date=3 February 2022}}</ref> The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree.

==Examples==
[[Image:Sierpinski deep.svg|thumb|250px|Dimension of a further [[fractal]] example. The [[Sierpinski triangle]], an object with Hausdorff dimension of log(3)/log(2)≈1.58.<ref name=ClaytonSCTPLS96/>]]
* [[Countable set]]s have Hausdorff dimension 0.<ref name="schleicher">{{cite journal |last1=Schleicher |first1=Dierk |title=Hausdorff Dimension, Its Properties, and Its Surprises |journal=The American Mathematical Monthly |date=June 2007 |volume=114 |issue=6 |pages=509–528 |doi=10.1080/00029890.2007.11920440 |language=en |issn=0002-9890|arxiv=math/0505099 |s2cid=9811750 }}</ref>
* The [[Euclidean space]] ℝ''n'' has Hausdorff dimension ''n'', and the circle '''S'''1 has Hausdorff dimension 1.<ref name="schleicher" />
* [[Fractal]]s often are spaces whose Hausdorff dimension strictly exceeds the [[topological dimension]].<ref name="mandelbrot" /> For example, the [[Cantor set]], a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63.<ref>{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}</ref> The [[Sierpinski triangle]] is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58.<ref name=CampbellAnnenberg15/> These Hausdorff dimensions are related to the "critical exponent" of the [[Master theorem (analysis of algorithms)|Master theorem]] for solving [[Recurrence relation|recurrence relations]] in the [[analysis of algorithms]].
* [[Space-filling curve]]s like the [[Peano curve]] have the same Hausdorff dimension as the space they fill.
* The trajectory of [[Brownian motion]] in dimension 2 and above is conjectured to be Hausdorff dimension 2.<ref>{{cite book | last=Morters | first=Peres | title= Brownian Motion | publisher=[[Cambridge University Press]] | year=2010 }}</ref>
[[image:Great Britain Hausdorff.svg|thumb|250px|Estimating the Hausdorff dimension of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coast of Great Britain]]]]
* [[Lewis Fry Richardson]] has performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of [[South Africa]] to 1.25 for the west coast of [[Great Britain]].<ref name="mandelbrot" />

==Properties of Hausdorff dimension==
{{refimprove section|date=March 2015}}

=== Hausdorff dimension and inductive dimension ===
Let ''X'' be an arbitrary [[Separable space|separable]] metric space. There is a [[topology|topological]] notion of [[inductive dimension]] for ''X'' which is defined recursively. It is always an integer (or +∞) and is denoted dimind(''X'').

'''Theorem'''. Suppose ''X'' is non-empty. Then
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
Moreover,
:<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
where ''Y'' ranges over metric spaces [[homeomorphic]] to ''X''. In other words, ''X'' and ''Y'' have the same underlying set of points and the metric ''d''''Y'' of ''Y'' is topologically equivalent to ''d''''X''.

These results were originally established by [[Edward Szpilrajn]] (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.{{full citation needed|date=March 2015}}

=== Hausdorff dimension and Minkowski dimension ===
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of [[rational number|rational]] points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.

=== Hausdorff dimensions and Frostman measures ===
If there is a [[measure (mathematics)|measure]] μ defined on [[Borel measure|Borel]] subsets of a metric space ''X'' such that ''μ''(''X'') > 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then dimHaus(''X'') ≥ ''s''. A partial converse is provided by [[Frostman's lemma]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>

=== Behaviour under unions and products ===
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then

:<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

This can be verified directly from the definition.

If ''X'' and ''Y'' are non-empty metric spaces, then the Hausdorff dimension of their product satisfies<ref>{{cite journal |author=Marstrand, J. M. |title=The dimension of Cartesian product sets |journal=Proc. Cambridge Philos. Soc. |volume=50 |issue=3 |pages=198–202 |year=1954 |doi=10.1017/S0305004100029236 |bibcode = 1954PCPS...50..198M }}</ref>

:<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.<ref>{{cite book | last = Falconer | first = Kenneth J. | title = Fractal geometry. Mathematical foundations and applications | publisher = John Wiley & Sons, Inc., Hoboken, New Jersey | year = 2003 }}</ref> In the opposite direction, it is known that when ''X'' and ''Y'' are Borel subsets of '''R'''''n'', the Hausdorff dimension of ''X'' × ''Y'' is bounded from above by the Hausdorff dimension of ''X'' plus the [[packing dimension|upper packing dimension]] of ''Y''. These facts are discussed in Mattila (1995).

==Self-similar sets==
{{refimprove section|date=March 2015}}
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set ''E'' is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(''E'') = ''E'', although the exact definition is given below.

<blockquote>'''Theorem'''. Suppose

:<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>

are [[Contraction mapping|contractive]] mappings on '''R'''''n'' with contraction constant ''rj'' < 1. Then there is a unique ''non-empty'' compact set ''A'' such that

:<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
</blockquote>

The theorem follows from [[Stefan Banach]]'s [[Contractive mapping theorem|contractive mapping fixed point theorem]] applied to the complete metric space of non-empty compact subsets of '''R'''''n'' with the [[Hausdorff distance]].<ref>{{cite book |author=Falconer, K. J. |title=The Geometry of Fractal Sets |publisher=Cambridge University Press |location=Cambridge, UK |year=1985 |isbn=0-521-25694-1 |chapter=Theorem 8.3}}</ref>

===The open set condition===
{{main|Open set condition}}
To determine the dimension of the self-similar set ''A'' (in certain cases), we need a technical condition called the ''open set condition'' (OSC) on the sequence of contractions ψ''i''.

There is a relatively compact open set ''V'' such that

:<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>

where the sets in union on the left are pairwise [[disjoint sets|disjoint]].

The open set condition is a separation condition that ensures the images ψ''i''(''V'') do not overlap "too much".

'''Theorem'''. Suppose the open set condition holds and each ψ''i'' is a similitude, that is a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is ''s'' where ''s'' is the unique solution of<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

The contraction coefficient of a similitude is the magnitude of the dilation.

In general, a set ''E'' which is a fixed point of a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections

:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''Hs'' denotes [[Hausdorff measure]]. This is clear in the case of the [[Sierpinski gasket]] (the intersections are just points), but is also true more generally:

'''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

==See also==
* [[List of fractals by Hausdorff dimension]] Examples of deterministic fractals, random and natural fractals.
* [[Assouad dimension]], another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
* [[Intrinsic dimension]]
* [[Packing dimension]]
* [[Fractal dimension]]

==References==
{{reflist}}

==Further reading==
* {{cite book |last1=Dodson |first1=M. Maurice |title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot |volume=72 |issue=1 |pages=305–347 |last2=Kristensen |first2=Simon |chapter=Hausdorff Dimension and Diophantine Approximation |date=June 12, 2003 |arxiv=math/0305399 |bibcode = 2003math......5399D |doi=10.1090/pspum/072.1/2112110|series=Proceedings of Symposia in Pure Mathematics |isbn=9780821836378 |s2cid=119613948 }}
* {{cite book |last1=Hurewicz |first1=Witold |author-link1=Witold Hurewicz |last2=Wallman |first2=Henry |author-link2=Henry Wallman |title=Dimension Theory |url=https://archive.org/details/in.ernet.dli.2015.84609 |publisher=Princeton University Press |year=1948 }}
* {{cite journal |author=E. Szpilrajn |author-link=Edward Marczewski |title=La dimension et la mesure |journal=Fundamenta Mathematicae |volume=28 |pages=81–9 |year=1937 }}
* {{cite journal
| last1=Marstrand
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
| volume=50
| issue=3
| pages=198–202
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
* {{Cite book
| last1=Mattila
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=[[Cambridge University Press]]
| isbn=978-0-521-65595-8 | year=1995}}
* {{cite journal |author=A. S. Besicovitch |author-link=A. S. Besicovitch |title=On Linear Sets of Points of Fractional Dimensions |journal=[[Mathematische Annalen]] |volume=101 |year=1929 | doi=10.1007/BF01454831| issue=1 |pages= 161–193|s2cid=125368661 }}
* {{cite journal |author1=A. S. Besicovitch |author-link1=A. S. Besicovitch |author2=H. D. Ursell |author-link2=H. D. Ursell |title=Sets of Fractional Dimensions |journal=Journal of the London Mathematical Society |volume=12 |year=1937 | issue=1 | doi=10.1112/jlms/s1-12.45.18 | pages=18–25 }} Several selections from this volume are reprinted in {{cite book |author=Edgar, Gerald A. |title=Classics on fractals |publisher=Addison-Wesley |location=Boston |year=1993 |isbn=0-201-58701-7}} See chapters 9,10,11
* {{cite journal |author=F. Hausdorff |author-link=F. Hausdorff |title=Dimension und äußeres Maß |journal=Mathematische Annalen |volume=79 |issue=1–2 |pages=157–179 |date=March 1919 |doi=10.1007/BF01457179|hdl=10338.dmlcz/100363 |s2cid=122001234 |url=http://dml.cz/bitstream/handle/10338.dmlcz/100363/CzechMathJ_09-1959-3_5.pdf }}
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}
*{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{Dimension topics}}

[[Category:Fractals]]
[[Category:Metric geometry]]
[[Category:Dimension theory]]

Open set condition

2022-02-04T00:26:47Z

IntegralPython: Changing short description from "Condition for self-similar fractals" to "Condition for fractals in math" (Shortdesc helper)

{{Short description|Condition for fractals in math}}
[[File:Open set condition.png|thumb|an open set covering of the [[sierpinski triangle]] along with one of its mappings ψ''i''.]]
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

==Computing Hausdorff dimension==
When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose [[Hausdorff dimension]] is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Strong open set condition==
The strong open set condition (SOSC) is an extension of the open set condition. A fractal F satisfies the SOSC if, in addition to satisfying the OSC, the intersection between F and the open set V is nonempty.<ref>{{Cite web | url=http://www.stat.uchicago.edu/~lalley/Papers/packing.pdf| title=The Packing and Covering Functions for Some Self-similar Fractals|last=Lalley|first=Steven|publisher=Purdue University|date=21 January 1988|access-date=2 February 2022}}</ref> The two conditions are equivalent for self-similar and self-conformal sets, but not for certain classes of other sets, such as function systems with infinite mappings and in non-euclidean metric spaces.<ref>{{Cite web| url=http://users.jyu.fi/~antakae/publications/preprints/009-controlled_moran.pdf| title=Separation Conditions on Controlled Moran Constructions| last1=Käenmäki| first1=Antti| last2=Vilppolainen| first2=Markku| access-date = 2 February 2022}}</ref><ref>{{Cite journal| last=Schief| first=Andreas| title=Self-similar Sets in Complete Metric Spaces| journal=Proceedings of the American Mathematical Society| volume=124| issue=2| year=1996| url=https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03158-9/S0002-9939-96-03158-9.pdf}}</ref> In these cases, SOCS is indeed a stronger condition.

==See also==
*[[Cantor set]]
*[[List of fractals by Hausdorff dimension]]
*[[Packing dimension]]

==References==
{{reflist}}

[[Category:Fractals]]
[[Category:Iterated function system fractals]]

Talk:Open set condition

2022-02-04T00:26:08Z

IntegralPython: create talk page with maths rating

Open set condition

2022-02-03T00:07:38Z

IntegralPython: add more blue-links

{{Short description|Condition for self-similar fractals}}
[[File:Open set condition.png|thumb|an open set covering of the [[sierpinski triangle]] along with one of its mappings ψ''i''.]]
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

==Computing Hausdorff dimension==
When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose [[Hausdorff dimension]] is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Strong open set condition==
The strong open set condition (SOSC) is an extension of the open set condition. A fractal F satisfies the SOSC if, in addition to satisfying the OSC, the intersection between F and the open set V is nonempty.<ref>{{Cite web | url=http://www.stat.uchicago.edu/~lalley/Papers/packing.pdf| title=The Packing and Covering Functions for Some Self-similar Fractals|last=Lalley|first=Steven|publisher=Purdue University|date=21 January 1988|access-date=2 February 2022}}</ref> The two conditions are equivalent for self-similar and self-conformal sets, but not for certain classes of other sets, such as function systems with infinite mappings and in non-euclidean metric spaces.<ref>{{Cite web| url=http://users.jyu.fi/~antakae/publications/preprints/009-controlled_moran.pdf| title=Separation Conditions on Controlled Moran Constructions| last1=Käenmäki| first1=Antti| last2=Vilppolainen| first2=Markku| access-date = 2 February 2022}}</ref><ref>{{Cite journal| last=Schief| first=Andreas| title=Self-similar Sets in Complete Metric Spaces| journal=Proceedings of the American Mathematical Society| volume=124| issue=2| year=1996| url=https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03158-9/S0002-9939-96-03158-9.pdf}}</ref> In these cases, SOCS is indeed a stronger condition.

==See also==
*[[Cantor set]]
*[[List of fractals by Hausdorff dimension]]
*[[Packing dimension]]

==References==
{{reflist}}

[[Category:Fractals]]
[[Category:Iterated function system fractals]]

Open set condition

2022-02-03T00:03:56Z

IntegralPython: Added strong open set condition

{{Short description|Condition for self-similar fractals}}
[[File:Open set condition.png|thumb|an open set covering of the sierpinski triangle along with one of its mappings ψ''i''.]]
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set V satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

==Computing Hausdorff measure==

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Strong open set condition==
The strong open set condition (SOSC) is an extension of the open set condition. A fractal F satisfies the SOSC if, in addition to satisfying the OSC, the intersection between F and the open set V is nonempty.<ref>{{Cite web | url=http://www.stat.uchicago.edu/~lalley/Papers/packing.pdf| title=The Packing and Covering Functions for Some Self-similar Fractals|last=Lalley|first=Steven|publisher=Purdue University|date=21 January 1988|access-date=2 February 2022}}</ref> The two conditions are equivalent for self-similar and self-conformal sets, but not for certain classes of other sets, such as function systems with infinite mappings and in non-euclidean metric spaces.<ref>{{Cite web| url=http://users.jyu.fi/~antakae/publications/preprints/009-controlled_moran.pdf| title=Separation Conditions on Controlled Moran Constructions| last1=Käenmäki| first1=Antti| last2=Vilppolainen| first2=Markku| access-date = 2 February 2022}}</ref><ref>{{Cite journal| last=Schief| first=Andreas| title=Self-similar Sets in Complete Metric Spaces| journal=Proceedings of the American Mathematical Society| volume=124| issue=2| year=1996| url=https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03158-9/S0002-9939-96-03158-9.pdf}}</ref> In these cases, SOCS is indeed a stronger condition.

==See also==
*[[List of fractals by Hausdorff dimension]]
*[[Packing dimension]]

==References==
{{reflist}}

[[Category:Fractals]]
[[Category:Iterated function system fractals]]

Open set condition

2022-02-02T11:33:08Z

IntegralPython: added categories

{{Short description|Condition for self-similar fractals}}
[[File:Open set covering.png|thumb|an open set covering of the sierpinski triangle along with one of its mappings ψ''i''.]]
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

==Computing Hausdorff measure==

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==See also==
*[[List of fractals by Hausdorff dimension]]
*[[Packing dimension]]

==References==
{{reflist}}

[[Category:Fractals]]
[[Category:Iterated function system fractals]]

User:IntegralPython

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Hi! I'm a [[Christians|Christian]] [[Wikipedia]] browser and recreational mathematician. My interests mainly lie in [[math]] and [[Science]], particularly in [[fractal]] analysis, [[quaternion]]s, [[hyperoperation]]s, and [[quantum physics]]. If I do anything stupid, please leave a long and angry comment on my talk page; make sure to include as many strongly worded critiques of me and my poor intelligence. Thanks!

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Hi! I'm a [[Christians|Christian]] [[Wikipedia]] browser and recreational mathematician. My interests mainly lie in [[math]] and [[Science]], particularly in [[fractal]] analysis, [[quaternion]]s, [[hyperoperation]]s, and [[quantum physics]]. If I do anything stupid, please leave a long and angry comment on my talk page; make sure to include as many strongly worded critiques of me and my poor intelligence. Thanks!

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Open set condition

2022-02-02T03:30:40Z

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Hausdorff dimension

2022-02-02T03:13:43Z

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{{short description|Invariant}}
[[File:KochFlake.svg|thumb|280px|Example of non-integer dimensions. The first four [[iteration]]s of the [[Koch snowflake|Koch curve]], where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at ''Annenberg Learner:MATHematics illuminated'', see [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref>]]

In [[mathematics]], '''Hausdorff dimension''' is a measure of ''roughness'', or more specifically, [[fractal dimension]], that was first introduced in 1918 by [[mathematician]] [[Felix Hausdorff]].<ref>{{Cite journal |arxiv = 1101.1444|doi = 10.1214/11-STS370|title = Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data|journal = Statistical Science|volume = 27|issue = 2|pages = 247–277|year = 2012|last1 = Gneiting|first1 = Tilmann|last2 = Ševčíková|first2 = Hana|last3 = Percival|first3 = Donald B.|s2cid = 88512325}}</ref> For instance, the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an [[integer]] agreeing with the usual sense of dimension, also known as the [[Inductive dimension|topological dimension]]. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of [[scaling (geometry)|scaling]] and [[self-similarity]], one is led to the conclusion that particular objects—including [[fractal]]s—have non-integer Hausdorff dimensions. Because of the significant technical advances made by [[Abram Samoilovitch Besicovitch]] allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the ''Hausdorff–Besicovitch dimension.''

More specifically, the Hausdorff dimension is a dimensional number associated with a [[metric space]], i.e. a set where the distances between all members are defined. The dimension is drawn from the [[Extended real number line|extended real numbers]], <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real [[vector space]]. That is, the Hausdorff dimension of an ''n''-dimensional [[inner product space]] equals ''n''. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that [[fractal|irregular sets]] can have noninteger Hausdorff dimensions. For instance, the [[Koch snowflake]] shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new [[equilateral]] triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.<ref>Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see [http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm], accessed 5 March 2015.</ref> That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.<ref name=CampbellAnnenberg15/> Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD.<ref name=ClaytonSCTPLS96>Keith Clayton, 1996, "Fractals and the Fractal Dimension," ''Basic Concepts in Nonlinear Dynamics and Chaos'' (workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see [http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html], accessed 5 March 2015.</ref> This equation is easily solved for D, yielding the ratio of logarithms (or [[natural logarithm]]s) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or [[Minkowski–Bouligand dimension]].

==Intuition==
{{refimprove section|date=March 2015}}
The intuitive concept of dimension of a geometric object ''X'' is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the [[cardinality]] of the [[real plane]] is equal to the cardinality of the [[real line]] (this can be seen by an [[Cantor's diagonal argument|argument]] involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a [[space-filling curve]] shows that one can even map the real line to the real plane [[Surjective function|surjectively]] (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and ''continuously'', so that a one-dimensional object completely fills up a higher-dimensional object.

Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called [[Lebesgue covering dimension]], explains why. This dimension is ''n'' if, in every covering of ''X'' by small open balls, there is at least one point where ''n'' + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension ''n'' = 1.

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A [[fractal]] has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the [[metric space|metric]]. Consider the number ''N''(''r'') of [[ball (mathematics)|balls]] of radius at most ''r'' required to cover ''X'' completely. When ''r'' is very small, ''N''(''r'') grows polynomially with 1/''r''. For a sufficiently well-behaved ''X'', the Hausdorff dimension is the unique number ''d'' such that N(''r'') grows as 1/''rd'' as ''r'' approaches zero. More precisely, this defines the [[Minkowski–Bouligand dimension|box-counting dimension]], which equals the Hausdorff dimension when the value ''d'' is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But [[Benoit Mandelbrot]] observed that [[fractal]]s, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:

<blockquote>Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.<ref name="mandelbrot">{{cite book | last = Mandelbrot | first = Benoît | author-link = Benoit Mandelbrot | title = The Fractal Geometry of Nature | publisher = W. H. Freeman | series = Lecture notes in mathematics 1358 | year = 1982 | isbn = 0-7167-1186-9 | url-access = registration | url = https://archive.org/details/fractalgeometryo00beno }}</ref></blockquote>

For fractals that occur in nature, the Hausdorff and [[Minkowski–Bouligand dimension|box-counting dimension]] coincide. The [[packing dimension]] is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.{{Example needed|s|date=January 2022}}

==Formal definitions==
{{unreferenced section|date=March 2015}}

===Hausdorff content===
Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞), the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by
:<math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a countable cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math>
In other words, <math>C_H^d(S)</math> is the [[infimum]] of the set of numbers <math>\delta \geq 0</math> such that there is some (indexed) collection of [[ball (mathematics)|ball]]s <math>\{B(x_i,r_i):i\in I\}</math> covering ''S'' with ''ri'' > 0 for each ''i'' ∈ ''I'' that satisfies <math>\sum_{i\in I} r_i^d<\delta </math>. (Here, we use the standard convention that [[infimum|inf Ø = ∞]].)

===Hausdorff measure===
The Hausdorff outer measure is different from the unbounded Hausdorff content in that rather than considering all possible coverings of ''S'', we see what happens when the sizes of the balls go to zero. For <math>d \geq 0 </math>, we define the ''d''-dimensional Hausdorff outer measure of ''S'' as
:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a countable cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>

===Hausdorff dimension===
The '''Hausdorff dimension''' of ''X'' is defined by
:<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>

Equivalently, dimH(''X'') may be defined as the [[infimum]] of the set of ''d'' ∈ [0, ∞) such that the ''d''-dimensional [[Hausdorff measure]] of ''X'' is zero. This is the same as the supremum of the set of ''d'' ∈ [0, ∞) such that the ''d''-dimensional Hausdorff measure of ''X'' is infinite (except that when this latter set of numbers ''d'' is empty the Hausdorff dimension is zero).

==Examples==
[[Image:Sierpinski deep.svg|thumb|250px|Dimension of a further [[fractal]] example. The [[Sierpinski triangle]], an object with Hausdorff dimension of log(3)/log(2)≈1.58.<ref name=ClaytonSCTPLS96/>]]
* [[Countable set]]s have Hausdorff dimension 0.<ref name="schleicher">{{cite journal |last1=Schleicher |first1=Dierk |title=Hausdorff Dimension, Its Properties, and Its Surprises |journal=The American Mathematical Monthly |date=June 2007 |volume=114 |issue=6 |pages=509–528 |doi=10.1080/00029890.2007.11920440 |language=en |issn=0002-9890|arxiv=math/0505099 |s2cid=9811750 }}</ref>
* The [[Euclidean space]] ℝ''n'' has Hausdorff dimension ''n'', and the circle '''S'''1 has Hausdorff dimension 1.<ref name="schleicher" />
* [[Fractal]]s often are spaces whose Hausdorff dimension strictly exceeds the [[topological dimension]].<ref name="mandelbrot" /> For example, the [[Cantor set]], a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63.<ref>{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}</ref> The [[Sierpinski triangle]] is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58.<ref name=CampbellAnnenberg15/> These Hausdorff dimensions are related to the "critical exponent" of the [[Master theorem (analysis of algorithms)|Master theorem]] for solving [[Recurrence relation|recurrence relations]] in the [[analysis of algorithms]].
* [[Space-filling curve]]s like the [[Peano curve]] have the same Hausdorff dimension as the space they fill.
* The trajectory of [[Brownian motion]] in dimension 2 and above is conjectured to be Hausdorff dimension 2.<ref>{{cite book | last=Morters | first=Peres | title= Brownian Motion | publisher=[[Cambridge University Press]] | year=2010 }}</ref>
[[image:Great Britain Hausdorff.svg|thumb|250px|Estimating the Hausdorff dimension of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coast of Great Britain]]]]
* [[Lewis Fry Richardson]] has performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of [[South Africa]] to 1.25 for the west coast of [[Great Britain]].<ref name="mandelbrot" />

==Properties of Hausdorff dimension==
{{refimprove section|date=March 2015}}

=== Hausdorff dimension and inductive dimension ===
Let ''X'' be an arbitrary [[Separable space|separable]] metric space. There is a [[topology|topological]] notion of [[inductive dimension]] for ''X'' which is defined recursively. It is always an integer (or +∞) and is denoted dimind(''X'').

'''Theorem'''. Suppose ''X'' is non-empty. Then
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
Moreover,
:<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
where ''Y'' ranges over metric spaces [[homeomorphic]] to ''X''. In other words, ''X'' and ''Y'' have the same underlying set of points and the metric ''d''''Y'' of ''Y'' is topologically equivalent to ''d''''X''.

These results were originally established by [[Edward Szpilrajn]] (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.{{full citation needed|date=March 2015}}

=== Hausdorff dimension and Minkowski dimension ===
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of [[rational number|rational]] points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.

=== Hausdorff dimensions and Frostman measures ===
If there is a [[measure (mathematics)|measure]] μ defined on [[Borel measure|Borel]] subsets of a metric space ''X'' such that ''μ''(''X'') > 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then dimHaus(''X'') ≥ ''s''. A partial converse is provided by [[Frostman's lemma]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>

=== Behaviour under unions and products ===
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then

:<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

This can be verified directly from the definition.

If ''X'' and ''Y'' are non-empty metric spaces, then the Hausdorff dimension of their product satisfies<ref>{{cite journal |author=Marstrand, J. M. |title=The dimension of Cartesian product sets |journal=Proc. Cambridge Philos. Soc. |volume=50 |issue=3 |pages=198–202 |year=1954 |doi=10.1017/S0305004100029236 |bibcode = 1954PCPS...50..198M }}</ref>

:<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.<ref>{{cite book | last = Falconer | first = Kenneth J. | title = Fractal geometry. Mathematical foundations and applications | publisher = John Wiley & Sons, Inc., Hoboken, New Jersey | year = 2003 }}</ref> In the opposite direction, it is known that when ''X'' and ''Y'' are Borel subsets of '''R'''''n'', the Hausdorff dimension of ''X'' × ''Y'' is bounded from above by the Hausdorff dimension of ''X'' plus the [[packing dimension|upper packing dimension]] of ''Y''. These facts are discussed in Mattila (1995).

==Self-similar sets==
{{refimprove section|date=March 2015}}
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set ''E'' is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(''E'') = ''E'', although the exact definition is given below.

<blockquote>'''Theorem'''. Suppose

:<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>

are [[Contraction mapping|contractive]] mappings on '''R'''''n'' with contraction constant ''rj'' < 1. Then there is a unique ''non-empty'' compact set ''A'' such that

:<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
</blockquote>

The theorem follows from [[Stefan Banach]]'s [[Contractive mapping theorem|contractive mapping fixed point theorem]] applied to the complete metric space of non-empty compact subsets of '''R'''''n'' with the [[Hausdorff distance]].<ref>{{cite book |author=Falconer, K. J. |title=The Geometry of Fractal Sets |publisher=Cambridge University Press |location=Cambridge, UK |year=1985 |isbn=0-521-25694-1 |chapter=Theorem 8.3}}</ref>

===The open set condition===
{{main|Open set condition}}
To determine the dimension of the self-similar set ''A'' (in certain cases), we need a technical condition called the ''open set condition'' (OSC) on the sequence of contractions ψ''i''.

There is a relatively compact open set ''V'' such that

:<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>

where the sets in union on the left are pairwise [[disjoint sets|disjoint]].

The open set condition is a separation condition that ensures the images ψ''i''(''V'') do not overlap "too much".

'''Theorem'''. Suppose the open set condition holds and each ψ''i'' is a similitude, that is a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is ''s'' where ''s'' is the unique solution of<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

The contraction coefficient of a similitude is the magnitude of the dilation.

In general, a set ''E'' which is a fixed point of a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections

:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''Hs'' denotes [[Hausdorff measure]]. This is clear in the case of the [[Sierpinski gasket]] (the intersections are just points), but is also true more generally:

'''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

==See also==
* [[List of fractals by Hausdorff dimension]] Examples of deterministic fractals, random and natural fractals.
* [[Assouad dimension]], another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
* [[Intrinsic dimension]]
* [[Packing dimension]]
* [[Fractal dimension]]

==References==
{{reflist}}

==Further reading==
* {{cite book |last1=Dodson |first1=M. Maurice |title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot |volume=72 |issue=1 |pages=305–347 |last2=Kristensen |first2=Simon |chapter=Hausdorff Dimension and Diophantine Approximation |date=June 12, 2003 |arxiv=math/0305399 |bibcode = 2003math......5399D |doi=10.1090/pspum/072.1/2112110|series=Proceedings of Symposia in Pure Mathematics |isbn=9780821836378 |s2cid=119613948 }}
* {{cite book |last1=Hurewicz |first1=Witold |author-link1=Witold Hurewicz |last2=Wallman |first2=Henry |author-link2=Henry Wallman |title=Dimension Theory |url=https://archive.org/details/in.ernet.dli.2015.84609 |publisher=Princeton University Press |year=1948 }}
* {{cite journal |author=E. Szpilrajn |author-link=Edward Marczewski |title=La dimension et la mesure |journal=Fundamenta Mathematicae |volume=28 |pages=81–9 |year=1937 }}
* {{cite journal
| last1=Marstrand
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
| volume=50
| issue=3
| pages=198–202
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
* {{Cite book
| last1=Mattila
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=[[Cambridge University Press]]
| isbn=978-0-521-65595-8 | year=1995}}
* {{cite journal |author=A. S. Besicovitch |author-link=A. S. Besicovitch |title=On Linear Sets of Points of Fractional Dimensions |journal=[[Mathematische Annalen]] |volume=101 |year=1929 | doi=10.1007/BF01454831| issue=1 |pages= 161–193|s2cid=125368661 }}
* {{cite journal |author1=A. S. Besicovitch |author-link1=A. S. Besicovitch |author2=H. D. Ursell |author-link2=H. D. Ursell |title=Sets of Fractional Dimensions |journal=Journal of the London Mathematical Society |volume=12 |year=1937 | issue=1 | doi=10.1112/jlms/s1-12.45.18 | pages=18–25 }} Several selections from this volume are reprinted in {{cite book |author=Edgar, Gerald A. |title=Classics on fractals |publisher=Addison-Wesley |location=Boston |year=1993 |isbn=0-201-58701-7}} See chapters 9,10,11
* {{cite journal |author=F. Hausdorff |author-link=F. Hausdorff |title=Dimension und äußeres Maß |journal=Mathematische Annalen |volume=79 |issue=1–2 |pages=157–179 |date=March 1919 |doi=10.1007/BF01457179|hdl=10338.dmlcz/100363 |s2cid=122001234 |url=http://dml.cz/bitstream/handle/10338.dmlcz/100363/CzechMathJ_09-1959-3_5.pdf }}
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}
*{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{Dimension topics}}

[[Category:Fractals]]
[[Category:Metric geometry]]
[[Category:Dimension theory]]

Hausdorff dimension

2022-02-02T03:12:33Z

IntegralPython: /* Self-similar sets */ moved some material into a new article open set condition

{{short description|Invariant}}
[[File:KochFlake.svg|thumb|280px|Example of non-integer dimensions. The first four [[iteration]]s of the [[Koch snowflake|Koch curve]], where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at ''Annenberg Learner:MATHematics illuminated'', see [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref>]]

In [[mathematics]], '''Hausdorff dimension''' is a measure of ''roughness'', or more specifically, [[fractal dimension]], that was first introduced in 1918 by [[mathematician]] [[Felix Hausdorff]].<ref>{{Cite journal |arxiv = 1101.1444|doi = 10.1214/11-STS370|title = Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data|journal = Statistical Science|volume = 27|issue = 2|pages = 247–277|year = 2012|last1 = Gneiting|first1 = Tilmann|last2 = Ševčíková|first2 = Hana|last3 = Percival|first3 = Donald B.|s2cid = 88512325}}</ref> For instance, the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an [[integer]] agreeing with the usual sense of dimension, also known as the [[Inductive dimension|topological dimension]]. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of [[scaling (geometry)|scaling]] and [[self-similarity]], one is led to the conclusion that particular objects—including [[fractal]]s—have non-integer Hausdorff dimensions. Because of the significant technical advances made by [[Abram Samoilovitch Besicovitch]] allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the ''Hausdorff–Besicovitch dimension.''

More specifically, the Hausdorff dimension is a dimensional number associated with a [[metric space]], i.e. a set where the distances between all members are defined. The dimension is drawn from the [[Extended real number line|extended real numbers]], <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real [[vector space]]. That is, the Hausdorff dimension of an ''n''-dimensional [[inner product space]] equals ''n''. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that [[fractal|irregular sets]] can have noninteger Hausdorff dimensions. For instance, the [[Koch snowflake]] shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new [[equilateral]] triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.<ref>Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see [http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm], accessed 5 March 2015.</ref> That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.<ref name=CampbellAnnenberg15/> Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD.<ref name=ClaytonSCTPLS96>Keith Clayton, 1996, "Fractals and the Fractal Dimension," ''Basic Concepts in Nonlinear Dynamics and Chaos'' (workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see [http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html], accessed 5 March 2015.</ref> This equation is easily solved for D, yielding the ratio of logarithms (or [[natural logarithm]]s) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or [[Minkowski–Bouligand dimension]].

==Intuition==
{{refimprove section|date=March 2015}}
The intuitive concept of dimension of a geometric object ''X'' is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the [[cardinality]] of the [[real plane]] is equal to the cardinality of the [[real line]] (this can be seen by an [[Cantor's diagonal argument|argument]] involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a [[space-filling curve]] shows that one can even map the real line to the real plane [[Surjective function|surjectively]] (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and ''continuously'', so that a one-dimensional object completely fills up a higher-dimensional object.

Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called [[Lebesgue covering dimension]], explains why. This dimension is ''n'' if, in every covering of ''X'' by small open balls, there is at least one point where ''n'' + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension ''n'' = 1.

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A [[fractal]] has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the [[metric space|metric]]. Consider the number ''N''(''r'') of [[ball (mathematics)|balls]] of radius at most ''r'' required to cover ''X'' completely. When ''r'' is very small, ''N''(''r'') grows polynomially with 1/''r''. For a sufficiently well-behaved ''X'', the Hausdorff dimension is the unique number ''d'' such that N(''r'') grows as 1/''rd'' as ''r'' approaches zero. More precisely, this defines the [[Minkowski–Bouligand dimension|box-counting dimension]], which equals the Hausdorff dimension when the value ''d'' is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But [[Benoit Mandelbrot]] observed that [[fractal]]s, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:

<blockquote>Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.<ref name="mandelbrot">{{cite book | last = Mandelbrot | first = Benoît | author-link = Benoit Mandelbrot | title = The Fractal Geometry of Nature | publisher = W. H. Freeman | series = Lecture notes in mathematics 1358 | year = 1982 | isbn = 0-7167-1186-9 | url-access = registration | url = https://archive.org/details/fractalgeometryo00beno }}</ref></blockquote>

For fractals that occur in nature, the Hausdorff and [[Minkowski–Bouligand dimension|box-counting dimension]] coincide. The [[packing dimension]] is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.{{Example needed|s|date=January 2022}}

==Formal definitions==
{{unreferenced section|date=March 2015}}

===Hausdorff content===
Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞), the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by
:<math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a countable cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math>
In other words, <math>C_H^d(S)</math> is the [[infimum]] of the set of numbers <math>\delta \geq 0</math> such that there is some (indexed) collection of [[ball (mathematics)|ball]]s <math>\{B(x_i,r_i):i\in I\}</math> covering ''S'' with ''ri'' > 0 for each ''i'' ∈ ''I'' that satisfies <math>\sum_{i\in I} r_i^d<\delta </math>. (Here, we use the standard convention that [[infimum|inf Ø = ∞]].)

===Hausdorff measure===
The Hausdorff outer measure is different from the unbounded Hausdorff content in that rather than considering all possible coverings of ''S'', we see what happens when the sizes of the balls go to zero. For <math>d \geq 0 </math>, we define the ''d''-dimensional Hausdorff outer measure of ''S'' as
:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a countable cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>

===Hausdorff dimension===
The '''Hausdorff dimension''' of ''X'' is defined by
:<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>

Equivalently, dimH(''X'') may be defined as the [[infimum]] of the set of ''d'' ∈ [0, ∞) such that the ''d''-dimensional [[Hausdorff measure]] of ''X'' is zero. This is the same as the supremum of the set of ''d'' ∈ [0, ∞) such that the ''d''-dimensional Hausdorff measure of ''X'' is infinite (except that when this latter set of numbers ''d'' is empty the Hausdorff dimension is zero).

==Examples==
[[Image:Sierpinski deep.svg|thumb|250px|Dimension of a further [[fractal]] example. The [[Sierpinski triangle]], an object with Hausdorff dimension of log(3)/log(2)≈1.58.<ref name=ClaytonSCTPLS96/>]]
* [[Countable set]]s have Hausdorff dimension 0.<ref name="schleicher">{{cite journal |last1=Schleicher |first1=Dierk |title=Hausdorff Dimension, Its Properties, and Its Surprises |journal=The American Mathematical Monthly |date=June 2007 |volume=114 |issue=6 |pages=509–528 |doi=10.1080/00029890.2007.11920440 |language=en |issn=0002-9890|arxiv=math/0505099 |s2cid=9811750 }}</ref>
* The [[Euclidean space]] ℝ''n'' has Hausdorff dimension ''n'', and the circle '''S'''1 has Hausdorff dimension 1.<ref name="schleicher" />
* [[Fractal]]s often are spaces whose Hausdorff dimension strictly exceeds the [[topological dimension]].<ref name="mandelbrot" /> For example, the [[Cantor set]], a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63.<ref>{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}</ref> The [[Sierpinski triangle]] is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58.<ref name=CampbellAnnenberg15/> These Hausdorff dimensions are related to the "critical exponent" of the [[Master theorem (analysis of algorithms)|Master theorem]] for solving [[Recurrence relation|recurrence relations]] in the [[analysis of algorithms]].
* [[Space-filling curve]]s like the [[Peano curve]] have the same Hausdorff dimension as the space they fill.
* The trajectory of [[Brownian motion]] in dimension 2 and above is conjectured to be Hausdorff dimension 2.<ref>{{cite book | last=Morters | first=Peres | title= Brownian Motion | publisher=[[Cambridge University Press]] | year=2010 }}</ref>
[[image:Great Britain Hausdorff.svg|thumb|250px|Estimating the Hausdorff dimension of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coast of Great Britain]]]]
* [[Lewis Fry Richardson]] has performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of [[South Africa]] to 1.25 for the west coast of [[Great Britain]].<ref name="mandelbrot" />

==Properties of Hausdorff dimension==
{{refimprove section|date=March 2015}}

=== Hausdorff dimension and inductive dimension ===
Let ''X'' be an arbitrary [[Separable space|separable]] metric space. There is a [[topology|topological]] notion of [[inductive dimension]] for ''X'' which is defined recursively. It is always an integer (or +∞) and is denoted dimind(''X'').

'''Theorem'''. Suppose ''X'' is non-empty. Then
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
Moreover,
:<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
where ''Y'' ranges over metric spaces [[homeomorphic]] to ''X''. In other words, ''X'' and ''Y'' have the same underlying set of points and the metric ''d''''Y'' of ''Y'' is topologically equivalent to ''d''''X''.

These results were originally established by [[Edward Szpilrajn]] (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.{{full citation needed|date=March 2015}}

=== Hausdorff dimension and Minkowski dimension ===
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of [[rational number|rational]] points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.

=== Hausdorff dimensions and Frostman measures ===
If there is a [[measure (mathematics)|measure]] μ defined on [[Borel measure|Borel]] subsets of a metric space ''X'' such that ''μ''(''X'') > 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then dimHaus(''X'') ≥ ''s''. A partial converse is provided by [[Frostman's lemma]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>

=== Behaviour under unions and products ===
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then

:<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>

This can be verified directly from the definition.

If ''X'' and ''Y'' are non-empty metric spaces, then the Hausdorff dimension of their product satisfies<ref>{{cite journal |author=Marstrand, J. M. |title=The dimension of Cartesian product sets |journal=Proc. Cambridge Philos. Soc. |volume=50 |issue=3 |pages=198–202 |year=1954 |doi=10.1017/S0305004100029236 |bibcode = 1954PCPS...50..198M }}</ref>

:<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.<ref>{{cite book | last = Falconer | first = Kenneth J. | title = Fractal geometry. Mathematical foundations and applications | publisher = John Wiley & Sons, Inc., Hoboken, New Jersey | year = 2003 }}</ref> In the opposite direction, it is known that when ''X'' and ''Y'' are Borel subsets of '''R'''''n'', the Hausdorff dimension of ''X'' × ''Y'' is bounded from above by the Hausdorff dimension of ''X'' plus the [[packing dimension|upper packing dimension]] of ''Y''. These facts are discussed in Mattila (1995).

==Self-similar sets==
{{refimprove section|date=March 2015}}
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set ''E'' is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(''E'') = ''E'', although the exact definition is given below.

<blockquote>'''Theorem'''. Suppose

:<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>

are [[Contraction mapping|contractive]] mappings on '''R'''''n'' with contraction constant ''rj'' < 1. Then there is a unique ''non-empty'' compact set ''A'' such that

:<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
</blockquote>

The theorem follows from [[Stefan Banach]]'s [[Contractive mapping theorem|contractive mapping fixed point theorem]] applied to the complete metric space of non-empty compact subsets of '''R'''''n'' with the [[Hausdorff distance]].<ref>{{cite book |author=Falconer, K. J. |title=The Geometry of Fractal Sets |publisher=Cambridge University Press |location=Cambridge, UK |year=1985 |isbn=0-521-25694-1 |chapter=Theorem 8.3}}</ref>

===The open set condition===
{{main|Open set condition}}
To determine the dimension of the self-similar set ''A'' (in certain cases), we need a technical condition called the ''open set condition'' (OSC) on the sequence of contractions ψ''i''.

There is a relatively compact open set ''V'' such that

:<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>

where the sets in union on the left are pairwise [[disjoint sets|disjoint]].

The open set condition is a separation condition that ensures the images ψ''i''(''V'') do not overlap "too much".

'''Theorem'''. Suppose the open set condition holds and each ψ''i'' is a similitude, that is a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is ''s'' where ''s'' is the unique solution of<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

The contraction coefficient of a similitude is the magnitude of the dilation.

We can use this theorem to compute the Hausdorff dimension of the [[Sierpinski triangle]]. In general, a set ''E'' which is a fixed point of a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections

:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''Hs'' denotes [[Hausdorff measure]]. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:

'''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

==See also==
* [[List of fractals by Hausdorff dimension]] Examples of deterministic fractals, random and natural fractals.
* [[Assouad dimension]], another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
* [[Intrinsic dimension]]
* [[Packing dimension]]
* [[Fractal dimension]]

==References==
{{reflist}}

==Further reading==
* {{cite book |last1=Dodson |first1=M. Maurice |title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot |volume=72 |issue=1 |pages=305–347 |last2=Kristensen |first2=Simon |chapter=Hausdorff Dimension and Diophantine Approximation |date=June 12, 2003 |arxiv=math/0305399 |bibcode = 2003math......5399D |doi=10.1090/pspum/072.1/2112110|series=Proceedings of Symposia in Pure Mathematics |isbn=9780821836378 |s2cid=119613948 }}
* {{cite book |last1=Hurewicz |first1=Witold |author-link1=Witold Hurewicz |last2=Wallman |first2=Henry |author-link2=Henry Wallman |title=Dimension Theory |url=https://archive.org/details/in.ernet.dli.2015.84609 |publisher=Princeton University Press |year=1948 }}
* {{cite journal |author=E. Szpilrajn |author-link=Edward Marczewski |title=La dimension et la mesure |journal=Fundamenta Mathematicae |volume=28 |pages=81–9 |year=1937 }}
* {{cite journal
| last1=Marstrand
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
| volume=50
| issue=3
| pages=198–202
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
* {{Cite book
| last1=Mattila
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=[[Cambridge University Press]]
| isbn=978-0-521-65595-8 | year=1995}}
* {{cite journal |author=A. S. Besicovitch |author-link=A. S. Besicovitch |title=On Linear Sets of Points of Fractional Dimensions |journal=[[Mathematische Annalen]] |volume=101 |year=1929 | doi=10.1007/BF01454831| issue=1 |pages= 161–193|s2cid=125368661 }}
* {{cite journal |author1=A. S. Besicovitch |author-link1=A. S. Besicovitch |author2=H. D. Ursell |author-link2=H. D. Ursell |title=Sets of Fractional Dimensions |journal=Journal of the London Mathematical Society |volume=12 |year=1937 | issue=1 | doi=10.1112/jlms/s1-12.45.18 | pages=18–25 }} Several selections from this volume are reprinted in {{cite book |author=Edgar, Gerald A. |title=Classics on fractals |publisher=Addison-Wesley |location=Boston |year=1993 |isbn=0-201-58701-7}} See chapters 9,10,11
* {{cite journal |author=F. Hausdorff |author-link=F. Hausdorff |title=Dimension und äußeres Maß |journal=Mathematische Annalen |volume=79 |issue=1–2 |pages=157–179 |date=March 1919 |doi=10.1007/BF01457179|hdl=10338.dmlcz/100363 |s2cid=122001234 |url=http://dml.cz/bitstream/handle/10338.dmlcz/100363/CzechMathJ_09-1959-3_5.pdf }}
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}
*{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{Dimension topics}}

[[Category:Fractals]]
[[Category:Metric geometry]]
[[Category:Dimension theory]]

Open set condition

2022-02-02T03:08:01Z

IntegralPython: Adding short description: "Condition for self-similar fractals" (Shortdesc helper)

{{Short description|Condition for self-similar fractals}}
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

==Computing Hausdorff measure==

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==See also==
*[[List of fractals by Hausdorff dimension]]
*[[Packing dimension]]

==References==
{{reflist}}

Open set condition

2022-02-02T03:07:23Z

IntegralPython: new article taking some material from Hausdorff dimension - Open Set Condition is studied in its own right and is now tied to other measures and conditions.

In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

==Computing Hausdorff measure==

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==See also==
*[[List of fractals by Hausdorff dimension]]
*[[Packing dimension]]

==References==
{{reflist}}

User:IntegralPython/sandbox

2022-02-02T03:00:54Z

IntegralPython: /* Open set condition */ reference clean

My sandbox, for drafting articles or saving them because I feel like it.

[[Meta:Meta:Meta]]

[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]

==Open set condition==
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15-23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

===Computing Hausdorff measure===

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Hand-eye calibration problem==
{{collapse top| Hand-eye calibration problem}}
In robotics, the '''hand-eye calibration problem''', or the '''robot-sensor calibration problem''', is the problem of determining the transformation between a robot [[end-effector]] and a camera or the transformation between a robot base and the world coordinate system.<ref> Amy Tabb, Khalil M. Ahmad Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] 29 Jul 2019.</ref> It takes the form of {{math|AX{{=}}ZB}}, where ''A'' and ''B'' are two systems, usually a robot base and a camera, and {{math|X}} and {{math|Z}} are unknown transformation matrices. A highly studied special case of the problem occurs where {{math|X{{=}}Z}}, taking the form of the problem {{math|AX{{=}}XB}}. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions".<ref>Mili I. Shah, Roger D. Eastman, Tsai Hong Hong. [https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."] 22 March, 2012</ref> The covariance of {{math|X}} in the equation can be calculated for any randomly perturbed matrices {{math|A}} and {{math|B}}.<ref>Huy Nguyen, Quang-Cuong Pham. [https://arxiv.org/pdf/1706.03498.pdf "On the covariance of X in AX = XB."] 12 June, 2017.</ref>

===Methods===
Many different methods and solutions developed to solve the problem, broadly defined as either Separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of [[quaternion]]s to represent rotation matrices.

====Separable solutions====
Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''A}} represents a 3×3 rotation matrix and {{math|'''t'''A}} a 3×1 translation vector, the equation can be broken into two parts:<ref>[https://arxiv.org/pdf/1907.12425.pdf]</ref>
:{{math|'''R'''A'''R'''X{{=}}'''R'''Z'''R'''B}}
:{{math|'''R'''A'''t'''X+'''t'''A{{=}}'''R'''Z'''t'''B+'''t'''Z}}
Equation 2 becomes linear if {{math|'''R'''Z}} is known. As such, the most frequent approach is to {{math|'''R'''x}} and {{math|'''R'''z}} using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using [[quaternion]]s, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector.<ref name="tsapps">Mili Shah, et al. [https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."]</ref> Other solutions avoid this problem.

====Simultaneous solutions====
Simultaneous solutions are based on solving for both {{math|X}} and {{math|Z}} at the same time (rather than basing the solution of one part off of the other as in seperable solutions), propogation of error is significantly reduced.<ref name="dual-quaternions"> Algo Li, et al. [https://pdfs.semanticscholar.org/225d/e4ea2d3f18b7743bfeabf925fa603fc47bcb.pdf "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product."] International Journal of the Physical Sciences Vol. 5(10), pp. 1530-1536, 4 September, 2010. </ref> By formulating the matrices as [[dual quaternion]]s, it is possible to get a linear equation by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/> An alternative way applies the [[least squares| least squares method]] to the [[Kronecker product]] of the matrices {{math|A⊗B}}. As confirmed by experimental results, simultaneous solutions have less error than seperable quaternion solutions.<ref name="dual-quaternions"/>

====Iterative solutions====
Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing {{math|{{!}}{{!}}AX−XB{{!}}{{!}}}}. As the program iterates, it will converge on a solution to {{math|X}} independent to the initial robot orientation of {{math|'''R'''B}}. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>Zhiqiang Zhang, et al. [https://link.springer.com/article/10.1007/s11548-017-1646-x "A computationally efficient method for hand–eye calibration."] 19 July, 2017.</ref> However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.<ref name="tsapps"/>

*[https://ieeexplore.ieee.org/abstract/document/8788685/keywords#keywords] - Octonion solution
{{collapse bottom}}

==What is algebra?==
{{collapse top| What is Algebra}}

Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them.<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of [[arithmetic]] but with variables that take on the properties of numbers.<ref name=citeboyer /> This allows proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
:<math>ax^2+bx+c=0,</math>
where <math>a, b, c</math> are any given numbers (except that <math>a</math> cannot be <math>0</math>), the [[quadratic formula]] can be used to find the two unique values of the unknown quantity <math>x</math> which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the [[quadratic equation]] above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.<ref>{{cite book |last=Gattengo |first=Caleb |year=2010 |title=The Common Sense of Teaching Mathematics |publisher=Educational Solutions Inc. |isbn=978-0878252206 }}</ref>

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of [[real number| real]] or [[complex number]]s. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. Abstract algebra is the study of these entities and more.<ref>http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018</ref>

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as [[constructible number]]s and [[singularity theory|singularities]]. Reducing properties of geometric structures into algebraic structures has created subjects such as [[algebraic geometry]], [[geometric algebra]], and [[algebraic topology]].

Today, the study of algebra includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|publisher=|accessdate=5 October 2014}}</ref> where none of the first level areas (two digit entries) is called ''algebra''. Algebra instead includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebra]]s, 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used in 14-[[Algebraic geometry]] and 11-[[Number theory]] via [[algebraic number theory]].
{{collapse bottom}}
==Antiassociative algebra==
{{collapse top|Antiassociative algebra}}

An algebra antiassociative if (xy)z = -x(yz) for every case of x,y, and z.<ref>[https://books.google.com/books?id=_PEWt18egGgC&pg=PA235&lpg=PA235&dq=%22antiassociative%22+algebra+aplications&source=bl&ots=Atxm0cdUVs&sig=OQjjF3ig6NYCQwP6O9P8fLgwSDE&hl=en&sa=X&ved=2ahUKEwix94P66LPdAhVIu1MKHckzBNQQ6AEwCHoECAYQAQ#v=onepage&q=%22antiassociative%22%20algebra%20aplications&f=false "Non-Associative Algebra and Its Applications."] Page 235.</ref>
{{collapse bottom}}
==Ugandan Knuckles==
{{collapse top | Ugandan knuckles}}

Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref><ref>{{Cite web|url=https://heavy.com/games/2018/01/controversial-ugandan-knuckles-meme/|title=Controversial ‘Ugandan Knuckles’ Meme Has Infested VRChat|last=MacGregor|first=Collin|date=9 January 2018|website=Heavy.com|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref> The meme was a significant trend followed by several news organisations, including ''[[USA Today]]''.<ref>https://www.usatoday.com/story/tech/news/2018/02/09/ugandan-knuckles-do-you-know-de-wey-meme-explained/307575002/ Retrieved October 9 2018</ref>

===History===
On February 20 2017, YouTuber Gregzilla uploaded a video on Sonic Lost World featuring a parody picture of Knuckles the Echidna. On December 22 2017, a 3D model of the Knuckles sketch was released on ''[[DeviantArt]]''. That day, YouTuber Stahlsby uploaded a video in which several ''VRChat'' players wearing the parody costume trolled others by making clicking noises and saying "You do not know the way".<ref>https://knowyourmeme.com/memes/ugandan-knuckles Retrieved October 9 2018</ref> After that, more and more people flooded ''VRChat'' to troll others as Ugandan Knuckles, leading to controversy, as the mock Ugandan accent and quotations used were widely regarded as racist. However, The meme continued to gain popularity until about mid-January 2018, but had mostly subsided by February.<ref>https://trends.google.com/trends/explore?q=Ugandan%20Knuckles&geo=US retrieved October 9 2018</ref>

===Controversy===
Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=‘Ugandan Knuckles’ is overtaking VRChat|last=Alexander|first=Julia|publisher=[[Vox Media, Inc.]]|work=Polygon|date=October 9, 2018|accessdate=January 9, 2018}}</ref> On January 27 2018, the company [[Razer Inc.|Razer]] was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.<ref>https://gizmodo.com/does-razer-know-it-posted-a-racist-meme-1822485212 Retrieved October 9 2018</ref>

The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’s ‘Ugandan Knuckles’ Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>
{{collapse bottom}}

==References==
{{reflist}}

User:IntegralPython/sandbox

2022-02-02T02:46:35Z

IntegralPython: /* Open set condition */ another use

My sandbox, for drafting articles or saving them because I feel like it.

[[Meta:Meta:Meta]]

[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]

==Open set condition==
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>Moran, P.A.P. (1946) Additive Functions of Intervals and Hausdorff Measure. Proceedings-Cambridge Philosophical Society, 42, 15-23.
https://doi.org/10.1017/S0305004100022684</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf</ref>

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf</ref>

===Computing Hausdorff measure===

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Hand-eye calibration problem==
{{collapse top| Hand-eye calibration problem}}
In robotics, the '''hand-eye calibration problem''', or the '''robot-sensor calibration problem''', is the problem of determining the transformation between a robot [[end-effector]] and a camera or the transformation between a robot base and the world coordinate system.<ref> Amy Tabb, Khalil M. Ahmad Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] 29 Jul 2019.</ref> It takes the form of {{math|AX{{=}}ZB}}, where ''A'' and ''B'' are two systems, usually a robot base and a camera, and {{math|X}} and {{math|Z}} are unknown transformation matrices. A highly studied special case of the problem occurs where {{math|X{{=}}Z}}, taking the form of the problem {{math|AX{{=}}XB}}. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions".<ref>Mili I. Shah, Roger D. Eastman, Tsai Hong Hong. [https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."] 22 March, 2012</ref> The covariance of {{math|X}} in the equation can be calculated for any randomly perturbed matrices {{math|A}} and {{math|B}}.<ref>Huy Nguyen, Quang-Cuong Pham. [https://arxiv.org/pdf/1706.03498.pdf "On the covariance of X in AX = XB."] 12 June, 2017.</ref>

===Methods===
Many different methods and solutions developed to solve the problem, broadly defined as either Separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of [[quaternion]]s to represent rotation matrices.

====Separable solutions====
Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''A}} represents a 3×3 rotation matrix and {{math|'''t'''A}} a 3×1 translation vector, the equation can be broken into two parts:<ref>[https://arxiv.org/pdf/1907.12425.pdf]</ref>
:{{math|'''R'''A'''R'''X{{=}}'''R'''Z'''R'''B}}
:{{math|'''R'''A'''t'''X+'''t'''A{{=}}'''R'''Z'''t'''B+'''t'''Z}}
Equation 2 becomes linear if {{math|'''R'''Z}} is known. As such, the most frequent approach is to {{math|'''R'''x}} and {{math|'''R'''z}} using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using [[quaternion]]s, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector.<ref name="tsapps">Mili Shah, et al. [https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."]</ref> Other solutions avoid this problem.

====Simultaneous solutions====
Simultaneous solutions are based on solving for both {{math|X}} and {{math|Z}} at the same time (rather than basing the solution of one part off of the other as in seperable solutions), propogation of error is significantly reduced.<ref name="dual-quaternions"> Algo Li, et al. [https://pdfs.semanticscholar.org/225d/e4ea2d3f18b7743bfeabf925fa603fc47bcb.pdf "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product."] International Journal of the Physical Sciences Vol. 5(10), pp. 1530-1536, 4 September, 2010. </ref> By formulating the matrices as [[dual quaternion]]s, it is possible to get a linear equation by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/> An alternative way applies the [[least squares| least squares method]] to the [[Kronecker product]] of the matrices {{math|A⊗B}}. As confirmed by experimental results, simultaneous solutions have less error than seperable quaternion solutions.<ref name="dual-quaternions"/>

====Iterative solutions====
Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing {{math|{{!}}{{!}}AX−XB{{!}}{{!}}}}. As the program iterates, it will converge on a solution to {{math|X}} independent to the initial robot orientation of {{math|'''R'''B}}. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>Zhiqiang Zhang, et al. [https://link.springer.com/article/10.1007/s11548-017-1646-x "A computationally efficient method for hand–eye calibration."] 19 July, 2017.</ref> However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.<ref name="tsapps"/>

*[https://ieeexplore.ieee.org/abstract/document/8788685/keywords#keywords] - Octonion solution
{{collapse bottom}}

==What is algebra?==
{{collapse top| What is Algebra}}

Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them.<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of [[arithmetic]] but with variables that take on the properties of numbers.<ref name=citeboyer /> This allows proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
:<math>ax^2+bx+c=0,</math>
where <math>a, b, c</math> are any given numbers (except that <math>a</math> cannot be <math>0</math>), the [[quadratic formula]] can be used to find the two unique values of the unknown quantity <math>x</math> which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the [[quadratic equation]] above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.<ref>{{cite book |last=Gattengo |first=Caleb |year=2010 |title=The Common Sense of Teaching Mathematics |publisher=Educational Solutions Inc. |isbn=978-0878252206 }}</ref>

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of [[real number| real]] or [[complex number]]s. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. Abstract algebra is the study of these entities and more.<ref>http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018</ref>

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as [[constructible number]]s and [[singularity theory|singularities]]. Reducing properties of geometric structures into algebraic structures has created subjects such as [[algebraic geometry]], [[geometric algebra]], and [[algebraic topology]].

Today, the study of algebra includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|publisher=|accessdate=5 October 2014}}</ref> where none of the first level areas (two digit entries) is called ''algebra''. Algebra instead includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebra]]s, 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used in 14-[[Algebraic geometry]] and 11-[[Number theory]] via [[algebraic number theory]].
{{collapse bottom}}
==Antiassociative algebra==
{{collapse top|Antiassociative algebra}}

An algebra antiassociative if (xy)z = -x(yz) for every case of x,y, and z.<ref>[https://books.google.com/books?id=_PEWt18egGgC&pg=PA235&lpg=PA235&dq=%22antiassociative%22+algebra+aplications&source=bl&ots=Atxm0cdUVs&sig=OQjjF3ig6NYCQwP6O9P8fLgwSDE&hl=en&sa=X&ved=2ahUKEwix94P66LPdAhVIu1MKHckzBNQQ6AEwCHoECAYQAQ#v=onepage&q=%22antiassociative%22%20algebra%20aplications&f=false "Non-Associative Algebra and Its Applications."] Page 235.</ref>
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==Ugandan Knuckles==
{{collapse top | Ugandan knuckles}}

Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref><ref>{{Cite web|url=https://heavy.com/games/2018/01/controversial-ugandan-knuckles-meme/|title=Controversial ‘Ugandan Knuckles’ Meme Has Infested VRChat|last=MacGregor|first=Collin|date=9 January 2018|website=Heavy.com|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref> The meme was a significant trend followed by several news organisations, including ''[[USA Today]]''.<ref>https://www.usatoday.com/story/tech/news/2018/02/09/ugandan-knuckles-do-you-know-de-wey-meme-explained/307575002/ Retrieved October 9 2018</ref>

===History===
On February 20 2017, YouTuber Gregzilla uploaded a video on Sonic Lost World featuring a parody picture of Knuckles the Echidna. On December 22 2017, a 3D model of the Knuckles sketch was released on ''[[DeviantArt]]''. That day, YouTuber Stahlsby uploaded a video in which several ''VRChat'' players wearing the parody costume trolled others by making clicking noises and saying "You do not know the way".<ref>https://knowyourmeme.com/memes/ugandan-knuckles Retrieved October 9 2018</ref> After that, more and more people flooded ''VRChat'' to troll others as Ugandan Knuckles, leading to controversy, as the mock Ugandan accent and quotations used were widely regarded as racist. However, The meme continued to gain popularity until about mid-January 2018, but had mostly subsided by February.<ref>https://trends.google.com/trends/explore?q=Ugandan%20Knuckles&geo=US retrieved October 9 2018</ref>

===Controversy===
Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=‘Ugandan Knuckles’ is overtaking VRChat|last=Alexander|first=Julia|publisher=[[Vox Media, Inc.]]|work=Polygon|date=October 9, 2018|accessdate=January 9, 2018}}</ref> On January 27 2018, the company [[Razer Inc.|Razer]] was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.<ref>https://gizmodo.com/does-razer-know-it-posted-a-racist-meme-1822485212 Retrieved October 9 2018</ref>

The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’s ‘Ugandan Knuckles’ Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>
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==References==
{{reflist}}

User:IntegralPython/sandbox

2022-02-02T02:39:35Z

IntegralPython: /* Open set condition */

My sandbox, for drafting articles or saving them because I feel like it.

[[Meta:Meta:Meta]]

[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]

==Open set condition==
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>Moran, P.A.P. (1946) Additive Functions of Intervals and Hausdorff Measure. Proceedings-Cambridge Philosophical Society, 42, 15-23.
https://doi.org/10.1017/S0305004100022684</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket.

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf</ref>

===Computing Hausdorff measure===

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Hand-eye calibration problem==
{{collapse top| Hand-eye calibration problem}}
In robotics, the '''hand-eye calibration problem''', or the '''robot-sensor calibration problem''', is the problem of determining the transformation between a robot [[end-effector]] and a camera or the transformation between a robot base and the world coordinate system.<ref> Amy Tabb, Khalil M. Ahmad Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] 29 Jul 2019.</ref> It takes the form of {{math|AX{{=}}ZB}}, where ''A'' and ''B'' are two systems, usually a robot base and a camera, and {{math|X}} and {{math|Z}} are unknown transformation matrices. A highly studied special case of the problem occurs where {{math|X{{=}}Z}}, taking the form of the problem {{math|AX{{=}}XB}}. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions".<ref>Mili I. Shah, Roger D. Eastman, Tsai Hong Hong. [https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."] 22 March, 2012</ref> The covariance of {{math|X}} in the equation can be calculated for any randomly perturbed matrices {{math|A}} and {{math|B}}.<ref>Huy Nguyen, Quang-Cuong Pham. [https://arxiv.org/pdf/1706.03498.pdf "On the covariance of X in AX = XB."] 12 June, 2017.</ref>

===Methods===
Many different methods and solutions developed to solve the problem, broadly defined as either Separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of [[quaternion]]s to represent rotation matrices.

====Separable solutions====
Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''A}} represents a 3×3 rotation matrix and {{math|'''t'''A}} a 3×1 translation vector, the equation can be broken into two parts:<ref>[https://arxiv.org/pdf/1907.12425.pdf]</ref>
:{{math|'''R'''A'''R'''X{{=}}'''R'''Z'''R'''B}}
:{{math|'''R'''A'''t'''X+'''t'''A{{=}}'''R'''Z'''t'''B+'''t'''Z}}
Equation 2 becomes linear if {{math|'''R'''Z}} is known. As such, the most frequent approach is to {{math|'''R'''x}} and {{math|'''R'''z}} using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using [[quaternion]]s, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector.<ref name="tsapps">Mili Shah, et al. [https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."]</ref> Other solutions avoid this problem.

====Simultaneous solutions====
Simultaneous solutions are based on solving for both {{math|X}} and {{math|Z}} at the same time (rather than basing the solution of one part off of the other as in seperable solutions), propogation of error is significantly reduced.<ref name="dual-quaternions"> Algo Li, et al. [https://pdfs.semanticscholar.org/225d/e4ea2d3f18b7743bfeabf925fa603fc47bcb.pdf "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product."] International Journal of the Physical Sciences Vol. 5(10), pp. 1530-1536, 4 September, 2010. </ref> By formulating the matrices as [[dual quaternion]]s, it is possible to get a linear equation by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/> An alternative way applies the [[least squares| least squares method]] to the [[Kronecker product]] of the matrices {{math|A⊗B}}. As confirmed by experimental results, simultaneous solutions have less error than seperable quaternion solutions.<ref name="dual-quaternions"/>

====Iterative solutions====
Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing {{math|{{!}}{{!}}AX−XB{{!}}{{!}}}}. As the program iterates, it will converge on a solution to {{math|X}} independent to the initial robot orientation of {{math|'''R'''B}}. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>Zhiqiang Zhang, et al. [https://link.springer.com/article/10.1007/s11548-017-1646-x "A computationally efficient method for hand–eye calibration."] 19 July, 2017.</ref> However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.<ref name="tsapps"/>

*[https://ieeexplore.ieee.org/abstract/document/8788685/keywords#keywords] - Octonion solution
{{collapse bottom}}

==What is algebra?==
{{collapse top| What is Algebra}}

Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them.<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of [[arithmetic]] but with variables that take on the properties of numbers.<ref name=citeboyer /> This allows proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
:<math>ax^2+bx+c=0,</math>
where <math>a, b, c</math> are any given numbers (except that <math>a</math> cannot be <math>0</math>), the [[quadratic formula]] can be used to find the two unique values of the unknown quantity <math>x</math> which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the [[quadratic equation]] above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.<ref>{{cite book |last=Gattengo |first=Caleb |year=2010 |title=The Common Sense of Teaching Mathematics |publisher=Educational Solutions Inc. |isbn=978-0878252206 }}</ref>

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of [[real number| real]] or [[complex number]]s. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. Abstract algebra is the study of these entities and more.<ref>http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018</ref>

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as [[constructible number]]s and [[singularity theory|singularities]]. Reducing properties of geometric structures into algebraic structures has created subjects such as [[algebraic geometry]], [[geometric algebra]], and [[algebraic topology]].

Today, the study of algebra includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|publisher=|accessdate=5 October 2014}}</ref> where none of the first level areas (two digit entries) is called ''algebra''. Algebra instead includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebra]]s, 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used in 14-[[Algebraic geometry]] and 11-[[Number theory]] via [[algebraic number theory]].
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==Antiassociative algebra==
{{collapse top|Antiassociative algebra}}

An algebra antiassociative if (xy)z = -x(yz) for every case of x,y, and z.<ref>[https://books.google.com/books?id=_PEWt18egGgC&pg=PA235&lpg=PA235&dq=%22antiassociative%22+algebra+aplications&source=bl&ots=Atxm0cdUVs&sig=OQjjF3ig6NYCQwP6O9P8fLgwSDE&hl=en&sa=X&ved=2ahUKEwix94P66LPdAhVIu1MKHckzBNQQ6AEwCHoECAYQAQ#v=onepage&q=%22antiassociative%22%20algebra%20aplications&f=false "Non-Associative Algebra and Its Applications."] Page 235.</ref>
{{collapse bottom}}
==Ugandan Knuckles==
{{collapse top | Ugandan knuckles}}

Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref><ref>{{Cite web|url=https://heavy.com/games/2018/01/controversial-ugandan-knuckles-meme/|title=Controversial ‘Ugandan Knuckles’ Meme Has Infested VRChat|last=MacGregor|first=Collin|date=9 January 2018|website=Heavy.com|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref> The meme was a significant trend followed by several news organisations, including ''[[USA Today]]''.<ref>https://www.usatoday.com/story/tech/news/2018/02/09/ugandan-knuckles-do-you-know-de-wey-meme-explained/307575002/ Retrieved October 9 2018</ref>

===History===
On February 20 2017, YouTuber Gregzilla uploaded a video on Sonic Lost World featuring a parody picture of Knuckles the Echidna. On December 22 2017, a 3D model of the Knuckles sketch was released on ''[[DeviantArt]]''. That day, YouTuber Stahlsby uploaded a video in which several ''VRChat'' players wearing the parody costume trolled others by making clicking noises and saying "You do not know the way".<ref>https://knowyourmeme.com/memes/ugandan-knuckles Retrieved October 9 2018</ref> After that, more and more people flooded ''VRChat'' to troll others as Ugandan Knuckles, leading to controversy, as the mock Ugandan accent and quotations used were widely regarded as racist. However, The meme continued to gain popularity until about mid-January 2018, but had mostly subsided by February.<ref>https://trends.google.com/trends/explore?q=Ugandan%20Knuckles&geo=US retrieved October 9 2018</ref>

===Controversy===
Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=‘Ugandan Knuckles’ is overtaking VRChat|last=Alexander|first=Julia|publisher=[[Vox Media, Inc.]]|work=Polygon|date=October 9, 2018|accessdate=January 9, 2018}}</ref> On January 27 2018, the company [[Razer Inc.|Razer]] was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.<ref>https://gizmodo.com/does-razer-know-it-posted-a-racist-meme-1822485212 Retrieved October 9 2018</ref>

The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’s ‘Ugandan Knuckles’ Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>
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==References==
{{reflist}}

User:IntegralPython/sandbox

2022-02-02T02:30:08Z

IntegralPython: /* Open set condition */ improve

My sandbox, for drafting articles or saving them because I feel like it.

[[Meta:Meta:Meta]]

[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]

==Open set condition==
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>Moran, P.A.P. (1946) Additive Functions of Intervals and Hausdorff Measure. Proceedings-Cambridge Philosophical Society, 42, 15-23.
https://doi.org/10.1017/S0305004100022684</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket.

===Computing Hausdorff measure===

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.

==Hand-eye calibration problem==
{{collapse top| Hand-eye calibration problem}}
In robotics, the '''hand-eye calibration problem''', or the '''robot-sensor calibration problem''', is the problem of determining the transformation between a robot [[end-effector]] and a camera or the transformation between a robot base and the world coordinate system.<ref> Amy Tabb, Khalil M. Ahmad Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] 29 Jul 2019.</ref> It takes the form of {{math|AX{{=}}ZB}}, where ''A'' and ''B'' are two systems, usually a robot base and a camera, and {{math|X}} and {{math|Z}} are unknown transformation matrices. A highly studied special case of the problem occurs where {{math|X{{=}}Z}}, taking the form of the problem {{math|AX{{=}}XB}}. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions".<ref>Mili I. Shah, Roger D. Eastman, Tsai Hong Hong. [https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."] 22 March, 2012</ref> The covariance of {{math|X}} in the equation can be calculated for any randomly perturbed matrices {{math|A}} and {{math|B}}.<ref>Huy Nguyen, Quang-Cuong Pham. [https://arxiv.org/pdf/1706.03498.pdf "On the covariance of X in AX = XB."] 12 June, 2017.</ref>

===Methods===
Many different methods and solutions developed to solve the problem, broadly defined as either Separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of [[quaternion]]s to represent rotation matrices.

====Separable solutions====
Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''A}} represents a 3×3 rotation matrix and {{math|'''t'''A}} a 3×1 translation vector, the equation can be broken into two parts:<ref>[https://arxiv.org/pdf/1907.12425.pdf]</ref>
:{{math|'''R'''A'''R'''X{{=}}'''R'''Z'''R'''B}}
:{{math|'''R'''A'''t'''X+'''t'''A{{=}}'''R'''Z'''t'''B+'''t'''Z}}
Equation 2 becomes linear if {{math|'''R'''Z}} is known. As such, the most frequent approach is to {{math|'''R'''x}} and {{math|'''R'''z}} using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using [[quaternion]]s, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector.<ref name="tsapps">Mili Shah, et al. [https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."]</ref> Other solutions avoid this problem.

====Simultaneous solutions====
Simultaneous solutions are based on solving for both {{math|X}} and {{math|Z}} at the same time (rather than basing the solution of one part off of the other as in seperable solutions), propogation of error is significantly reduced.<ref name="dual-quaternions"> Algo Li, et al. [https://pdfs.semanticscholar.org/225d/e4ea2d3f18b7743bfeabf925fa603fc47bcb.pdf "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product."] International Journal of the Physical Sciences Vol. 5(10), pp. 1530-1536, 4 September, 2010. </ref> By formulating the matrices as [[dual quaternion]]s, it is possible to get a linear equation by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/> An alternative way applies the [[least squares| least squares method]] to the [[Kronecker product]] of the matrices {{math|A⊗B}}. As confirmed by experimental results, simultaneous solutions have less error than seperable quaternion solutions.<ref name="dual-quaternions"/>

====Iterative solutions====
Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing {{math|{{!}}{{!}}AX−XB{{!}}{{!}}}}. As the program iterates, it will converge on a solution to {{math|X}} independent to the initial robot orientation of {{math|'''R'''B}}. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>Zhiqiang Zhang, et al. [https://link.springer.com/article/10.1007/s11548-017-1646-x "A computationally efficient method for hand–eye calibration."] 19 July, 2017.</ref> However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.<ref name="tsapps"/>

*[https://ieeexplore.ieee.org/abstract/document/8788685/keywords#keywords] - Octonion solution
{{collapse bottom}}

==What is algebra?==
{{collapse top| What is Algebra}}

Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them.<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of [[arithmetic]] but with variables that take on the properties of numbers.<ref name=citeboyer /> This allows proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
:<math>ax^2+bx+c=0,</math>
where <math>a, b, c</math> are any given numbers (except that <math>a</math> cannot be <math>0</math>), the [[quadratic formula]] can be used to find the two unique values of the unknown quantity <math>x</math> which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the [[quadratic equation]] above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.<ref>{{cite book |last=Gattengo |first=Caleb |year=2010 |title=The Common Sense of Teaching Mathematics |publisher=Educational Solutions Inc. |isbn=978-0878252206 }}</ref>

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of [[real number| real]] or [[complex number]]s. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. Abstract algebra is the study of these entities and more.<ref>http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018</ref>

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as [[constructible number]]s and [[singularity theory|singularities]]. Reducing properties of geometric structures into algebraic structures has created subjects such as [[algebraic geometry]], [[geometric algebra]], and [[algebraic topology]].

Today, the study of algebra includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|publisher=|accessdate=5 October 2014}}</ref> where none of the first level areas (two digit entries) is called ''algebra''. Algebra instead includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebra]]s, 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used in 14-[[Algebraic geometry]] and 11-[[Number theory]] via [[algebraic number theory]].
{{collapse bottom}}
==Antiassociative algebra==
{{collapse top|Antiassociative algebra}}

An algebra antiassociative if (xy)z = -x(yz) for every case of x,y, and z.<ref>[https://books.google.com/books?id=_PEWt18egGgC&pg=PA235&lpg=PA235&dq=%22antiassociative%22+algebra+aplications&source=bl&ots=Atxm0cdUVs&sig=OQjjF3ig6NYCQwP6O9P8fLgwSDE&hl=en&sa=X&ved=2ahUKEwix94P66LPdAhVIu1MKHckzBNQQ6AEwCHoECAYQAQ#v=onepage&q=%22antiassociative%22%20algebra%20aplications&f=false "Non-Associative Algebra and Its Applications."] Page 235.</ref>
{{collapse bottom}}
==Ugandan Knuckles==
{{collapse top | Ugandan knuckles}}

Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref><ref>{{Cite web|url=https://heavy.com/games/2018/01/controversial-ugandan-knuckles-meme/|title=Controversial ‘Ugandan Knuckles’ Meme Has Infested VRChat|last=MacGregor|first=Collin|date=9 January 2018|website=Heavy.com|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref> The meme was a significant trend followed by several news organisations, including ''[[USA Today]]''.<ref>https://www.usatoday.com/story/tech/news/2018/02/09/ugandan-knuckles-do-you-know-de-wey-meme-explained/307575002/ Retrieved October 9 2018</ref>

===History===
On February 20 2017, YouTuber Gregzilla uploaded a video on Sonic Lost World featuring a parody picture of Knuckles the Echidna. On December 22 2017, a 3D model of the Knuckles sketch was released on ''[[DeviantArt]]''. That day, YouTuber Stahlsby uploaded a video in which several ''VRChat'' players wearing the parody costume trolled others by making clicking noises and saying "You do not know the way".<ref>https://knowyourmeme.com/memes/ugandan-knuckles Retrieved October 9 2018</ref> After that, more and more people flooded ''VRChat'' to troll others as Ugandan Knuckles, leading to controversy, as the mock Ugandan accent and quotations used were widely regarded as racist. However, The meme continued to gain popularity until about mid-January 2018, but had mostly subsided by February.<ref>https://trends.google.com/trends/explore?q=Ugandan%20Knuckles&geo=US retrieved October 9 2018</ref>

===Controversy===
Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=‘Ugandan Knuckles’ is overtaking VRChat|last=Alexander|first=Julia|publisher=[[Vox Media, Inc.]]|work=Polygon|date=October 9, 2018|accessdate=January 9, 2018}}</ref> On January 27 2018, the company [[Razer Inc.|Razer]] was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.<ref>https://gizmodo.com/does-razer-know-it-posted-a-racist-meme-1822485212 Retrieved October 9 2018</ref>

The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’s ‘Ugandan Knuckles’ Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>
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==References==
{{reflist}}

User:IntegralPython/sandbox

2022-02-02T02:22:04Z

IntegralPython: started on new article

My sandbox, for drafting articles or saving them because I feel like it.

[[Meta:Meta:Meta]]

[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]

==Open set condition==
In fractal geometry, the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an iterative function system of contractive mappings fi, the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.

The open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket.

https://www.jstor.org/stable/4097989?read-now=1&refreqid=excelsior%3A19b81930eca74e0a264d556ab56211ae&seq=1#page_scan_tab_contents

===Computing Hausdorff measure===

When the open set condition holds and each ψ''i'' is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>

:<math> \sum_{i=1}^m r_i^s = 1. </math>

where ri is the magnitude of the dilation of the similitude.

With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''1, ''a''2, ''a''3 in the plane '''R'''2 and let ψ''i'' be the dilation of ratio 1/2 around ''ai''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a set ''E'' which is a fixed point of a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections

:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''Hs'' denotes [[Hausdorff measure]]. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally. Indeed, under the same conditions as above, the unique fixed point of ψ is self-similar.

==Hand-eye calibration problem==
{{collapse top| Hand-eye calibration problem}}
In robotics, the '''hand-eye calibration problem''', or the '''robot-sensor calibration problem''', is the problem of determining the transformation between a robot [[end-effector]] and a camera or the transformation between a robot base and the world coordinate system.<ref> Amy Tabb, Khalil M. Ahmad Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] 29 Jul 2019.</ref> It takes the form of {{math|AX{{=}}ZB}}, where ''A'' and ''B'' are two systems, usually a robot base and a camera, and {{math|X}} and {{math|Z}} are unknown transformation matrices. A highly studied special case of the problem occurs where {{math|X{{=}}Z}}, taking the form of the problem {{math|AX{{=}}XB}}. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions".<ref>Mili I. Shah, Roger D. Eastman, Tsai Hong Hong. [https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."] 22 March, 2012</ref> The covariance of {{math|X}} in the equation can be calculated for any randomly perturbed matrices {{math|A}} and {{math|B}}.<ref>Huy Nguyen, Quang-Cuong Pham. [https://arxiv.org/pdf/1706.03498.pdf "On the covariance of X in AX = XB."] 12 June, 2017.</ref>

===Methods===
Many different methods and solutions developed to solve the problem, broadly defined as either Separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of [[quaternion]]s to represent rotation matrices.

====Separable solutions====
Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''A}} represents a 3×3 rotation matrix and {{math|'''t'''A}} a 3×1 translation vector, the equation can be broken into two parts:<ref>[https://arxiv.org/pdf/1907.12425.pdf]</ref>
:{{math|'''R'''A'''R'''X{{=}}'''R'''Z'''R'''B}}
:{{math|'''R'''A'''t'''X+'''t'''A{{=}}'''R'''Z'''t'''B+'''t'''Z}}
Equation 2 becomes linear if {{math|'''R'''Z}} is known. As such, the most frequent approach is to {{math|'''R'''x}} and {{math|'''R'''z}} using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using [[quaternion]]s, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector.<ref name="tsapps">Mili Shah, et al. [https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."]</ref> Other solutions avoid this problem.

====Simultaneous solutions====
Simultaneous solutions are based on solving for both {{math|X}} and {{math|Z}} at the same time (rather than basing the solution of one part off of the other as in seperable solutions), propogation of error is significantly reduced.<ref name="dual-quaternions"> Algo Li, et al. [https://pdfs.semanticscholar.org/225d/e4ea2d3f18b7743bfeabf925fa603fc47bcb.pdf "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product."] International Journal of the Physical Sciences Vol. 5(10), pp. 1530-1536, 4 September, 2010. </ref> By formulating the matrices as [[dual quaternion]]s, it is possible to get a linear equation by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/> An alternative way applies the [[least squares| least squares method]] to the [[Kronecker product]] of the matrices {{math|A⊗B}}. As confirmed by experimental results, simultaneous solutions have less error than seperable quaternion solutions.<ref name="dual-quaternions"/>

====Iterative solutions====
Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing {{math|{{!}}{{!}}AX−XB{{!}}{{!}}}}. As the program iterates, it will converge on a solution to {{math|X}} independent to the initial robot orientation of {{math|'''R'''B}}. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>Zhiqiang Zhang, et al. [https://link.springer.com/article/10.1007/s11548-017-1646-x "A computationally efficient method for hand–eye calibration."] 19 July, 2017.</ref> However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.<ref name="tsapps"/>

*[https://ieeexplore.ieee.org/abstract/document/8788685/keywords#keywords] - Octonion solution
{{collapse bottom}}

==What is algebra?==
{{collapse top| What is Algebra}}

Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them.<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of [[arithmetic]] but with variables that take on the properties of numbers.<ref name=citeboyer /> This allows proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
:<math>ax^2+bx+c=0,</math>
where <math>a, b, c</math> are any given numbers (except that <math>a</math> cannot be <math>0</math>), the [[quadratic formula]] can be used to find the two unique values of the unknown quantity <math>x</math> which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the [[quadratic equation]] above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.<ref>{{cite book |last=Gattengo |first=Caleb |year=2010 |title=The Common Sense of Teaching Mathematics |publisher=Educational Solutions Inc. |isbn=978-0878252206 }}</ref>

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of [[real number| real]] or [[complex number]]s. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. Abstract algebra is the study of these entities and more.<ref>http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018</ref>

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as [[constructible number]]s and [[singularity theory|singularities]]. Reducing properties of geometric structures into algebraic structures has created subjects such as [[algebraic geometry]], [[geometric algebra]], and [[algebraic topology]].

Today, the study of algebra includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|publisher=|accessdate=5 October 2014}}</ref> where none of the first level areas (two digit entries) is called ''algebra''. Algebra instead includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebra]]s, 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used in 14-[[Algebraic geometry]] and 11-[[Number theory]] via [[algebraic number theory]].
{{collapse bottom}}
==Antiassociative algebra==
{{collapse top|Antiassociative algebra}}

An algebra antiassociative if (xy)z = -x(yz) for every case of x,y, and z.<ref>[https://books.google.com/books?id=_PEWt18egGgC&pg=PA235&lpg=PA235&dq=%22antiassociative%22+algebra+aplications&source=bl&ots=Atxm0cdUVs&sig=OQjjF3ig6NYCQwP6O9P8fLgwSDE&hl=en&sa=X&ved=2ahUKEwix94P66LPdAhVIu1MKHckzBNQQ6AEwCHoECAYQAQ#v=onepage&q=%22antiassociative%22%20algebra%20aplications&f=false "Non-Associative Algebra and Its Applications."] Page 235.</ref>
{{collapse bottom}}
==Ugandan Knuckles==
{{collapse top | Ugandan knuckles}}

Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref><ref>{{Cite web|url=https://heavy.com/games/2018/01/controversial-ugandan-knuckles-meme/|title=Controversial ‘Ugandan Knuckles’ Meme Has Infested VRChat|last=MacGregor|first=Collin|date=9 January 2018|website=Heavy.com|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref> The meme was a significant trend followed by several news organisations, including ''[[USA Today]]''.<ref>https://www.usatoday.com/story/tech/news/2018/02/09/ugandan-knuckles-do-you-know-de-wey-meme-explained/307575002/ Retrieved October 9 2018</ref>

===History===
On February 20 2017, YouTuber Gregzilla uploaded a video on Sonic Lost World featuring a parody picture of Knuckles the Echidna. On December 22 2017, a 3D model of the Knuckles sketch was released on ''[[DeviantArt]]''. That day, YouTuber Stahlsby uploaded a video in which several ''VRChat'' players wearing the parody costume trolled others by making clicking noises and saying "You do not know the way".<ref>https://knowyourmeme.com/memes/ugandan-knuckles Retrieved October 9 2018</ref> After that, more and more people flooded ''VRChat'' to troll others as Ugandan Knuckles, leading to controversy, as the mock Ugandan accent and quotations used were widely regarded as racist. However, The meme continued to gain popularity until about mid-January 2018, but had mostly subsided by February.<ref>https://trends.google.com/trends/explore?q=Ugandan%20Knuckles&geo=US retrieved October 9 2018</ref>

===Controversy===
Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=‘Ugandan Knuckles’ is overtaking VRChat|last=Alexander|first=Julia|publisher=[[Vox Media, Inc.]]|work=Polygon|date=October 9, 2018|accessdate=January 9, 2018}}</ref> On January 27 2018, the company [[Razer Inc.|Razer]] was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.<ref>https://gizmodo.com/does-razer-know-it-posted-a-racist-meme-1822485212 Retrieved October 9 2018</ref>

The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’s ‘Ugandan Knuckles’ Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>
{{collapse bottom}}

==References==
{{reflist}}

Scorigami

2021-12-14T00:49:57Z

IntegralPython: /* History */ more specific, again.

{{short description|A scoring combination that has never happened before in a sport or league's history.}}
In sports, a '''Scorigami''' (a [[portmanteau]] of ''[[score (sport)|score]]'' and ''[[origami]]'') is a scoring combination that has never happened before in a sport or league's history.

==History==
The term was coined by [[SB Nation]] sportswriter [[Jon Bois]] in 2016 and most commonly refers to scores in [[American football]], particularly in the [[National Football League]] (NFL), due to the unusual point values in football compared to other team sports.{{efn|Under current NFL rules, certain scores, such as 1–1, 5–1, and 7–1, are impossible.<ref name="chart">{{Cite web |last=Day |first=Lewin |date=January 22, 2020 |title=Scorigami Bot Charts NFL History In The Making |url=https://hackaday.com/2020/01/22/scorigami-bot-charts-nfl-history-in-the-making/ |access-date=December 14, 2020 |website=[[Hackaday]] |language=en-US}}</ref>}}<ref name="JonBois">{{Cite web |last=Bois |first=Jon |date=December 7, 2016 |title=Chart Party: Scorigami, or the story of every NFL final score that has ever happened |url=https://www.sbnation.com/2016/12/7/13856392/chart-party-scorigami |access-date=December 20, 2020 |website=[[SB Nation]] |authorlink=Jon Bois}}</ref> However, other sports have referenced the term, as well.<ref name="mlbn" />

Since the term's inception, an internet program has tracked every Scorigami in NFL history,<ref name="chart" /> with 1,071 unique scores as of January 11, 2021.<ref name="pit-brown">{{Cite web |date=11 January 2021 |title=Browns' 48–37 Victory Over Steelers is Unique in NFL History |url=https://touchdownwire.usatoday.com/2021/01/11/cleveland-browns-48-37-victory-over-pittsburgh-steelers-is-unique-in-nfl-history/ |access-date=5 September 2021}}</ref> As of December 12th, 2021, the most recent Scorigami for an NFL game was when, that day, the [[Kansas City Chiefs]] defeated the [[Las Vegas Raiders]] by a score of 48–9.<ref>{{Cite web|last=Kerr|first=Jeff|date=December 12, 2021|title=Chiefs' victory over Raiders ends in a final score never before seen history of NFL|url=https://www.cbssports.com/nfl/news/chiefs-victory-over-raiders-ends-in-a-final-score-never-before-seen-in-nfl-history/|url-status=live|archive-url=https://web.archive.org/web/20211212222118/https://www.cbssports.com/nfl/news/chiefs-victory-over-raiders-ends-in-a-final-score-never-before-seen-in-nfl-history/|archive-date=December 12, 2021|access-date=December 13, 2021|website=CBS Sports}}</ref>

Bois and other media observers have noted the tendency of the [[Seattle Seahawks]] under head coach [[Pete Carroll]] to create Scorigamis; Bois dubbed Carroll "the wizard of modern Scorigami, without question".<ref name=JonBois /> From [[2011 NFL season|2011]] to [[2018 NFL season|2018]], the Seahawks had at least one Scorigami every season.<ref>{{Cite web |last=Musgrove |first=Kole |date=December 3, 2018 |title=Seahawks continue bizarre 'Scorigami' streak under Pete Carroll |url=https://seahawkswire.usatoday.com/2018/12/03/seahawks-continue-bizarre-scorigami-streak-under-pete-carroll/ |access-date=December 14, 2020 |website=[[USA Today]] |language=en-US}}</ref><ref name="sbn">{{Cite news |last=Alexander |first=Mookie |date=January 18, 2020 |title=The "Scorigami" streak is over for the Seahawks |work=[[SB Nation]] |url=https://www.fieldgulls.com/2020/1/18/21072023/scorigami-streak-seattle-seahawks-nfl |access-date=December 14, 2020}}</ref> One of the Seahawks' Scorigamis during this period occurred in [[Super Bowl XLVIII]], when they defeated the [[Denver Broncos]] 43–8.<ref>{{Cite web |last=Whitney |first=Ched |date=January 31, 2019 |title=Will Super Bowl Scorigami Happen Again? |url=https://www.gamingtoday.com/race_sports/nfl/article/80618-Will_Super_Bowl_Scorigami_Happen_Again |access-date=December 14, 2020 |website=Gaming Today |language=en}}</ref> Carroll himself has acknowledged his team's frequent Scorigamis, joking to reporters after another game with a unique score: <blockquote>"That's ridiculous. I don't know how that happens. I'm thrilled that that happened again, for no reason. It's just something we've been working on in the offseason."<ref>{{Cite web |title=Seattle Seahawks head coach Pete Carroll jokes about scorigami: 'It's something we've been working on in the offseason' |url=https://www.nfl.com/videos/seattle-seahawks-head-coach-pete-carroll-jokes-about-scorigami-it-s-somet-287701 |access-date=December 20, 2020 |website=[[National Football League|NFL]]}}</ref></blockquote>

On September 9, 2020, [[Major League Baseball]] (MLB) had its first Scorigami in 20 years, a 29–9 victory by the [[Atlanta Braves]] over the [[Miami Marlins]]—the most recent Scorigami for an MLB game had been a 24–12 win by the [[Cincinnati Reds]] over the [[Colorado Rockies]] on May 19, 1999.<ref name="mlbn">{{Cite news |last=Werle |first=Andy |date=September 10, 2020 |title=For 1st time since '99, a score not seen before |work=[[Major League Baseball|MLB]] |url=https://www.mlb.com/news/marlins-braves-first-mlb-game-with-29-9-score |access-date=December 14, 2020}}</ref>

== See also ==
* [[Conversion safety]], for discussion of how a team in American football can score just one point in a game

== Notes ==
{{reflist|group=lower-alpha}}

== References ==
{{reflist}}

== External links ==
* [https://nflscorigami.com/ NFL Scorigami website]
* [https://www.sbnation.com/2016/12/7/13856392/chart-party-scorigami Chart Party: Scorigami, or the story of every NFL final score that has ever happened]
* [https://www.espn.com/video/clip/_/id/32153397 ESPN video on Scorigami]

[[Category:2016 neologisms]]
[[Category:National Football League culture]]
[[Category:Portmanteaus]]
[[Category:SB Nation]]
[[Category:Sports records and statistics]]
[[Category:Statistical analysis]]

Scorigami

2021-12-14T00:44:44Z

IntegralPython: /* History */ style

{{short description|A scoring combination that has never happened before in a sport or league's history.}}
In sports, a '''Scorigami''' (a [[portmanteau]] of ''[[score (sport)|score]]'' and ''[[origami]]'') is a scoring combination that has never happened before in a sport or league's history.

==History==
The term was coined by [[SB Nation]] sportswriter [[Jon Bois]] in 2016 and most commonly refers to scores in [[American football]], particularly in the [[National Football League]] (NFL), due to the unusual point values in football compared to other team sports.{{efn|Under current NFL rules, certain scores, such as 1–1, 5–1, and 7–1, are impossible.<ref name="chart">{{Cite web |last=Day |first=Lewin |date=January 22, 2020 |title=Scorigami Bot Charts NFL History In The Making |url=https://hackaday.com/2020/01/22/scorigami-bot-charts-nfl-history-in-the-making/ |access-date=December 14, 2020 |website=[[Hackaday]] |language=en-US}}</ref>}}<ref name="JonBois">{{Cite web |last=Bois |first=Jon |date=December 7, 2016 |title=Chart Party: Scorigami, or the story of every NFL final score that has ever happened |url=https://www.sbnation.com/2016/12/7/13856392/chart-party-scorigami |access-date=December 20, 2020 |website=[[SB Nation]] |authorlink=Jon Bois}}</ref> However, other sports have referenced the term, as well.<ref name="mlbn" />

Since the term's inception, an internet program has tracked every Scorigami in NFL history,<ref name="chart" /> with 1,071 unique scores to date.<ref name="pit-brown">{{Cite web |date=11 January 2021 |title=Browns' 48–37 Victory Over Steelers is Unique in NFL History |url=https://touchdownwire.usatoday.com/2021/01/11/cleveland-browns-48-37-victory-over-pittsburgh-steelers-is-unique-in-nfl-history/ |access-date=5 September 2021}}</ref>

As of December 12th, 2021, the most recent Scorigami for an NFL game was when, that day, the [[Kansas City Chiefs]] defeated the [[Las Vegas Raiders]] by a score of 48–9.<ref>{{Cite web|last=Kerr|first=Jeff|date=December 12, 2021|title=Chiefs' victory over Raiders ends in a final score never before seen history of NFL|url=https://www.cbssports.com/nfl/news/chiefs-victory-over-raiders-ends-in-a-final-score-never-before-seen-in-nfl-history/|url-status=live|archive-url=https://web.archive.org/web/20211212222118/https://www.cbssports.com/nfl/news/chiefs-victory-over-raiders-ends-in-a-final-score-never-before-seen-in-nfl-history/|archive-date=December 12, 2021|access-date=December 13, 2021|website=CBS Sports}}</ref>

Bois and other media observers have noted the tendency of the [[Seattle Seahawks]] under head coach [[Pete Carroll]] to create Scorigamis; Bois dubbed Carroll "the wizard of modern Scorigami, without question".<ref name=JonBois /> From [[2011 NFL season|2011]] to [[2018 NFL season|2018]], the Seahawks had at least one Scorigami every season.<ref>{{Cite web |last=Musgrove |first=Kole |date=December 3, 2018 |title=Seahawks continue bizarre 'Scorigami' streak under Pete Carroll |url=https://seahawkswire.usatoday.com/2018/12/03/seahawks-continue-bizarre-scorigami-streak-under-pete-carroll/ |access-date=December 14, 2020 |website=[[USA Today]] |language=en-US}}</ref><ref name="sbn">{{Cite news |last=Alexander |first=Mookie |date=January 18, 2020 |title=The "Scorigami" streak is over for the Seahawks |work=[[SB Nation]] |url=https://www.fieldgulls.com/2020/1/18/21072023/scorigami-streak-seattle-seahawks-nfl |access-date=December 14, 2020}}</ref> One of the Seahawks' Scorigamis during this period occurred in [[Super Bowl XLVIII]], when they defeated the [[Denver Broncos]] 43–8.<ref>{{Cite web |last=Whitney |first=Ched |date=January 31, 2019 |title=Will Super Bowl Scorigami Happen Again? |url=https://www.gamingtoday.com/race_sports/nfl/article/80618-Will_Super_Bowl_Scorigami_Happen_Again |access-date=December 14, 2020 |website=Gaming Today |language=en}}</ref> Carroll himself has acknowledged his team's frequent Scorigamis, joking to reporters after another game with a unique score: <blockquote>"That's ridiculous. I don't know how that happens. I'm thrilled that that happened again, for no reason. It's just something we've been working on in the offseason."<ref>{{Cite web |title=Seattle Seahawks head coach Pete Carroll jokes about scorigami: 'It's something we've been working on in the offseason' |url=https://www.nfl.com/videos/seattle-seahawks-head-coach-pete-carroll-jokes-about-scorigami-it-s-somet-287701 |access-date=December 20, 2020 |website=[[National Football League|NFL]]}}</ref></blockquote>

On September 9, 2020, [[Major League Baseball]] (MLB) had its first Scorigami in 20 years, a 29–9 victory by the [[Atlanta Braves]] over the [[Miami Marlins]]—the most recent Scorigami for an MLB game had been a 24–12 win by the [[Cincinnati Reds]] over the [[Colorado Rockies]] on May 19, 1999.<ref name="mlbn">{{Cite news |last=Werle |first=Andy |date=September 10, 2020 |title=For 1st time since '99, a score not seen before |work=[[Major League Baseball|MLB]] |url=https://www.mlb.com/news/marlins-braves-first-mlb-game-with-29-9-score |access-date=December 14, 2020}}</ref>

== See also ==
* [[Conversion safety]], for discussion of how a team in American football can score just one point in a game

== Notes ==
{{reflist|group=lower-alpha}}

== References ==
{{reflist}}

== External links ==
* [https://nflscorigami.com/ NFL Scorigami website]
* [https://www.sbnation.com/2016/12/7/13856392/chart-party-scorigami Chart Party: Scorigami, or the story of every NFL final score that has ever happened]
* [https://www.espn.com/video/clip/_/id/32153397 ESPN video on Scorigami]

[[Category:2016 neologisms]]
[[Category:National Football League culture]]
[[Category:Portmanteaus]]
[[Category:SB Nation]]
[[Category:Sports records and statistics]]
[[Category:Statistical analysis]]

Scorigami

2021-12-14T00:44:01Z

IntegralPython: /* History */ more precise language

{{short description|A scoring combination that has never happened before in a sport or league's history.}}
In sports, a '''Scorigami''' (a [[portmanteau]] of ''[[score (sport)|score]]'' and ''[[origami]]'') is a scoring combination that has never happened before in a sport or league's history.

==History==
The term was coined by [[SB Nation]] sportswriter [[Jon Bois]] in 2016 and most commonly refers to scores in [[American football]], particularly in the [[National Football League]] (NFL), due to the unusual point values in football compared to other team sports.{{efn|Under current NFL rules, certain scores, such as 1–1, 5–1, and 7–1, are impossible.<ref name="chart">{{Cite web |last=Day |first=Lewin |date=January 22, 2020 |title=Scorigami Bot Charts NFL History In The Making |url=https://hackaday.com/2020/01/22/scorigami-bot-charts-nfl-history-in-the-making/ |access-date=December 14, 2020 |website=[[Hackaday]] |language=en-US}}</ref>}}<ref name="JonBois">{{Cite web |last=Bois |first=Jon |date=December 7, 2016 |title=Chart Party: Scorigami, or the story of every NFL final score that has ever happened |url=https://www.sbnation.com/2016/12/7/13856392/chart-party-scorigami |access-date=December 20, 2020 |website=[[SB Nation]] |authorlink=Jon Bois}}</ref> However, other sports have referenced the term, as well.<ref name="mlbn" />

Since the term's inception, an internet program has tracked every Scorigami in NFL history,<ref name="chart" /> with 1,071 unique scores to date.<ref name="pit-brown">{{Cite web |date=11 January 2021 |title=Browns' 48–37 Victory Over Steelers is Unique in NFL History |url=https://touchdownwire.usatoday.com/2021/01/11/cleveland-browns-48-37-victory-over-pittsburgh-steelers-is-unique-in-nfl-history/ |access-date=5 September 2021}}</ref>

As of December 12th, 2021, the most recent Scorigami for an NFL game occurred on December 12, 2021, when the [[Kansas City Chiefs]] defeated the [[Las Vegas Raiders]] by a score of 48–9.<ref>{{Cite web|last=Kerr|first=Jeff|date=December 12, 2021|title=Chiefs' victory over Raiders ends in a final score never before seen history of NFL|url=https://www.cbssports.com/nfl/news/chiefs-victory-over-raiders-ends-in-a-final-score-never-before-seen-in-nfl-history/|url-status=live|archive-url=https://web.archive.org/web/20211212222118/https://www.cbssports.com/nfl/news/chiefs-victory-over-raiders-ends-in-a-final-score-never-before-seen-in-nfl-history/|archive-date=December 12, 2021|access-date=December 13, 2021|website=CBS Sports}}</ref>

Bois and other media observers have noted the tendency of the [[Seattle Seahawks]] under head coach [[Pete Carroll]] to create Scorigamis; Bois dubbed Carroll "the wizard of modern Scorigami, without question".<ref name=JonBois /> From [[2011 NFL season|2011]] to [[2018 NFL season|2018]], the Seahawks had at least one Scorigami every season.<ref>{{Cite web |last=Musgrove |first=Kole |date=December 3, 2018 |title=Seahawks continue bizarre 'Scorigami' streak under Pete Carroll |url=https://seahawkswire.usatoday.com/2018/12/03/seahawks-continue-bizarre-scorigami-streak-under-pete-carroll/ |access-date=December 14, 2020 |website=[[USA Today]] |language=en-US}}</ref><ref name="sbn">{{Cite news |last=Alexander |first=Mookie |date=January 18, 2020 |title=The "Scorigami" streak is over for the Seahawks |work=[[SB Nation]] |url=https://www.fieldgulls.com/2020/1/18/21072023/scorigami-streak-seattle-seahawks-nfl |access-date=December 14, 2020}}</ref> One of the Seahawks' Scorigamis during this period occurred in [[Super Bowl XLVIII]], when they defeated the [[Denver Broncos]] 43–8.<ref>{{Cite web |last=Whitney |first=Ched |date=January 31, 2019 |title=Will Super Bowl Scorigami Happen Again? |url=https://www.gamingtoday.com/race_sports/nfl/article/80618-Will_Super_Bowl_Scorigami_Happen_Again |access-date=December 14, 2020 |website=Gaming Today |language=en}}</ref> Carroll himself has acknowledged his team's frequent Scorigamis, joking to reporters after another game with a unique score: <blockquote>"That's ridiculous. I don't know how that happens. I'm thrilled that that happened again, for no reason. It's just something we've been working on in the offseason."<ref>{{Cite web |title=Seattle Seahawks head coach Pete Carroll jokes about scorigami: 'It's something we've been working on in the offseason' |url=https://www.nfl.com/videos/seattle-seahawks-head-coach-pete-carroll-jokes-about-scorigami-it-s-somet-287701 |access-date=December 20, 2020 |website=[[National Football League|NFL]]}}</ref></blockquote>

On September 9, 2020, [[Major League Baseball]] (MLB) had its first Scorigami in 20 years, a 29–9 victory by the [[Atlanta Braves]] over the [[Miami Marlins]]—the most recent Scorigami for an MLB game had been a 24–12 win by the [[Cincinnati Reds]] over the [[Colorado Rockies]] on May 19, 1999.<ref name="mlbn">{{Cite news |last=Werle |first=Andy |date=September 10, 2020 |title=For 1st time since '99, a score not seen before |work=[[Major League Baseball|MLB]] |url=https://www.mlb.com/news/marlins-braves-first-mlb-game-with-29-9-score |access-date=December 14, 2020}}</ref>

== See also ==
* [[Conversion safety]], for discussion of how a team in American football can score just one point in a game

== Notes ==
{{reflist|group=lower-alpha}}

== References ==
{{reflist}}

== External links ==
* [https://nflscorigami.com/ NFL Scorigami website]
* [https://www.sbnation.com/2016/12/7/13856392/chart-party-scorigami Chart Party: Scorigami, or the story of every NFL final score that has ever happened]
* [https://www.espn.com/video/clip/_/id/32153397 ESPN video on Scorigami]

[[Category:2016 neologisms]]
[[Category:National Football League culture]]
[[Category:Portmanteaus]]
[[Category:SB Nation]]
[[Category:Sports records and statistics]]
[[Category:Statistical analysis]]

Wikipedia:In the news/Candidates

2021-12-10T21:31:20Z

IntegralPython: /* World Chess Championship 2021 */ support

{{Short description|Page for suggesting items for "In the news"}}
{{notice|<big>Welcome to ''In the news''. Please '''[[Wikipedia:In the news|read the guidelines]]'''. Admin instructions are '''[[Wikipedia:In the news/Administrator instructions|here]]'''.</big>}}{{Wikipedia:In the news/Candidates/header}}
{{Skip to top and bottom}}

== Archives ==
{{Wikipedia:In the news/Candidates/Archives}}
{{Anchor|Suggestions}}

== December 10 ==
{{cot|[[Portal:Current events/2021 December 10]]}}
{{Portal:Current events/2021 December 10}}
{{cob}}
----

==== RD: Michael Nesmith ====
{{ITN candidate
| article = Michael Nesmith 
| recent deaths = yes
| sources = [https://www.rollingstone.com/music/music-news/monkees-michael-nesmith-dead-1270079/]
| updated = yes 
| nominator = ArsenalGhanaPartey 
| updaters = Jkaharper, Moncrief
| nom cmt = American Musician
| sign = [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 18:06, 10 December 2021 (UTC) 
}}
*'''Oppose''' with regret; too much unsourced information. Not just a musician, a seminal one from what was never intended to be one of the greatest 60s groups ever, but amazingly turned out to be just that. Stick on "Circle Sky" from the ''Head'' soundtrack. [[User:Ritchie333|Ritchie333]] [[User talk:Ritchie333|(talk)]] [[Special:Contributions/Ritchie333|(cont)]] 18:34, 10 December 2021 (UTC)
*'''Oppose''' as Ritchie333... Of the Monkees, Nesmith was probably the most musically relevant; a stellar guitarist, singer and songwriter, he had the most impactful career after the Monkees relevance ended... His work as a songwriter ("Different Drum") is stellar, and [[The First National Band]] was one of the seminal acts in alt-country ("Joanne"). Still, the article is a trainwreck of inadequate referencing. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 19:05, 10 December 2021 (UTC)
*'''Oppose''' Too much tags. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 20:00, 10 December 2021 (UTC)

==== World Chess Championship 2021 ====
{{ITN candidate
| article = World Chess Championship 2021
| blurb = [[Magnus Carlsen]] (pictured) defends his '''[[World Chess Championship 2021|World Chess Championship]]''' title, defeating [[Ian Nepomniachtchi]].
| image = FIDE World FR Chess Championship 2019 - Magnus Carlsen (cropped).jpg
| recent deaths = no 
| ongoing = no 
| ITNR = yes
| altblurb = [[Magnus Carlsen]] (pictured) defeats [[Ian Nepomniachtchi]] to defend his '''[[World Chess Championship 2021|World Chess Championship]]''' title. 
| altblurb2 = 
| sources = [https://www.chess.com/news/view/fide-world-chess-championship-2021-game-11 Chess.com], [https://www.wsj.com/articles/magnus-carlsen-ian-nepomniachtchi-world-chess-championship-computer-analysis-11639003641 WSJ], [https://www.espn.com/chess/story/_/id/32837284/magnus-carlsen-wins-fide-world-chess-championship-ian-nepomniachtchi ESPN]
| updated = 
| nominator = Kndimov 
| creator = 
| updaters = 
| nom cmt = 
| sign = [[User:Kndimov|Kndimov]] ([[User talk:Kndimov|talk]]) 15:56, 10 December 2021 (UTC) 
}}
*'''Support''' The section for game 11 at the bottom of the article needs a source. Otherwise, this looks like a well written and decently referenced article. It would also be good to get some coverage for something outside of the usual sporting activities we tend to cover. -[[User:Ad Orientem|Ad Orientem]] ([[User talk:Ad Orientem|talk]]) 16:50, 10 December 2021 (UTC)
*'''Conditional Support''' The article needs a small expansion of Game 11 prose and a proper source for that section, but it looks good otherwise. Please do that before posting this. Also, the event should be bolded in the blurb instead of the winner. That is our standard practice. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 16:57, 10 December 2021 (UTC)
*'''Comment'''. Is this in a mainstream news outlet? [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 16:59, 10 December 2021 (UTC)
** I've added more sources to the nom. [[User:力]] (powera, [[User talk:力|π]], [[Special:Contributions/力|ν]]) 17:11, 10 December 2021 (UTC)
*'''Comment''' the summaries are way too technical, listing every move in a format not known to the average reader. As a result, I learned very little from reading those game summaries. And some of the matches need some sources too. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 17:02, 10 December 2021 (UTC)
* '''Comment''' as noted, the Game 11 summary isn't ready yet. [[User:力]] (powera, [[User talk:力|π]], [[Special:Contributions/力|ν]]) 17:11, 10 December 2021 (UTC)
*'''Comment''' This is...an interesting one. It's well-written, but far too technical with the game sections. Perhaps a rewrite is needed to turn this into a more readable format. [[User:Heythereimaguy|Heythereimaguy]] ([[User talk:Heythereimaguy|talk]]) 17:14, 10 December 2021 (UTC)
*'''Support and Comment''' WRT the article being "too technical", this is how articles on chess games [https://en.wikipedia.org/wiki/World_Chess_Championship_2016 are] [https://en.wikipedia.org/wiki/World_Chess_Championship_2006 written] (these are two articles previously on ITN, but there are plenty more examples). If someone can't read chess notation that's unfortunate but there's not much to be done. Articles on football matches don't explain every positional abbreviation, they just list them and move on. [[User:BSMRD|BSMRD]] ([[User talk:BSMRD|talk]]) 20:10, 10 December 2021 (UTC)
*'''Support''' the summaries are short but give the gist of each match well, and accurately highlight what made each game a win or draw. There's not much more to wish for than that. '''[[User:IntegralPython| Integral Python]]''[[User talk:IntegralPython| click here to argue with me]]''''' 21:31, 10 December 2021 (UTC)

==== RD: Oded Muhammad Danial ====
{{ITN candidate
| article = Oded Muhammad Danial
| recent deaths = yes
| sources = [https://jakartaglobe.id/news/bandung-mayor-dies-after-collapsing-during-friday-prayer]
| updated = yes 
| nominator = Juxlos 
| updaters = 
| nom cmt = - 
| sign = [[User:Juxlos|Juxlos]] ([[User talk:Juxlos|talk]]) 09:04, 10 December 2021 (UTC) 
}}

* '''Support''': A notable person and his death was sudden, and also appeared everywhere around the national media. The article could have a bit more expansion but otherwise good for RD. [[User:Nyanardsan|Nyanardsan]] ([[User talk:Nyanardsan|talk]]) 12:41, 10 December 2021 (UTC)

== December 9 ==
{{cot|[[Portal:Current events/2021 December 9]]}}
{{Portal:Current events/2021 December 9}}
{{cob}}
----
==== RD: Al Unser (Sr.) ====
{{ITN candidate
| article = Al Unser
| recent deaths = yes
| sources = [https://deadline.com/2021/12/al-unser-sr-dead-indianapolis-50-winner-was-82-1234888698/ Deadline Hollywood]
| updated = yes
| nominator = Masem 
| updaters = 
| nom cmt = Successful racecar driver. Article needs lots of sourcing. If posted, I recommend adding Sr. as to make sure we're not talking about his still-living son [[Al Unser Jr.]]
| sign = [[User:Masem|Masem]] ([[User Talk:Masem|t]]) 16:45, 10 December 2021 (UTC) 
}}
*'''Oppose''' It's a shame that the article about such a legend in the sport is in so piss-poor of a shape. It's a disgrace, really. It's not like he's that obscure. Book-length biographies exist about him and his family. I'm not a motorsports fan in any way, but even I know about Al Unser. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 17:20, 10 December 2021 (UTC)
*:Yup. A household name in his day, though mainly in the U.S. From a famous racing family. (And I'm not a motorsports fan, either.) – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 19:43, 10 December 2021 (UTC)

==== RD: Lina Wertmüller ====
{{ITN candidate
| article = Lina Wertmüller
| recent deaths = yes
| sources =[https://edition.cnn.com/2021/12/09/entertainment/lina-wertmuller-death/index.html CNN], [https://www.bbc.com/news/world-europe-59599270 BBC], [https://www.washingtonpost.com/obituaries/2021/12/09/lina-wertmuller-dead/ Washington Post], [https://www.nytimes.com/2021/12/09/movies/lina-wertmuller-dead.html NY Times], [https://www.theguardian.com/film/2021/dec/09/lina-wertmuller-first-woman-to-be-nominated-for-best-director-oscar-has-died-aged-93 The Guardian], [https://www.latimes.com/entertainment-arts/movies/story/2021-12-09/lina-wertmuller-dead-obit LA Times]
| updated = yes
| nominator = Kacamata 
| creator = Jleybov
| updaters = 
| nom cmt = Very important filmmaker. The article, unfortunately, is tagged and has several issues.
| sign = --[[User:Kacamata|Kacamata!]] [[User_talk:Kacamata|Dimmi!!!]] 06:22, 10 December 2021 (UTC)
}}

==== The Game Awards 2021 ====
{{ITN candidate
| article = The Game Awards 2021
| image = Josef Fares, vinnare av Nordiska radets filmpris 2006 (cropped).jpg
| blurb = At '''[[The Game Awards 2021]]''', ''[[It Takes Two (video game)|It Takes Two]]'' (game director [[Josef Fares]] pictured) wins [[The Game Award for Game of the Year|game of the year]].
| recent deaths = no 
| ongoing = no 
| ITNR = no 
| altblurb = 
| altblurb2 = 
| sources = [https://www.washingtonpost.com/video-games/2021/12/09/game-awards-announcements-highlights/ Washington Post], [https://variety.com/2021/digital/news/the-game-awards-winners-list-1235129480/ Variety], [https://deadline.com/2021/12/the-game-awards-winners-list-it-takes-two-scores-game-of-the-year-deathloop-kena-1234887555/ Deadline Hollywood], [https://www.bbc.com/news/entertainment-arts-59606985 BBC]
| updated = yes
| nominator = NorthernFalcon 
| creator = Dissident93
| updaters = Rhain, Masem
| nom cmt = Why nominate this? It's viewed in mainstream media as [https://www.scotsman.com/culture/gaming/game-awards-2021-when-are-the-game-awards-2021-date-of-game-awards-nominees-and-how-to-watch-live-in-uk-3482238 the Oscars of the video game industry]; it gets coverage in mainstream media sources (will add more sources to the sources line once the articles come out); the people who organize these awards have been doing this annually since 2003, suggesting they are here to stay; and some cool people on Wikipedia ensure that there's a good article to go with this. It has not been posted in ITN before, however this could be a good time to change that. If not now, when?
| sign = [[User:NorthernFalcon|NorthernFalcon]] ([[User talk:NorthernFalcon|talk]]) 06:18, 10 December 2021 (UTC) 
}}

* As a comment, I would like to make sure that more mainstream sources cover this to be assured that this was considered important outside of the video game circle (I absolutely know it is within it). But the article I know is sources up to including what we can of the ceremony, so its arguably in the right shape for posting. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 06:33, 10 December 2021 (UTC)
** To add, if we're talking the top award for video games, it is either this or the BAFTA Game Awards (We also have DICE Awards and GDC Awards but they tend to be more from the industry focus side. The Game Awards are from the media's POV, while BAFTA tends to be non-video game critics looking in, hence why those are better, in addition to having more non-VG mainstream coverage) --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 06:35, 10 December 2021 (UTC)
***[[Japan Game Awards]] should be considered along with those. [[User:TarkusAB|'''TarkusAB''']][[User talk:TarkusAB|'''talk''']]/[[Special:Contributions/TarkusAB|'''contrib''']] 10:09, 10 December 2021 (UTC)
****That doesn't get anywhere near the coverage that these two get. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 14:40, 10 December 2021 (UTC)
** Adding non-VG sources that are reporting on it this morning --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 14:40, 10 December 2021 (UTC)
*'''Support''' Obviously significance is a spectrum not a binary. Our criteria specifically direct that any relative deficiency in significance can be made up for in the quality of the update. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 12:07, 10 December 2021 (UTC)
*'''Oppose for now''' So far, I couldn't find any mainstream coverage on the event besides Washington Post and USA Today. If more articles appear in the media, I might reconsider. [[User:Scaramouche33|Scaramouche33]] ([[User talk:Scaramouche33|talk]]) 13:16, 10 December 2021 (UTC)
* '''Oppose''' – Absent from all main RS sites' general news/features presentations; not prominently in the news. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 14:03, 10 December 2021 (UTC)
*'''Oppose''' not covered in many mainstream news/websites (other than the 2 listed). Therefore topic is too niche for ITN, same as how we rejected esports competitions a few months ago as all the sources on it are niche. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 14:08, 10 December 2021 (UTC)
*Sooo...subjectively, as someone who denounces the existence of e-sport (I use games instead), I would follow the folks above me who oppose. But objectively I have to admit a '''Support'''. The event is also covered in the RSes of my country Indonesia ([https://tekno.kompas.com/read/2021/12/10/16310057/daftar-pemenang-the-game-awards-2021-it-takes-two-sabet-gelar-bergengsi Kompas], [https://tekno.tempo.co/read/1537867/pemenang-the-game-awards-2021-diumumkan-ini-daftar-lengkapnya Tempo], [https://tasikmalaya.pikiran-rakyat.com/teknologi/pr-063201530/franchise-sonic-tampil-di-the-game-awards-2021-dengan-menampilkan-trailer-film-dan-gim Pikiran Rakyat], etc etc]. I really believe those who oppose should check on the RSes of their respective countries. --Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 15:57, 10 December 2021 (UTC)
*:Have done. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 16:45, 10 December 2021 (UTC)
*::I also have done, it's not on the main page of BBC (it's only on the specific Entertainment & Arts section), and not in other sources prominently. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 16:58, 10 December 2021 (UTC)
*:::There is zero requirement that it be "front page" news, just that it has more widestream coverage than the niche of video games. At least four non-VG sources have covered it now, so that's demonstrating being beyond niche, though I'm still watching for more. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 17:39, 10 December 2021 (UTC)
*: Bold of you to assume that there are reliable sources from my country. [[User:Scaramouche33|Scaramouche33]] ([[User talk:Scaramouche33|talk]]) 18:08, 10 December 2021 (UTC)
*'''Support''' Decent article, the kind of quality we should be highlighting, and covered by enough RS for me. [[User:Pawnkingthree|Pawnkingthree]] ([[User talk:Pawnkingthree|talk]]) 17:45, 10 December 2021 (UTC)
*'''Support''' As per {{user|Masem}}, this is covered in non-VG reliable sources, enough for notability. [[User:Heythereimaguy|Heythereimaguy]] ([[User talk:Heythereimaguy|talk]]) 18:27, 10 December 2021 (UTC)
* '''Comment''' – Totally a niche topic; lacks wider news significance. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 19:51, 10 December 2021 (UTC)
::''PS:'' – One must question [https://deadline.com/2021/12/the-game-awards-winners-list-it-takes-two-scores-game-of-the-year-deathloop-kena-1234887555/ ''Deadline Hollywood'']'s status as a reliable source – in the ITN sense of the term. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 19:56, 10 December 2021 (UTC)
:::Video games are not niche, though I will agree I want to make sure these awards are reasonably covered in major sources just as some of the more niches sporting events like gaelic football. And to question the reliability of a source normally considered reliable to challenge a ITN nom is really not appropriate. RS/N is that way. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 20:11, 10 December 2021 (UTC)
::::Nothing could be more niche than video games. Considered by whom? Never saw it cited on ITN before.– [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 20:40, 10 December 2021 (UTC)
:::::So niche that [https://www.marketwatch.com/story/videogames-are-a-bigger-industry-than-sports-and-movies-combined-thanks-to-the-pandemic-11608654990 it's a bigger industry than global movies and US sports combined.] [[User:Pawnkingthree|Pawnkingthree]] ([[User talk:Pawnkingthree|talk]]) 21:16, 10 December 2021 (UTC)

==== (Posted) Chiapas truck crash ====
{{ITN candidate
| article = December 2021 Chiapas truck crash
| recent deaths = no
| image =
| sources = [https://www.nytimes.com/2021/12/09/world/americas/mexico-migrants-killed-accident.html NYT] [https://www.bbc.com/news/world-latin-america-59603801 BBC] [https://www.theguardian.com/world/2021/dec/10/dozens-killed-after-truck-packed-with-migrants-crashes-in-mexico The Guardian] [https://www.msn.com/en-us/news/world/mexico-road-accident-dozens-killed/ar-AARFmDJ?ocid=msedgdhp&pc=U531 CNN], [https://apnews.com/article/mexico-accidents-smuggling-4a101767083046f872999fe26c788c47 AP]
| updated = yes
| blurb = At least 49 people are killed and 58 people are injured after '''[[December 2021 Chiapas truck crash|a truck crashes]]''' in [[Chiapas]], Mexico.
| altblurb = Over 50 people, mostly [[Guatemalans]] and [[Hondurans]], are killed by a [[December 2021 Chiapas truck crash|truck crash]] in [[Chiapas]], Mexico.
| altblurb2 =
| nominator = Destroyeraa 
| updaters = Mooonswimmer, Garmin21
| nom cmt = Needs expansion, this could be blurb-worthy. Per Guardian "It was one of the worst single-day death tolls for migrants in Mexico since the 2010 massacre of 72 migrants by the Zetas drug cartel". 
| sign = [[User:Destroyeraa|Destroyeraa]] ([[User:Destroyeraa-alt|Alternate account]]) 02:41, 10 December 2021 (UTC)
}}
*'''Support''' once expanded. Definitely notable enough for a blurb. -- [[User:Rockstone35|Rockstone]][[User talk:Rockstone35|Send me a message!]] 06:34, 10 December 2021 (UTC)
*'''Oppose on quality''' event looks ITN-worthy, but article is a stub. Consider this a support once article quality is fixed. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 10:49, 10 December 2021 (UTC)
* '''Comment''' – Toll has risen to [https://apnews.com/article/mexico-accidents-smuggling-4a101767083046f872999fe26c788c47 53]. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:07, 10 December 2021 (UTC)
*'''Support''' because it's easily notable enough & the article is good enough. [[User:Jim Michael|Jim Michael]] ([[User talk:Jim Michael|talk]]) 14:33, 10 December 2021 (UTC)
*'''Oppose''' Article is not detailed enough for main page posting. Seven sentences as of my writing this. That's not good enough. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 14:36, 10 December 2021 (UTC)
*'''Support''' I have expanded and updated the article. Perhaps someone could add a map to the infobox. Moonswimmer [[User:Mooonswimmer|Mooonswimmer]] 15:20, 10 December 2021 (UTC)
*'''Support''' Article updated now. [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 15:56, 10 December 2021 (UTC)
* '''Comment''' – "... victims ''are believed to have been'' Central American migrants from Guatemala." Too iffy. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 16:52, 10 December 2021 (UTC)
{{ec}}
*'''Posted'''. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 16:48, 10 December 2021 (UTC)

==== RD: Julie Brougham ====
{{ITN candidate
| article = Julie Brougham
| recent deaths = yes
| sources =[https://www.stuff.co.nz/sport/women-in-sport/127240352/rio-olympics-equestrian-julie-brougham-dies]
| updated = yes
| nominator = TJMSmith 
| creator = Schwede66
| updaters = Strattonsmith, HenryCrun15
| nom cmt = New Zealand equestrian. Death announced on this date.
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 14:21, 9 December 2021 (UTC)
}}
*'''Weak support''' It's not great, such as long gaps in her biography, but for someone who became famous so late in life, that's at least a little understandable. What is there covers why she is notable, and its all referenced. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 15:11, 9 December 2021 (UTC)

==== (Posted) RD: Demaryius Thomas ====
{{ITN candidate
| article = Demaryius Thomas 
| recent deaths = yes
| sources = [https://kdvr.com/sports/denver-broncos/demaryius-thomas-dead-at-33/ KDVR Fox 31] [https://www.espn.com/nfl/story/_/id/32834405/former-denver-broncos-wr-demaryius-thomas-33-found-dead-home-police-say ESPN] [https://www.denverpost.com/2021/12/09/demaryius-thomas-broncos-dies-age-33/ Denver Post]
| updated = 
| nominator = The Kip 
| updaters = 
| nom cmt = Former NFL wide receiver. 5-time Pro Bowler, Super Bowl champion, two-time All-Pro, Broncos legend. 
| sign = [[User:The Kip|The Kip]] ([[User talk:The Kip|talk]]) 04:47, 10 December 2021 (UTC) 
}}
*'''Support''' Good depth of coverage, referenced (career stats needs a ref but should be available from the ESPN stats link in the external links section). '''[[User:Spencer|Spencer]]'''[[User talk:Spencer|T•]][[Special:Contributions/Spencer|C]] 05:28, 10 December 2021 (UTC)
*'''Support''' Excellent quality for an article of that length. Meets minimum requirements for RD. [[User:NorthernFalcon|NorthernFalcon]] ([[User talk:NorthernFalcon|talk]]) 06:25, 10 December 2021 (UTC)
*'''Support''' very good article, everything sourced. Marked as ready. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 10:47, 10 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 11:37, 10 December 2021 (UTC)

== December 8 ==
{{cot|[[Portal:Current events/2021 December 8]]}}
{{Portal:Current events/2021 December 8}}
{{cob}}
----
==== Sokoto bus massacre ====
{{ITN candidate
| article = Nigerian bandit conflict
| recent deaths = no
| image =
| sources = [https://www.reuters.com/world/africa/gunmen-torch-bus-kill-30-passengers-nigerias-sokoto-state-2021-12-07/ Reuters]
| updated = no
| blurb = Gunmen burn down a bus in [[Sokoto]], [[Nigeria]], killing 30 passengers inside.
| altblurb = Gunmen torch a bus in [[Sokoto]], [[Nigeria]], killing 30 passengers including children.
| altblurb2 = A bus is burned down by bandits in [[Sokoto]], [[Nigeria]], killing 30 passengers.
| nominator = ArsenalGhanaPartey 
| updaters = 
| nom cmt = After an update, this could be blurb-worthy. 
| sign = [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 19:04, 8 December 2021 (UTC)
}}
*'''Oppose''' Literally zero updates to the target article. What I am assessing the quality of if Wikipedia has no information on the topic? Where are we directing readers to learn more about the topic? Let me know when I have something to assess.--[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 19:09, 8 December 2021 (UTC)
*'''Oppose''' – Absent from Thursday's RS coverage. Reuters story cited above merely corrects an article from Tuesday. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:03, 9 December 2021 (UTC)
*'''Oppose''' whilst it doesn't have it's own article & isn't mentioned on the article about the conflict. [[User:Jim Michael|Jim Michael]] ([[User talk:Jim Michael|talk]]) 18:04, 9 December 2021 (UTC)
*'''Oppose''' This isn't out of the ordinary for northern Nigeria. We generally don't blurb school shootings in the United States because they're relatively common; same principle applies here. [[User:Mlb96|Mlb96]] ([[User talk:Mlb96|talk]]) 04:22, 10 December 2021 (UTC)
::We'd certainly quickly blurb a mass shooting in the US whose death toll were that high. [[User:Jim Michael|Jim Michael]] ([[User talk:Jim Michael|talk]]) 14:36, 10 December 2021 (UTC)
*'''Oppose''' the fact it's not even notable enough for a separate article indicates it's not ITN-worthy. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 10:48, 10 December 2021 (UTC)
::It is notable enough for its own article, the problem is that it hasn't been created. [[User:Jim Michael|Jim Michael]] ([[User talk:Jim Michael|talk]]) 14:30, 10 December 2021 (UTC)

==== (Posted) RD: Greg Tate ====
{{ITN candidate
| article = Greg Tate
| recent deaths = yes
| sources =[https://www.nbcnews.com/news/amp/rcna7948 NBC News] ; [https://www.npr.org/2021/12/07/1062137384/greg-tate-music-critic-author-journalist-dead NPR] ; [https://www.artnews.com/art-news/news/greg-tate-dead-1234612574/amp/ ARTnews]
| updated = yes
| nominator = 2600:1004:B0C4:7412:B173:C9AA:6318:1713 
| updaters = Innisfree987, Strattonsmith, AleatoryPonderings 
| nom cmt = Greg Tates’s ''Recent Death'' status '''is''' confirmed by reliable sources. But the ''exact'' day has ''not'' been confirmed yet according to the updaters listed above.
| sign = [[Special:Contributions/2600:1004:B0C4:7412:B173:C9AA:6318:1713|2600:1004:B0C4:7412:B173:C9AA:6318:1713]] ([[User talk:2600:1004:B0C4:7412:B173:C9AA:6318:1713|talk]]) 19:03, 8 December 2021 (UTC) 
}}
*'''Support''' Article is short but sufficient and well referenced, death appears to have been first reported December 7; the report date is usually our secondary method of chronologizing deaths if it was not reported immediately. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 19:12, 8 December 2021 (UTC)
*'''Support''' Sufficient, though the proseline treatment of his writings is less than ideal. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 20:59, 8 December 2021 (UTC)
::Yes, I agree. Had planned to continue working on the entry and waiting for an official date of death before nominating. Will do my best. [[User:Innisfree987|Innisfree987]] ([[User talk:Innisfree987|talk]]) 21:12, 8 December 2021 (UTC)
:::In better shape now. [[User:Innisfree987|Innisfree987]] ([[User talk:Innisfree987|talk]]) 21:04, 9 December 2021 (UTC)
*'''Support''' Article is referenced and long enough. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 23:44, 8 December 2021 (UTC)
*'''Support'''. Well it’s no GA, which I regret, but I think it hits most of the high notes now and the date of death was confirmed by NYT. So it’s probably about as ready as I can quickly make it. [[User:Innisfree987|Innisfree987]] ([[User talk:Innisfree987|talk]]) 23:56, 8 December 2021 (UTC)
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 08:08, 9 December 2021 (UTC)
*'''Posted''' [[User:Stephen|Step]][[User talk:Stephen|hen]] 21:57, 9 December 2021 (UTC)

==== (Posted) RD/Blurb: Bipin Rawat ====
{{ITN candidate
| article = Bipin Rawat
| recent deaths = yes
| image = Bipin Rawat Chief of Defence Staff (CDS).jpg
| sources = [https://www.ndtv.com/india-news/defence-chief-general-bipin-rawat-dies-in-chopper-crash-in-tamil-nadu-2642615][https://twitter.com/IAF_MCC/status/1468559355868028936]
| updated = yes
| blurb = Indian [[Chief of Defence Staff (India)|Chief of Defence Staff]] General [[Bipin Rawat]] dead in a [[2021 Indian Air Force Mil Mi-17 crash|'''helicopter crash''']]
| altblurb = Indian [[Chief of Defence Staff (India)|Chief of Defence Staff]] General [[Bipin Rawat]] among 13 dead in a [[2021 Indian Air Force Mil Mi-17 crash|'''helicopter crash''']]
| altblurb2 = Indian [[Chief of Defence Staff (India)|Chief of Defence Staff]] General [[Bipin Rawat]] along with 12 others dead in a [[2021 Indian Air Force Mil Mi-17 crash|'''helicopter crash''']]
| altblurb3 = [[Chief of Defence Staff (India)|Chief of the Defence Staff of India]] '''[[Bipin Rawat]]''' and 12 others die in a '''[[2021 Indian Air Force Mil Mi-17 crash|helicopter crash]]'''.
| nominator = Venkat TL 
| updaters = Venkat TL
| nom cmt = Good sourcing in the article. RIP General. 
| sign = [[User:Venkat TL|Venkat TL]] ([[User talk:Venkat TL|talk]]) 12:47, 8 December 2021 (UTC) 
}}
*<s>'''Oppose''' for now. Several sections lack cites. That needs fixing before this can be posted. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 13:27, 8 December 2021 (UTC)</s>
*:'''Support'''. Article is in good shape now. Agnostic on RD or Blurb, but either way it's good enough for the main page. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 13:48, 8 December 2021 (UTC)
*::Thank you @[[User:Jayron32|Jayron32]], @[[User:GreatCaesarsGhost|GreatCaesarsGhost]]. Total 13 people died. Should this number 13 be included in the blurb? His Wife and staff were among the dead. NDTV headline says, "General Bipin Rawat, Wife Among 13 Killed In Chopper Crash" [[User:Venkat TL|Venkat TL]] ([[User talk:Venkat TL|talk]]) 13:57, 8 December 2021 (UTC)
*<s>'''Wait'''</s> handful of CN tags will need to be resolved first. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 13:33, 8 December 2021 (UTC)
*'''Support''' per above. [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 14:38, 8 December 2021 (UTC)
*'''Support Blurb''' while military chopper crashes are unfortunately common, the death of the highest ranking officer is not. Rawat has also played significant roles during his tenure, which makes it all the more notable. [[Special:Contributions/180.151.20.32|180.151.20.32]] ([[User talk:180.151.20.32|talk]]) 14:53, 8 December 2021 (UTC)
*'''Support Blurb''' The death of 13 people in a single accident alone is enough for a blurb, plus the death of the CJCS. --Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 15:26, 8 December 2021 (UTC)
*'''Support Blurb''' 13 deaths including the [[Chief of Defence Staff (India)|Chief of Defence Staff]] -- the highest-ranking officer of the Indian Armed Forces. The articles look to be in good shape. – [[User:SD0001|SD0001]] ([[User talk:SD0001|talk]]) 16:16, 8 December 2021 (UTC)
*'''Comment''' proposing blurb – Indian [[Chief of the Defence Staff (India)|Chief of Defence Staff]] General [[Bipin Rawat]], is killed along with 12 others in a [[2021 Indian Air Force Mi-17 crash|'''helicopter crash''']] in the state of Tamil Nadu. [[Special:Contributions/2405:201:4013:8162:99FC:DC7C:6496:6DBD|2405:201:4013:8162:99FC:DC7C:6496:6DBD]] ([[User talk:2405:201:4013:8162:99FC:DC7C:6496:6DBD|talk]]) 16:32, 8 December 2021 (UTC)
*'''Comment''' I have added a blurb box with pic and 2 alternate blurbs proposed till now. [[User:Venkat TL|Venkat TL]] ([[User talk:Venkat TL|talk]]) 16:47, 8 December 2021 (UTC)
*'''Comment''' I've added another alternate blurb, though this one targets Rawat's article rather than the crash.--[[User:Sunshineisles2|Sunshineisles2]] ([[User talk:Sunshineisles2|talk]]) 17:09, 8 December 2021 (UTC)
*'''Support Blurb''' - death of 13 people in this accident is enough for a blurb. [[User:BabbaQ|BabbaQ]] ([[User talk:BabbaQ|talk]]) 17:40, 8 December 2021 (UTC)
*'''Support Blurb''' Important general. Plus, article is well sourced. [[User:Pyramids09|Pyramids09]] ([[User talk:Pyramids09|talk]]) 18:09, 8 December 2021 (UTC)
*'''Support Alt3''' with both bolded. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 18:44, 8 December 2021 (UTC)
*'''Posted''' Alt3. Put the photo in for protection but also figured the Scholz photo could remain a bit. [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 18:54, 8 December 2021 (UTC)
** I was thinking of swapping in Bipin Rawat's pic when it's daytime in India. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 18:57, 8 December 2021 (UTC)
*** Makes sense to me. [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 19:01, 8 December 2021 (UTC)
**** Pic swapped. It's now 6 am in India and in the middle of the night in Germany. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 00:31, 9 December 2021 (UTC)

==== (Posted) New Chancellor of Germany ====

{{ITN candidate
| article = Olaf Scholz
| image = Olaf Scholz 2021-12-07 Unterzeichnung des Koalitionsvertrages der 20. Wahlperiode des Bundestages (cropped for ITN).jpg
| blurb = '''[[Olaf Scholz]]''' is elected [[Chancellor of Germany]] at the head of a [[Scholz cabinet|coalition government]] of his [[Social Democratic Party of Germany|Social Democrats]], the [[Alliance 90/The Greens|Greens]] and the [[Free Democratic Party (Germany)|Free Democrats]].
| recent deaths = no 
| ongoing = no 
| ITNR = yes 
| altblurb = '''[[Olaf Scholz]]''' replaces [[Angela Merkel]] as the [[Chancellor of Germany]].
| altblurb2 = 
| sources = {{Cite news|last=Bennhold|first=Katrin|date=2021-12-08|title=Germany Live Updates: Parliament Approves Scholz as Chancellor, Ending Merkel Era|language=en-US|work=The New York Times|url=https://www.nytimes.com/live/2021/12/08/world/germany-scholz-merkel|access-date=2021-12-08|issn=0362-4331}}, [https://apnews.com/article/climate-elections-europe-angela-merkel-european-union-675ee26e99ce6e7f57d0f61989b05a16 AP], [https://www.bbc.com/news/world-europe-59575773 BBC], [https://www.theguardian.com/world/2021/dec/08/olaf-scholz-elected-succeed-angela-merkel-german-chancellor Guardian], [https://www.dpa-international.com/topic/social-democrat-olaf-scholz-becomes-germany-new-chancellor-urn%3Anewsml%3Adpa.com%3A20090101%3A211208-99-301532 dpa], [https://www.reuters.com/world/europe/social-democrat-olaf-scholz-elected-german-chancellor-2021-12-08/ Reuters]
| updated = yes
| nominator = Sandstein 
| creator = 
| updaters = Tobby72, SoWhy, Whoisjohngalt
| nom cmt = Change of the chief executive of a major country, [[WP:ITNR]].
| sign = [[User:Sandstein|''' Sandstein ''']] 10:14, 8 December 2021 (UTC) 
}}
* Technically, we already posted the election results, but this may be additionally notable because it means Germany will get a new chancellor after [[Angela Merkel]]'s 16 years in office. --'''[[User:Tone|Tone]]''' 10:39, 8 December 2021 (UTC)
:* Germany has a parliamentary system and the result of the chancellor election was open until this day. "except when that change was already posted as part of a general election" does not apply here. [[User:LenaAvrelia|LenaAvrelia]] ([[User talk:LenaAvrelia|talk]]) 10:41, 8 December 2021 (UTC)
*'''Support''' per LenaAvrelia. In line with other such events (see Austria, which had three chancellors in three months). Article looks to be in a good shape. Regards [[User:SoWhy|So]][[User talk:SoWhy|Why]] 11:15, 8 December 2021 (UTC)
*:Can we please use [[:File:Olaf Scholz 1984.jpg|this photo]] instead though? 😅 Regards [[User:SoWhy|So]][[User talk:SoWhy|Why]] 11:19, 8 December 2021 (UTC)
*Since a coalition had to be formed, it was not a guarantee he was going to be Chancellor after the general election, so I think this qualifies for posting. [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 11:17, 8 December 2021 (UTC)
*'''Oppose''' article is BLP and contains many unreferenced claims. [[User:The Rambling Man|The Rambling Man]] ([[User talk:The Rambling Man|Keep wearing the mask...]]) 11:58, 8 December 2021 (UTC)
*'''Support''' - as 331dot says, the general election result did not guarantee Scholz would become chancellor - a "Jamaica coalition" of CDU/CSU, Greens and FDP would also have had a solid majority, but Greens and FDP decided to support Scholz (SPD) instead of Laschet (CDU). --[[Special:Contributions/141.100.201.16|141.100.201.16]] ([[User talk:141.100.201.16|talk]]) 12:03, 8 December 2021 (UTC)
*<s>'''Oppose''' due to CNs. </s> I concur with this being tagged as ITNR. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 12:14, 8 December 2021 (UTC)
*: Withdrawn. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 17:43, 8 December 2021 (UTC)
*<s>'''Oppose''' about 10 CN tags and an entire section orange tagged as needing more citations. That needs to be fixed if this is to be posted. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 12:47, 8 December 2021 (UTC)</s>
*:'''Support''' Article is in good shape now. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 14:19, 8 December 2021 (UTC)
*<s>'''Comment''' I would support if the citation needed tags get fixed.</s> [[User:Heythereimaguy|Heythereimaguy]] ([[User talk:Heythereimaguy|talk]]) 13:05, 8 December 2021 (UTC)
:*Seeing how there are no more citation needed tags, I change my opinion to a big '''support'''. [[User:Heythereimaguy|Heythereimaguy]] ([[User talk:Heythereimaguy|talk]]) 15:18, 8 December 2021 (UTC)
* '''Comment''' – In French, German, Swedish and Norwegian (''Bokmål'') versions of ITN. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:15, 8 December 2021 (UTC)
*:How does that fact cause citations to appear in the article? --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 13:28, 8 December 2021 (UTC)
*::Indeed, that is completely irrelevant as to whether this article should be posted here.-- [[User:Pawnkingthree|Pawnkingthree]] ([[User talk:Pawnkingthree|talk]]) 14:27, 8 December 2021 (UTC)
*:::''Schönen Tag noch ihr Kollegen, und alles gute zur Weinachtszeit.'' – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 14:40, 8 December 2021 (UTC)
*::::Indeed, those particular Wikipedias have no interest at all in verification or BLPs, they just promote whatever is happening in whatever state to the main page, hardly something for this Wikipedia to care about, let alone be encouraged to mimic. [[User:The Rambling Man|The Rambling Man]] ([[User talk:The Rambling Man|Keep wearing the mask...]]) 16:06, 9 December 2021 (UTC)

*{{ping|Heythereimaguy|GreatCaesarsGhost|Jayron32}} I've added citations for all the {{tl|cn}} tags and removed some uncited material. In my view, the article is now reasonably well sourced for a BLP. [[User:Sandstein|''' Sandstein ''']] 14:13, 8 December 2021 (UTC)

*'''Support''' Looks to be ready now that CN tags have been fixed.-- [[User:Pawnkingthree|Pawnkingthree]] ([[User talk:Pawnkingthree|talk]]) 14:30, 8 December 2021 (UTC)

*'''Comment''' I tend to think we should mention Merkel in here somewhere b/c her leadership was long & notable. Anyone have thoughts on this? Btw, I support a blurb. — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:TenorTwelve|TenorTwelve]] ([[User talk:TenorTwelve#top|talk]] • [[Special:Contributions/TenorTwelve|contribs]]) 15:11, 8 December 2021 (UTC)
*:With all due respect to Merkel, she didn't resign or was defeated, she just didn't run again and now left office as people do who have served. Plus, the blurb is already pretty long and adding Merkel would make it too long imho. Regards [[User:SoWhy|So]][[User talk:SoWhy|Why]] 15:28, 8 December 2021 (UTC)
*'''Support''' Decent enough article, a change of government is already currently on ITN. [[User:Llewee|Llewee]] ([[User talk:Llewee|talk]]) 16:14, 8 December 2021 (UTC)
*'''Comment''' - proposed altblurb, simpler prose and more worth mentioning Angela Merkel I think. - [[User:Indefensible|Indefensible]] ([[User talk:Indefensible|talk]]) 16:55, 8 December 2021 (UTC)
:*That Merkel leaves isn't the news; everybody knew that a year ago. The news is that a new coalition now leads Germany. [[User:Sandstein|''' Sandstein ''']] 17:23, 8 December 2021 (UTC)
::*{{u|Sandstein}}, Yes but it probably is the most significant thing about this to a typical English-speaking reader. [[User:Llewee|Llewee]] ([[User talk:Llewee|talk]]) 17:38, 8 December 2021 (UTC)
:::* That Merkel officially ceases to be chancellor is in the news. However much it is foreseen. It's like saying that neither is the re-election of certain politicians like Putin or Ortega. And yes, this should be included in the blurb. [[User:Alsoriano97|_-_Alsoriano97]] ([[User talk:Alsoriano97|talk]]) 17:48, 8 December 2021 (UTC)
* '''Posted''': {{tq|A [[Scholz cabinet|coalition government]] led by [[Chancellor of Germany|Chancellor]] '''[[Olaf Scholz]]''' ''(pictured)'' is formed in Germany.}} --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 18:24, 8 December 2021 (UTC)
*:Since we're freelancing on the blurb now, I've added {{tq|newly-elected}} before the word {{tq|Chancellor}}, as all prior proposed blurbs and most of this discussion focused on his winning said election, as do many news sources. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 18:54, 8 December 2021 (UTC)
*::Looks fine. We didn't need tons of detail since this topic (& Merkel) has been so heavily in the news of late. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 23:27, 8 December 2021 (UTC)
*'''Comment''' I'd prefer a blurb that includes Merkel as well. Merkel staying in power for 16 years in a democratic country is very unusual and notable. [[User:NorthernFalcon|NorthernFalcon]] ([[User talk:NorthernFalcon|talk]]) 23:45, 8 December 2021 (UTC)
*:Much though I admire ''Mutti,'' I'm Merkeled out. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:07, 9 December 2021 (UTC)

== December 7 ==
{{cot|[[Portal:Current events/2021 December 7]]}}
{{Portal:Current events/2021 December 7}}
{{cob}}
----
==== RD: Suresh Jadhav (biotechnology executive) ====
{{ITN candidate
| article = Suresh Jadhav (biotechnology executive)
| recent deaths = yes
| sources = [https://economictimes.indiatimes.com/industry/healthcare/biotech/pharmaceuticals/suresh-jadhav-key-executive-of-sii-doyen-of-vaccine-industry-passed-away/articleshow/88170637.cms Economic Times]
| updated = yes
| nominator = Ktin
| creator =
| updaters = Ktin
| nom cmt = Indian biotechnology executive. Article has shaped alright. A bit on the smaller side, start-class biography, meets basic hygiene expectations for homepage / RD.
| sign = [[User:Ktin|Ktin]] ([[User talk:Ktin|talk]]) 06:51, 9 December 2021 (UTC)
}}
*'''Support''' - article is referenced. Great job starting this article! [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 13:58, 9 December 2021 (UTC)

==== RD: Farida Mammadova ====
{{ITN candidate
| article = Farida Mammadova
| recent deaths = yes
| sources = [https://report.az/elm-ve-tehsil-xeberleri/feride-memmedova-salyanda-defn-olunacaq/ The Report]
| updated = yes
| nominator = TJMSmith
| creator = Dacy69
| updaters = TarPas, CAWylie
| nom cmt = Azerbaijani historian who specialized in the history of ancient Caucasian Albania.
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 23:37, 8 December 2021 (UTC)
}}
*'''Support''', looks ok, though reception section is [[WP:UNDUE|too critical]]. RIP. [[User:Brandmeister|Brandmeister]][[User talk:Brandmeister|talk]] 09:35, 9 December 2021 (UTC)

==== (Posted) RD: Carol Jenkins Barnett ====
{{ITN candidate
| article = Carol Jenkins Barnett
| recent deaths = yes
| sources = [https://www.theledger.com/story/news/local/2021/12/08/philanthropist-publix-heiress-carol-jenkins-barnett-dies-65/6431461001/ The Ledger]
| updated = yes
| nominator = TJMSmith
| creator = Jumplike23
| updaters = 
| nom cmt = American philanthropist and businesswoman.
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 23:31, 8 December 2021 (UTC)
}}
*'''Support''' Looks good and ready. [[User:Ktin|Ktin]] ([[User talk:Ktin|talk]]) 01:42, 10 December 2021 (UTC)
*'''Support''' The article is in good shape and well-referenced. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 02:36, 10 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 14:19, 10 December 2021 (UTC)

==== (Posted) Burundi prison fire====
{{ITN candidate
| article = Gitega prison fire
| blurb = At least 38 inmates die and over 60 are injured after a '''[[Gitega prison fire|fire breaks]]''' out in an overcrowded prison in [[Gitega]], [[Burundi]]
| recent deaths = no 
| ongoing = no 
| ITNR = no 
| altblurb = A '''[[Gitega prison fire|prison fire]]''' kills at least 38 inmates and injures over 69 others in [[Gitega]], [[Burundi]].
| altblurb2 = 
| sources = [https://www.bbc.com/news/world-africa-59560444 BBC], [https://www.france24.com/en/africa/20211207-dozens-killed-in-massive-pre-dawn-fire-at-overcrowded-burundi-prison France24], [https://www.dw.com/en/burundi-prison-fire-kills-dozens/a-60049039 DW], [https://www.theguardian.com/world/2021/dec/07/burundi-prison-gitega-fire-overcrowded Guardian], [https://www.reuters.com/world/africa/burundi-prison-fire-kills-38-inmates-injures-dozens-more-vice-president-2021-12-07/ Reuters], [https://www.aljazeera.com/news/2021/12/7/fire-tears-through-burundi-prison-kills-dozens AlJazeera]
| updated = yes
| nominator = Scaramouche33 
| creator = Scaramouche33
| updaters =
| nom cmt = Fairly large death toll, potential human rights abuses involved
| sign = [[User:Scaramouche33]] ([[User talk: Scaramouche33|talk]]) 15:26, 7 December 2021 (UTC) 
}}
*<s>'''Oppose''' barely more than a stub.</s> [[User:Heythereimaguy|Heythereimaguy]] ([[User talk:Heythereimaguy|talk]]) 17:06, 7 December 2021 (UTC)
:*Okay, seeing how there seems to be more sources, I am changing my vote to '''weak support'''. [[User:Heythereimaguy|Heythereimaguy]] ([[User talk:Heythereimaguy|talk]]) 15:16, 8 December 2021 (UTC)
*'''Support''' in principle. '''Oppose''' in condition.--Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 17:13, 7 December 2021 (UTC)
*'''Comment''' I tried expanding the article a little bit. We could also add a map and a table.[[User:Scaramouche33|Scaramouche33]] ([[User talk:Scaramouche33|talk]]) 18:47, 7 December 2021 (UTC)
*'''Support''' article is start-class at 349 words and well-sourced. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 18:59, 7 December 2021 (UTC)
*'''Support''' Article is short, but sufficient, level of coverage by reputable news sources demonstrate significance. -[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 19:08, 7 December 2021 (UTC)
*'''Oppose''' The description of the event totals 168 words, or half a stub. The rest is BG & reax. Too thin for MP promotion. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 19:13, 7 December 2021 (UTC)
*'''Support''' substantial death toll. [[User:CaffeinAddict|CaffeinAddict]] ([[User talk:CaffeinAddict|talk]]) 04:29, 8 December 2021 (UTC)
*'''Comment''' I think I added everything I could find in English language sources. There could be more info in French sources but I don't speak the language.[[User:Scaramouche33|Scaramouche33]] ([[User talk:Scaramouche33|talk]]) 06:44, 8 December 2021 (UTC)
*'''Weak Support''' Passable, but barely over a stub. [[User:Gotitbro|Gotitbro]] ([[User talk:Gotitbro|talk]]) 10:33, 8 December 2021 (UTC)
*'''Support''' because it's important enough & the article just about good enough. [[User:Jim Michael|Jim Michael]] ([[User talk:Jim Michael|talk]]) 15:17, 8 December 2021 (UTC)
*'''Support''' barely qualifies but include for encyclopedic coverage; proposed shorter altblurb. - [[User:Indefensible|Indefensible]] ([[User talk:Indefensible|talk]]) 17:03, 8 December 2021 (UTC)
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 18:10, 8 December 2021 (UTC)
* Posting. --'''[[User:Tone|Tone]]''' 18:16, 8 December 2021 (UTC)

== December 6 ==
{{cot|[[Portal:Current events/2021 December 6]]}}
{{Portal:Current events/2021 December 6}}
{{cob}}
----
==== (Posted) RD: Olha Ilkiv ====
{{ITN candidate
| article = Olha Ilkiv
| recent deaths = yes
| sources =[https://www.pravda.com.ua/news/2021/12/6/7316379/ Ukrayinska Pravda]
| updated = yes
| nominator = TJMSmith
| creator = Thriley
| updaters = Yulia Romero, Curbon7
| nom cmt = Ukrainian partisan and liaison officer of the Ukrainian Insurgent Army. She was a centenarian. 
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 00:34, 10 December 2021 (UTC)
}}
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 06:47, 10 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 17:01, 10 December 2021 (UTC)

==== (Posted) RD: Masayuki Uemura ====
{{ITN candidate
| article = Masayuki Uemura
| recent deaths = yes
| sources =[https://www.nintendolife.com/news/2021/12/masayuki-uemura-creator-of-the-nes-and-snes-has-passed-away Nintendo Life] ; [https://kotaku.com/masayuki-uemura-creator-of-the-nes-and-snes-dies-at-7-1848184264 Kotaku] ; [http://www.nintendoworldreport.com/news/59101/masayuki-uemura-1943-2021 Nintendo World Report]
| updated = yes
| nominator = TarkusAB 
| updaters = TarkusAB
| nom cmt = Died on December 6, Announced today. Lead hardware architect of the Nintendo Entertainment System and Super NES.
| sign = [[User:TarkusAB|'''TarkusAB''']][[User talk:TarkusAB|'''talk''']]/[[Special:Contributions/TarkusAB|'''contrib''']] 11:54, 9 December 2021 (UTC)
}}
*'''Support''' Nice article. Well written and referenced. [[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 13:02, 9 December 2021 (UTC)
*'''Support''' Meets minimum RD quality requirements. [[User:NorthernFalcon|NorthernFalcon]] ([[User talk:NorthernFalcon|talk]]) 17:41, 9 December 2021 (UTC)
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 19:19, 9 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 23:31, 9 December 2021 (UTC)

==== (Posted) RD: Ebrahim Ismail Ebrahim ====
{{ITN candidate
| article = Ebrahim Ismail Ebrahim
| recent deaths = yes
| sources = [https://www.ndtv.com/world-news/s-africa-mourns-death-of-indian-origin-anti-apartheid-veteran-ebrahim-ebrahim-2640092 NDTV]
| updated = yes
| nominator = Ktin 
| creator =
| updaters = Ktin 
| nom cmt = South African [[anti-apartheid]] activist. Article has shaped into a nice C-class biography. Meets hygiene expectations for homepage / RD. I can do with some assistance on the images. But, this is good to go.
| sign = [[User:Ktin|Ktin]] ([[User talk:Ktin|talk]]) 03:32, 8 December 2021 (UTC)
}}
*'''Support''' Per nom. C-class fully sourced article. Good enough for RD.--[[User:Kacamata|Kacamata!]] [[User_talk:Kacamata|Dimmi!!!]] 03:42, 8 December 2021 (UTC)
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 07:46, 8 December 2021 (UTC)
*'''Support''' Well written comprehensive article. Off topic: there are now two South African activitists on the candidate page. What are the odds. [[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 08:44, 8 December 2021 (UTC)
*'''Posted '''to RD. Great work on the new article. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 15:18, 8 December 2021 (UTC)

==== (Posted) RD: Lindiwe Mabuza ====
{{ITN candidate
| article = Lindiwe Mabuza
| recent deaths = yes
| sources = [https://www.citizen.co.za/news/south-africa/2937397/poet-and-freedom-fighter-lindiwe-mabuza-passes-away/ The Citizen]
| updated = 
| nominator = TJMSmith
| creator = Mdu13081984
| updaters = KittenKlub
| nom cmt = South African politician, diplomat, poet, academic, journalist, and cultural activist.
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 12:48, 7 December 2021 (UTC)
}}
*'''Support''' Comprehensive article, well written and referenced. [[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 13:26, 7 December 2021 (UTC)
*'''Support''' Fully sourced and good enough for RD.--[[User:Kacamata|Kacamata!]] [[User_talk:Kacamata|Dimmi!!!]] 03:26, 8 December 2021 (UTC)
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 07:32, 8 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 23:05, 8 December 2021 (UTC)

==== (Posted) RD: Marjorie Tallchief ====
{{ITN candidate
| article = Marjorie Tallchief
| recent deaths = yes
| sources = [https://www.oklahoman.com/story/entertainment/2021/12/06/marjorie-tallchief-last-oklahomas-five-moons-native-ballerinas-dies/8837729002/ The Oklahoman]
| updated = 
| nominator = TJMSmith
| creator = T. Anthony
| updaters = Jkaharper , Innisfree987 
| nom cmt = American ballerina and the first Native American to be named "première danseuse étoile" in the [[Paris Opera Ballet]]. Death published on December 6.
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 12:44, 7 December 2021 (UTC)
}}
*'''Support''' I added a couple of references, seems in good shape now. [[User:Innisfree987|Innisfree987]] ([[User talk:Innisfree987|talk]]) 01:11, 8 December 2021 (UTC)
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 08:16, 8 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 18:39, 8 December 2021 (UTC)

==== (Closed) Sarco suicide device passes legal review for use in Switzerland ====
{{atop|Consensus will not develop to post. [[User:Pawnkingthree|Pawnkingthree]] ([[User talk:Pawnkingthree|talk]]) 13:43, 7 December 2021 (UTC)}}
{{ITN candidate
| article = Sarco device
| image = 
| blurb = [[Sarco device|Sarco suicide device]] passes legal review for use in Switzerland
| recent deaths = no 
| ongoing = no 
| ITNR = no 
| altblurb = 
| altblurb2 = 
| sources = [https://www.swissinfo.ch/eng/sarco-suicide-capsule--passes-legal-review--in-switzerland/46966510 SWI]
| updated = yes
| nominator = Count Iblis 
| creator = 
| updaters = 
| nom cmt = 
| sign = [[User:Count Iblis|Count Iblis]] ([[User talk:Count Iblis|talk]]) 01:20, 7 December 2021 (UTC) 
}}

*'''Oppose''', per the sourced article "Some 1,300 people died by assisted suicide in Switzerland in 2020 using the services of the country’s two largest assisted suicide organisations, Exit (no connection to Exit International) and Dignitas." This is not the first such means of assisted suicide in Switzerland; it may have been more newsworthy if this made Switzerland the first country to approve of assisted suicide. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 01:27, 7 December 2021 (UTC)
*'''Support''' This has made the news, although could be more suitable for DYK instead. [[Special:Contributions/99.247.176.90|99.247.176.90]] ([[User talk:99.247.176.90|talk]]) 02:29, 7 December 2021 (UTC)
*'''Oppose''' -- this is interesting, but I fail to see how this is worthy of ITN at all. Switzerland already has legal euthanasia. -- [[User:Rockstone35|Rockstone]][[User talk:Rockstone35|Send me a message!]] 04:19, 7 December 2021 (UTC)
*'''Oppose''' both on quality ({{tq|Nitschke plans to release the open source plans for the Sarco by 2019.}}) and importance. [[User:力]] (powera, [[User talk:力|π]], [[Special:Contributions/力|ν]]) 04:27, 7 December 2021 (UTC)
*'''Oppose.''' Minor story. The blurb is misleading, as it implies official approval for the device; in reality, as quoted in the cited Swissinfo article, the inventor merely told a journalist that he asked unnamed lawyers about the legality of the device and that he is "very pleased with the result", whatever that means. [[User:Sandstein|''' Sandstein ''']] 11:16, 7 December 2021 (UTC)
*'''Oppose'''. Aside from what has been stated, I would wonder if we should post information about suicide devices on the Main Page, even if legal in the country involved. Assisted suicide isn't legal in many places. [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 13:24, 7 December 2021 (UTC)
{{abot}}

==== (Posted) New Chancellor of Austria====
{{ITN candidate
| article = Karl Nehammer
| image = File:2020 Karl Nehammer Ministerrat am 8.1.2020 (49351366976) (cropped).jpg

| blurb = '''[[Karl Nehammer]]''' (pictured) becomes the new [[Chancellor of Austria]] following the resignation of [[Alexander Schallenberg]].
|altblurb =
| recent deaths = no
| ongoing = no
| sources = [https://abcnews.go.com/International/wireStory/nehammer-sworn-austrias-chancellor-months-81581062 (ABC News)], [https://www.reuters.com/world/europe/austrias-third-leader-two-months-takes-office-seeking-stability-2021-12-06/ Reuters], [https://www.dw.com/en/austria-karl-nehammer-sworn-in-as-new-chancellor/a-60032938 DW]

|ITNR=yes
| updated = yes
| nominator = Alsoriano97
| creator =
| updaters =
| nom cmt = Article looks pretty good.
| sign = [[User:Alsoriano97|_-_Alsoriano97]] ([[User talk:Alsoriano97|talk]]) 23:03, 6 December 2021 (UTC)
}}
* The blurb should include "following the resignation of [[Alexander Schallenberg]]" and some more context on the replacement would be helpful in the article, it is a bit short at the moment on that part. --'''[[User:Tone|Tone]]''' 23:08, 6 December 2021 (UTC)
:*Done. Thanks! [[User:Alsoriano97|_-_Alsoriano97]] ([[User talk:Alsoriano97|talk]]) 23:15, 6 December 2021 (UTC)
*'''Support''' Looks adequate. One minor change made. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 23:13, 6 December 2021 (UTC)
*'''Support''' per Sca. Quality looks acceptable [[User:Canadianerk|Canadianerk]] ([[User talk:Canadianerk|talk]]) 02:19, 7 December 2021 (UTC)
*'''Support''' - seems to meet requirements, although the chancellor article needs ref improvement. - [[User:Indefensible|Indefensible]] ([[User talk:Indefensible|talk]]) 04:57, 7 December 2021 (UTC)
*'''Support''' Looks good to go. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 09:46, 7 December 2021 (UTC)
*'''Posted'''. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 17:17, 7 December 2021 (UTC)

====(Posted) RD: Medina Spirit ====
{{ITN candidate
| article = Medina Spirit
| recent deaths = yes
| sources = [https://theathletic.com/news/medina-spirit-2021-kentucky-derby-winner-dies-after-workout-at-santa-anita/tkOk7f0ncWII/ The Athletic], [https://www.theguardian.com/sport/2021/dec/06/medina-spirit-horse-at-heart-of-kentucky-derby-controversy-dies-in-training The Guardian]
| updated = 
| nominator = Rawmustard 
| creator = Strattonsmith
| updaters = Dahossindamile 
| nom cmt = Controversial 2021 Kentucky Derby winner. The article looks fully referenced throughout.
| sign = [[User:Rawmustard|rawmustard]] ([[User talk:Rawmustard|talk]]) 20:19, 6 December 2021 (UTC) 
}}

*'''Support''' I was just about to nominate this. Article looks to be in good shape. [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 20:22, 6 December 2021 (UTC)
*'''Support''' the article is thorough and sourced. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 22:19, 6 December 2021 (UTC)
*'''Posted to RD'''. '''[[User:Spencer|Spencer]]'''[[User talk:Spencer|T•]][[Special:Contributions/Spencer|C]] 06:38, 7 December 2021 (UTC)

==== RD: Marvin Morgan ====
{{ITN candidate
| article = Marvin Morgan
| recent deaths = yes
| sources = [https://www.skysports.com/football/news/11095/12488366/marvin-morgan-former-striker-turned-entrepreneur-and-campaigner-dies-aged-38 Sky Sports]
| updated = yes
| nominator = ArsenalGhanaPartey 
| updaters = Matthew Salisbury, LBLM9253, Robby.is.on
| nom cmt = Former footballer. Should be ready.
| sign = [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 20:00, 6 December 2021 (UTC)
}}
*'''Comment:''' Close but needs more citations. '''[[User:Spencer|Spencer]]'''[[User talk:Spencer|T•]][[Special:Contributions/Spencer|C]] 05:51, 7 December 2021 (UTC)
* @[[User:Spencer|Spencer]] I added a few more citations to the article, would it be safe to say it's ready now? [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 12:24, 7 December 2021 (UTC)
*'''Support'''- citations where expected. Decently sized article. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 17:33, 8 December 2021 (UTC)
* I inserted a few footnotes and fixed some links, too. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 09:05, 9 December 2021 (UTC)

====(Posted) RD: Fred Hiatt ====
{{ITN candidate
| article = Fred Hiatt
| recent deaths = yes
| sources = [https://www.washingtonpost.com/obituaries/2021/12/06/fred-hiatt-dies/ WaPo]
| updated = 
| nominator = Muboshgu 
| creator = Fuzheado
| updaters = Muboshgu, Sunshineisles2
| nom cmt = <s>Needs some sourcing, but not far from ready</s> Should be ready
| sign = – [[User:Muboshgu|Muboshgu]] ([[User talk:Muboshgu#top|talk]]) 19:11, 6 December 2021 (UTC) 
}}
*'''Support''' Appropriate depth of coverage, referenced. Marking ready. '''[[User:Spencer|Spencer]]'''[[User talk:Spencer|T•]][[Special:Contributions/Spencer|C]] 05:51, 7 December 2021 (UTC)
*'''Posted '''to RD. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 17:12, 7 December 2021 (UTC)

==== Aung San Suu Kyi sentenced to four years in prison====
{{ITN candidate
| article = Aung San Suu Kyi
| image = File:Aung San Suu Kyi visited the European Commission, met with Federica Mogherini (14) (cropped).jpg
| blurb = Former [[State Counsellor of Myanmar]] '''[[Aung San Suu Kyi]]''' is sentenced to four years in prison on charges of [[dissent|inciting dissent]] and breaking [[COVID-19 pandemic in Myanmar|COVID-19 rules]] under a natural disasters law.
|altblurb = Former [[State Counsellor of Myanmar]] '''[[Aung San Suu Kyi]]''' is sentenced to four years in prison on charges of [[dissent|inciting dissent]], the sentence being cut to two years in a partial pardon.
|altblurb2 = Former [[State Counsellor of Myanmar]] '''[[Aung San Suu Kyi]]''' is sentenced to prison on charges of [[dissent|inciting dissent]].
| recent deaths = no
| ongoing = no
| sources = [https://www.bbc.com/news/world-asia-59544484 BBC], [https://edition.cnn.com/2021/12/06/asia/suu-kyi-verdict-sentence-myanmar-intl-hnk/index.html CNN], [https://apnews.com/article/coronavirus-pandemic-health-elections-voting-myanmar-986be24c558b54c6ebdb20aa272657ca AP], [https://www.theguardian.com/world/2021/dec/06/aung-san-suu-kyi-sentenced-to-four-years-in-prison-for-incitement Guardian], [https://www.reuters.com/world/asia-pacific/myanmar-court-give-first-rulings-suu-kyi-trial-2021-12-05/ Reuters], [https://www.aljazeera.com/news/2021/12/6/aung-san-suu-kyi-sentenced-to-x AlJazeera]

| updated = yes
| nominator = Kiril Simeonovski 
| creator =
| updaters =
| nom cmt = This is a notable ruling against a famous political figure and a Nobel Peace Prize laureate.
| sign = --[[User:Kiril Simeonovski|Kiril Simeonovski]] ([[User talk:Kiril Simeonovski|talk]]) 08:21, 6 December 2021 (UTC)
}}
*'''Question''' Should we wait? There are more charges against her and it can add up to life in prison. [[User:Scaramouche33|Scaramouche33]] ([[User talk:Scaramouche33|talk]]) 08:49, 6 December 2021 (UTC)
*'''Oppose''' Significance is somewhat diminished as we already posted the coup in February, and the section on this period needs a rewrite as it is currently proseline with lots of extraneous detail. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 13:06, 6 December 2021 (UTC)
* '''Support''' – Long-term political cause célèbre comes to a head. Widely and prominently covered, with much criticism of Myanmar military regime. Favor '''Alt1'''. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:37, 6 December 2021 (UTC)
*'''Support''' The coup being posted shouldn't diminish someone going to prison [again] when they received international coverage for efforts to come out of it. [[User:Gotitbro|Gotitbro]] ([[User talk:Gotitbro|talk]]) 15:00, 6 December 2021 (UTC)
::Can do with a better blurb though, too lengthy and incoherent for the average reader. [[User:Gotitbro|Gotitbro]] ([[User talk:Gotitbro|talk]]) 20:30, 6 December 2021 (UTC)
*<s>'''Comment'''</s> '''Support'''. She received a partial pardon and will serve two years ([https://www.aljazeera.com/news/2021/12/6/aung-san-suu-kyi-sentenced-to-x Al Jazeera] and all sources above). -[[User:SusanLesch|SusanLesch]] ([[User talk:SusanLesch|talk]]) 18:21, 6 December 2021 (UTC)
*'''Support''' because it's easily important enough. If she's later convicted of anything else, the blurb can be adjusted accordingly. [[User:Jim Michael|Jim Michael]] ([[User talk:Jim Michael|talk]]) 20:05, 6 December 2021 (UTC)
*'''Support''', It is an important news and will start other events as well. [[User:Alex-h|Alex-h]] ([[User talk:Alex-h|talk]]) 21:11, 6 December 2021 (UTC)
*:Blurbed in French, German versions of ITN. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 23:15, 6 December 2021 (UTC)
*Is anyone actually reviewing the article? ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 00:00, 7 December 2021 (UTC)
** I just glanced through and there's a fair number of unsourced statements near the bottom half of the article. I'd also argue there's far too much excessive detail on her history as related to any litigation or complaints, but that's not an ITN item as long as it appears neutrally written. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 01:29, 7 December 2021 (UTC)
*'''Support''' per Alex-h, Gotitbro, and Jim Michael. [[User:Fakescientist8000|Fakescientist8000]] ([[User talk:Fakescientist8000|talk]]) 01:38, 7 December 2021 (UTC)
*'''Comment''': The [[Aung San Suu Kyi]] article has about 20 {cn} tags, and the 2021 arrest and trial section consists of mostly proseline. Please fix. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 05:51, 7 December 2021 (UTC)
*'''Oppose''' Too much tags. [[User:Hanamanteo|Hanamanteo]] ([[User talk:Hanamanteo|talk]]) 08:15, 7 December 2021 (UTC)
*'''Oppose''' There are too many tags for this to be ready on time. It is also a borderline case for me in terms of significance of news given that we already posted the coup in February. So lets wait for more charges against her so that there is more time for the article tags to be fixed. [[User:Tradedia|Tradedia]][[User talk:Tradedia|talk]] 12:34, 7 December 2021 (UTC)
::::Asleep at the switch. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:44, 7 December 2021 (UTC)
*<s>'''Oppose''' Unfortunately, the article has far too many CN tags to be posted on the main page. If someone wanted this posted on the main page, they would work on fixing that problem. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 19:12, 7 December 2021 (UTC)</s>
:*'''Support''' Citation issue has been fixed. It's good for the main page now. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 11:50, 8 December 2021 (UTC)
*Have fixed all the tags that I could find. {{re|Jayron32|Sca|Tradedia}} please review the votes. [[User:Gotitbro|Gotitbro]] ([[User talk:Gotitbro|talk]]) 11:13, 8 December 2021 (UTC)
:*Proseline issues remain. There's a graph about her being moved, one about the trial beginning, and another about a delay in the trial when she got sick. Collectively, this reads very poorly. Our goal here is to encourage quality updates by featuring them; this is not quality. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 12:09, 8 December 2021 (UTC)
:::Alas, getting stale. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:10, 9 December 2021 (UTC)
::::''PS:'' Would someone care to define ''proseline?'' It's not in Oxford, Webster, Wiktionary or even Urban Dictionary. I suggest we not gabble in jargon. Thanks. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 16:00, 9 December 2021 (UTC)
:::::I think you would benefit more from the definition of [[WP:POINT|disruptive editing]], with four off-topic comments in this nom alone. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 19:19, 9 December 2021 (UTC)
:::::{{ping|Sca}} See [[WP:PROSELINE]]. If you had already fixed this, this would have already been posted. You should spend more time fixing articles and less time complaining that nothing you want is being done. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 19:52, 9 December 2021 (UTC)
::::::Jargon. This user prefers English. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 19:57, 9 December 2021 (UTC)
:::::::Use English to fix the problems with the article if you want it posted. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 20:02, 9 December 2021 (UTC)

==== (Posted) RD: John Miles (musician)====

{{ITN candidate
| article = John Miles (musician)
| recent deaths = yes
| sources = [https://www.bbc.com/news/entertainment-arts-59553438 BBC]
| updated = no
| nominator = Davidstewartharvey 
| creator = Deleteme42
| updaters = Davidstewartharvey, Britmax
| nom cmt = Needs some work but one of the most underrated musicians of all time. Music is one of the great 70s anthems.
| sign = [[User:Davidstewartharvey|Davidstewartharvey]] ([[User talk:Davidstewartharvey|talk]]) 19:46, 6 December 2021 (UTC)
}}
*'''Comment''' I have now upgraded and improved the article. Think there is more to be added with paper refs as stuff like Melody Maker etc aren't digitalised.[[User:Davidstewartharvey|Davidstewartharvey]] ([[User talk:Davidstewartharvey|talk]]) 14:18, 7 December 2021 (UTC)
*'''Support''' Article looks good to me. RIP. I always loved the song, but didn't know it was called Music or who John Miles was... [[User:Tradedia|Tradedia]][[User talk:Tradedia|talk]] 21:41, 7 December 2021 (UTC)
*'''Posted '''to RD. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 17:36, 8 December 2021 (UTC)

== December 5 ==
{{cot|[[Portal:Current events/2021 December 5]]}}
{{Portal:Current events/2021 December 5}}
{{cob}}
----
==== (Posted) RD: M. Sarada Menon ====
{{ITN candidate
| article = M. Sarada Menon
| recent deaths = yes
| sources = [https://thefederal.com/states/south/tamil-nadu/dr-sarada-menon-indias-first-lady-of-mental-health-passes-away-in-chennai/ The Federal]
| updated = y
| nominator = TJMSmith
| creator = Tachs
| updaters = Jkaharper
| nom cmt = Indian psychiatrist, social worker and the founder of [[Schizophrenia Research Foundation]].
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 12:52, 7 December 2021 (UTC)
}}
*'''Comment''' - The ref number 13, referecing the fact that she was "a recipient of the Lifetime Achievement Award of Madras Neuro Trust" is dead. It seems the only issue in the article.--[[User:Kacamata|Kacamata!]] [[User_talk:Kacamata|Dimmi!!!]] 03:35, 8 December 2021 (UTC)
** The deadlink has been revived with an updated URL. Lucky Ref.#13! --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 23:17, 8 December 2021 (UTC)
* '''Support'''. Article looks good to me. Should be ready. [[User:Ktin|Ktin]] ([[User talk:Ktin|talk]]) 01:35, 10 December 2021 (UTC)
* '''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 03:58, 10 December 2021 (UTC)

==== (Posted) 2021 Nagaland killings ====
{{ITN candidate
| article = 2021 Oting Nagaland Massacre
| image = Kohima Candlelight Vigil Service 2021.jpg
| blurb = '''[[2021 Nagaland killings|Fourteen civillians are killed]]''' by [[Indian Army]] forces in failed ambush and subsequent violence in [[Nagaland]], India.
| recent deaths = no 
| ongoing = no 
| ITNR = no 
| altblurb = '''[[2021 Nagaland killings|Violent confrontations]]''' with [[Indian Army]] forces result in the death of at least fourteen civilians and one soldier in [[Nagaland]].
| altblurb2 = 
| sources = [https://www.bbc.com/news/world-asia-india-59544599 BBC], [https://www.thehindu.com/news/national/other-states/16-dead-after-armys-nagaland-ambush/article37849057.ece The Hindu]
| updated = yes
| nominator = Nizil Shah 
| creator = The Anonymous Earthling
| updaters = Nizil Shah
| nom cmt = Need move to [[2021 Nagaland killings]], see talkpage. Redirect deletion requested. Propose alt blurb if any. Caption: Candlelight vigil organised in [[Kohima]].
| sign = [[User:Nizil Shah|Nizil]] ([[User talk:Nizil Shah|talk]]) 07:55, 7 December 2021 (UTC) 
}}
*'''Support''' - an escalation in the ongoing insurgency with a significant no. of deaths. [[User:MasterOfMetaverse|MasterOfMetaverse]] ([[User talk:MasterOfMetaverse|talk]]) 13:44, 7 December 2021 (UTC)
*'''Support pending''' cleanup noted by nom. There are a few minor issues with delineating the initial attack from subsequent killings. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 14:12, 7 December 2021 (UTC)
*'''Support''' number of civilians death ae high.-[[User:امین اکبر|Ameen Akbar]] ([[User talk:امین اکبر|talk]]) 16:55, 7 December 2021 (UTC)
*'''Support''' article is sufficiently comprehensive and well referenced, topic has been covered appropriately by reliable news sources, demonstrating significance. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 19:14, 7 December 2021 (UTC)
* Added altblurb. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 01:44, 8 December 2021 (UTC)
*'''Posted''' [[User:Stephen|Step]][[User talk:Stephen|hen]] 03:00, 8 December 2021 (UTC)

==== RD: Jacques Tits ====
{{ITN candidate
| article = Jacques Tits
| recent deaths = yes
| sources = [https://smf.emath.fr/actualites-smf/deces-de-jacques-tits Société mathématique de France]
| updated = no
| nominator = Davey2116
| updaters =
| nom cmt = French mathematician, laureate of the [[Abel Prize]], dies at age 91. Article needs some work.
| sign = [[User:Davey2116|Davey2116]] ([[User talk:Davey2116|talk]]) 22:36, 6 December 2021 (UTC)
}}
* Please add more refs, particularly to the "Life and career" and "Contributions" sections. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 01:33, 8 December 2021 (UTC)

==== (Posted) RD: Christine Haidegger‎ ====
{{ITN candidate
| article = Christine Haidegger‎
| recent deaths = yes
| sources = [https://www.derstandard.de/story/2000131679498/autorin-christine-haidegger-ist-verstorben Der Standard]
| updated = yes
| nominator = Grimes2
| creator = T. Anthony
| updaters = Grimes2, Gerda Arendt
| nom cmt = Austrian writer, poet. Was influential in the Salzburg literary scene.
| sign = [[User:Grimes2|Grimes2]] ([[User talk:Grimes2|talk]]) 13:38, 6 December 2021 (UTC)
}}
*'''Support''' Well sourced little article. [[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 14:34, 6 December 2021 (UTC)
*'''Posted'''. That was a nice and quick 5x expansion. Good luck at DYK. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 18:40, 6 December 2021 (UTC)

==== (Posted) RD: Bob Dole ====
{{ITN candidate
| article = Bob Dole
| recent deaths = yes
| sources = [https://apnews.com/article/bob-dole-dead-kansas-republican-737084f4e606c10a33a15384fbe05967 AP], [https://www.bbc.com/news/world-us-canada-45667690 BBC], [https://www.theguardian.com/us-news/2021/dec/05/bob-dole-republican-presidential-nominee-senator-dies-aged Guardian], [https://www.npr.org/2021/12/05/123981928/bob-dole-dies-at-98 NPR], [https://www.pbs.org/newshour/politics/former-senate-majority-leader-presidential-candidate-bob-dole-dies PBS], [https://www.nbcnews.com/news/us-news/bob-dole-wwii-hero-former-republican-presidential-candidate-dies-98-n953981 NBC], [https://twitter.com/DoleFoundation/status/1467533869905817602 Twitter]
| updated = 
| nominator = 力 
| creator = RobLa
| updaters = The Image Editor, Dylansh99, Ser!
| nom cmt = Former US Senator and presidential candidate. Article needs some work but is close to ready.
| sign = [[User:力]] (powera, [[User talk:力|π]], [[Special:Contributions/力|ν]]) 16:51, 5 December 2021 (UTC) 
}}

Think it could work as a blurb, Bob Dole was a massive figure in American politics for over two decades. [[User:HadesTTW|HadesTTW]] (he/him • [[User talk:HadesTTW|talk]]) 17:18, 5 December 2021 (UTC)
* '''Comment''' – Likely blurb. Very widely covered, much subject info. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 17:38, 5 December 2021 (UTC)
*'''Oppose blurb''' It wasn't his run for president that made him notable, it was his role in the Senate, and that section of the article is far far too short to show that (that's orange tagged right now). The article is ready for RD outside that orange tag. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 17:42, 5 December 2021 (UTC)
*:Household name for decades. Suggest we '''wait''' at least a while for article development. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 17:51, 5 December 2021 (UTC)
*::In the US, absolutely, but not so much worldwide politics. Not a strong reason to post as a blurb (particularly with the article's lack of his political record outside of Presidental runs). --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 18:07, 5 December 2021 (UTC)
*:::"2. oppose an item because the event is only relating to a single country, or failing to relate to one. This applies to a high percentage of the content we post and is unproductive." I don't know if this applies here but to me it kind of does. '''[[User:KingOfAllThings|KingOf'''AllThings''']]''' [[User_talk:KingOfAllThings|(thou shalt chatter!)]] 00:26, 6 December 2021 (UTC)
*::::My original oppose is not related to just being the US, but pointing out that the above "household name" is relying on being a US centric aspect. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 00:33, 6 December 2021 (UTC)
*'''Tentative support RD, Oppose blurb''' Article does have an expansion tag so I would support once that has been sorted. As for the blurb, to be blunt (as ''[[Futurama]]'' once put it) he was a "Presidential loser". We don't tend to do blurbs for people who lost their one and only major election for head of state. '''[[User:The C of E|The C of E God Save the Queen!]]''' ([[User talk:The C of E|talk]]) 17:57, 5 December 2021 (UTC)
*'''Support RD''' pending removal of citation needed tags and accompanying orange banner in awards section. Tag on Senate section is no reason to hold back though. Bob Dole... Bob Dole... bob... dole... - '''[[User:Floydian|Floydian]]''' [[User_talk:Floydian|τ]] [[Special:Contributions/Floydian|¢]] 18:20, 5 December 2021 (UTC)
*'''Oppose blurb''' per Masem. He's as important as you say he is in the United States, but that doesn't mean it's important beyond its borders. And that's not the case. He doesn't pass the Thatcher/Mandela filter and that he was a failed candidate for the presidency of that country and a senator for many years IMO is no reason for him to be blurb-worthy. One section is orange-tagged. [[User:Alsoriano97|_-_Alsoriano97]] ([[User talk:Alsoriano97|talk]]) 18:34, 5 December 2021 (UTC)
*'''Oppose blurb'''. Worldwide, Dole is probably mostly known for being lampooned in ''The Simpsons''. Never served in a cabinet, and I can't imagine us considering posting a blurb for a backbencher from another country. [[User:Grapple X|ᵹʀᴀᴘᴘʟᴇ]] [[User talk:Grapple X|ꭗ]] 18:43, 5 December 2021 (UTC)
*'''Oppose blurb''' If we go by precedent, the 2008 runner-up with a long Congressional history was also RD only [https://en.wikipedia.org/w/index.php?title=Template:In_the_news&oldid=856567161]. No disrespect to the deceased, but even in America's closest ally, he's mainly known as the guy speaking in the third person who got abducted by [[Kang and Kodos]]. [[User:Unknown Temptation|Unknown Temptation]] ([[User talk:Unknown Temptation|talk]]) 18:57, 5 December 2021 (UTC)
*'''{{s|Photo}} Plain RD''' If anybody had a name that still speaks for itself, it's Bob Dole....Bob Dole! [[User:InedibleHulk|InedibleHulk]] ([[User talk:InedibleHulk|talk]]) 19:03, 5 December 2021 (UTC)
** That's a poor reason to include an image. We've only done that when the blurbs have gotten stale, but with the volcano, we clearly have a notable image. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 19:05, 5 December 2021 (UTC)
***Oh, right. Hadn't thought about what we'd be losing. A striking visage, but clearly no volcano. [[User:InedibleHulk|InedibleHulk]] ([[User talk:InedibleHulk|talk]]) 19:12, 5 December 2021 (UTC)
*'''Oppose blurb, oppose photo for RD''' Not a head of state, not a head of government, not sufficiently notable. [[User:Chrisclear|Chrisclear]] ([[User talk:Chrisclear|talk]]) 19:07, 5 December 2021 (UTC)
*'''Support RD''' notable person. [[:User:Wizzito|wizzito]] | [[:User talk:Wizzito|say hello!]] 22:20, 5 December 2021 (UTC)
*'''Support blurb''', futile as this may be. Someone has to stand up for the notability of the former Majority Leader of the United States Senate, who effectively shares leadership of the federal legislative branch of the country, which is technically coequal with the executive and judicial branches. I would suggest that someone who holds this position ''and'' becomes the leading contender for the U.S. presidency is blurb-worthy. [[User:BD2412|'''''BD2412''''']] [[User talk:BD2412|'''T''']] 22:47, 5 December 2021 (UTC)
**Does the same rationale apply to someone who holds/held the position of leadership of the federal legislative branch of another country, ''and'' becomes the leading contender for the head of government of that country? Or does it only apply to [[Wikipedia:BIAS|US Presidential candidates]]? [[User:Chrisclear|Chrisclear]] ([[User talk:Chrisclear|talk]]) 23:30, 5 December 2021 (UTC)
*** I think it would depend in part on the structure of the government, and the means by which the position is contended. In parliamentary systems, the titular leader of the legislature often ''is'' the head of government. In countries where the legislature is clearly a subservient entity under the control of the executive, it wouldn't matter, and in countries where the head of government is not democratically selected, the meaning of being a "leading contender" for that position is also too shrouded in mystery to matter. I think there are very few countries that actually have three legally coequal branches of government, and a system wherein the leader of one branch can become a leading contender to be democratically elected to head another. [[User:BD2412|'''''BD2412''''']] [[User talk:BD2412|'''T''']] 23:37, 5 December 2021 (UTC)
**** There's a lot of country where the three branches of government coexist equally. See [[Separation of powers]]. UK, Norway, Sweden, Denmark, etc. --Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 01:18, 6 December 2021 (UTC)
***** How many instances are there of leaders of one branch being selected by their party to credibly contend for leadership of another branch? [[User:BD2412|'''''BD2412''''']] [[User talk:BD2412|'''T''']] 01:20, 6 December 2021 (UTC)
****** IDK about other country, but the [https://books.google.co.id/books?id=kc4qAAAAMAAJ&pg=RA3-PA34 leader of the armed forces group parliament in my country] was [[Susilo Bambang Yudhoyono|elected into presidency]] in a {{abbr|[https://apjjf.org/2014/12/10/David-Adam-Stott/4087/article.html free and fair]|"After the fall of the authoritarian Suharto regime (1967-98) it successfully conducted free and fair elections in 1999, 2004 and 2009, becoming arguably the most politically free country in Southeast Asia."}} [[2004 Indonesian presidential election|election]] [[2009 Indonesian presidential election|twice]]. --Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 01:27, 6 December 2021 (UTC)
******* I'm pretty sure they'll get a blurb, then. [[User:BD2412|'''''BD2412''''']] [[User talk:BD2412|'''T''']] 01:40, 6 December 2021 (UTC)
*'''Support RD''' now that all of the missing citations have been added, as far as I could see. Oppose blurb per above: someone in Bob Dole's position would never get a blurb in any other country. We wouldn't blurb William Hague if he died today, why would we blurb Bob Dole? [[User:NorthernFalcon|NorthernFalcon]] ([[User talk:NorthernFalcon|talk]]) 23:51, 5 December 2021 (UTC)
*'''Posted''' [[User:Stephen|Step]][[User talk:Stephen|hen]] 02:07, 6 December 2021 (UTC)
* '''Post-posting oppose''' His senate section is extremely short for a man who has been described the “Lion of the Senate”. I feel that some who approved this were discussing his nobility rather than the state of the article. I’d recommend pulling this until the section (rather important one) is expanded. Once the section is thoroughly expanded then I’ll support a possible blurb. [[User:TDKR Chicago 101|TDKR Chicago 101]] ([[User talk:TDKR Chicago 101|talk]]) 05:38, 6 December 2021 (UTC)
** I see no sense in asserting that the article is incomplete because it doesn't live up to a purported nickname that doesn't even appear in the article. [[Lion of the Senate]] points to someone else. If the section can be expanded, by all means do so, but there is no concrete basis here for asserting that it is lacking. [[User:BD2412|'''''BD2412''''']] [[User talk:BD2412|'''T''']] 05:53, 6 December 2021 (UTC)
:*{{ping|BD2412}} However all obits cite his long senate career that lasted for decades. If he was a one term senator then sure a short senate section makes sense, but Dole was a Senate leader and longtime GOP Senator. It’s like having an influential actor with a short career section. His senate career section lacks information of his long tenure in the senate. [[User:TDKR Chicago 101|TDKR Chicago 101]] ([[User talk:TDKR Chicago 101|talk]]) 15:49, 6 December 2021 (UTC)
:** You are conflating ''longevity'' with ''productivity'', without justification, and your conclusions are therefore incorrect. His senate career section does specify the ''length of time'' for which he served in the senate. Show me a source that says that he was known for getting signature legislation passed or the like, and you will have a case for saying there is something missing. [[User:BD2412|'''''BD2412''''']] [[User talk:BD2412|'''T''']] 16:51, 6 December 2021 (UTC)
* '''Post-posting Support''' - Definitely for RD. Article looks decent enough for posting.[[User:BabbaQ|BabbaQ]] ([[User talk:BabbaQ|talk]]) 12:40, 6 December 2021 (UTC)

==== (Posted) RD: Peter Cundall ====
{{ITN candidate
| article = Peter Cundall
| recent deaths = yes
| sources = 
| updated = yes
| nominator = Stephen 
| updaters = 
| nom cmt = Australian gardening personality
| sign = [[User:Stephen|Step]][[User talk:Stephen|hen]] 08:47, 5 December 2021 (UTC) 
}}
*'''Support''' Comprehensive article which is well written and referenced.[[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 09:40, 5 December 2021 (UTC)
::Not that it matters, but this is the first time I've heard a nominee described as a "gardening personality." Cheers. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:42, 5 December 2021 (UTC)
:::Cundall can also be described as a bloody commie gardener, but that violates NPOV.[[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 14:15, 5 December 2021 (UTC)
:::In the UK there's been [[Percy Thrower]], [[Alan Titchmarsh]], [[Monty Don]]... c'mon, there must be some US gardening personalities. [[User:Pawnkingthree|Pawnkingthree]] ([[User talk:Pawnkingthree|talk]]) 22:56, 5 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 16:54, 5 December 2021 (UTC)

== December 4 ==
{{cot|[[Portal:Current events/2021 December 4]]}}
{{Portal:Current events/2021 December 4}}
{{cob}}
----
==== RD: Stonewall Jackson ====
{{ITN candidate
| article = Stonewall Jackson (musician)
| recent deaths = yes
| sources = [https://www.foxnews.com/entertainment/stonewall-jackson-grant-ole-opry-dead-89 Fox News], [https://www.nytimes.com/2021/12/05/obituaries/stonewall-jackson-dead.html New York Times]
| updated = No
| nominator = CoatCheck 
| updaters = 
| nom cmt = Grand Ole Opry inductee, [[Waterloo (Stonewall Jackson song) | Waterloo]] country crossover hit and 43 other charting songs.
| sign = [[User:CoatCheck|CoatCheck]] ([[User talk:CoatCheck|talk]]) 01:48, 8 December 2021 (UTC) 
}}
* The Discography section is unsourced. The prose has a few {cn} tags, too. Please add more refs. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 03:13, 8 December 2021 (UTC)

==== (Posted) 2021 Gambian presidential election ====
{{ITN candidate
| article = 2021 Gambian presidential election
| image = Adama Barrow 2018.jpg
| blurb = [[Adama Barrow]] ''(pictured)'' is '''[[2021 Gambian presidential election|re-elected]]''' [[President of the Gambia]].
| recent deaths = no
| ongoing = no
| ITNR = yes
| sources = [https://www.bbc.com/news/world-africa-59542813 BBC], [https://www.aljazeera.com/news/2021/12/6/the-gambias-barrow-wins-second-term-opposition-reject-results Al Jazeera], [https://www.dw.com/pt-002/adama-barrow-reeleito-presidente-da-g%C3%A2mbia/a-60029470 DW]
| updated = yes
| nominator = Kacamata
| creator = Thomascampbell123
| updaters = Lbynch , Number 57 , Joofjoof
| nom cmt = The article is sourced and updated.
| sign = --[[User:Kacamata|Kacamata!]] [[User_talk:Kacamata|Dimmi!!!]] 05:48, 7 December 2021 (UTC)
}}
*'''Support''' The article seems ready, <s>although it doesn’t have an "Aftermath" or "Reactions" section.</s> I would change the proposed photo to the one on his wikibio, which looks more up to date. [[User:Alsoriano97|_-_Alsoriano97]] ([[User talk:Alsoriano97|talk]]) 21:59, 7 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 01:13, 8 December 2021 (UTC)

==== RD: Vinod Dua ====
{{ITN candidate
| article = Vinod Dua
| recent deaths = yes
| sources = [https://www.ndtv.com/india-news/veteran-journalist-vinod-dua-dies-at-67-2637145 NDTV]
| updated = No
| nominator = Ktin 
| updaters = To be updated.
| nom cmt = Indian journalist. Orange tagged. Will require some work.
| sign = [[User:Ktin|Ktin]] ([[User talk:Ktin|talk]]) 19:11, 4 December 2021 (UTC)
}}
* '''Oppose:''' The whole career is covered under early life. Unnecessary controversy section ([[WP:CSECTION]]), with undue expanded statements; needs to be heavily condensed and incorporated into a career section. [[User:Gotitbro|Gotitbro]] ([[User talk:Gotitbro|talk]]) 13:18, 5 December 2021 (UTC)
* Too much footnote-free materials. Please add more refs. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 06:04, 9 December 2021 (UTC)

==== RD: Claude Humphrey ====
{{ITN candidate
| article = Claude Humphrey
| recent deaths = yes
| sources = [https://www.nfl.com/news/pro-football-hall-of-famer-claude-humphrey-dies-at-77]
| updated = Yes
| nominator = ArsenalGhanaPartey 
| updaters = KittenKlub
| nom cmt = National Football League hall of famer. Article could use a bit of work.
| sign = [[User:ArsenalGhanaPartey|ArsenalGhanaPartey]] ([[User talk:ArsenalGhanaPartey|talk]]) 17:21, 4 December 2021 (UTC) 
}}
* Too many footnote-free paragraphs. Please add more refs. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 01:28, 8 December 2021 (UTC)

==== (Posted) RD: Eileen Ash ====
{{ITN candidate
| article = Eileen Ash
| recent deaths = yes
| sources = [https://www.bbc.co.uk/sport/cricket/59533539 BBC], [https://www.ecb.co.uk/news/2392824/ecb-pays-tribute-to-eileen-ash-the-worlds-oldest-ever-test-cricketer ECB]
| updated = Yes
| nominator = Lugnuts 
| creator = Nick mallory
| updaters = Kirubar
| nom cmt = World's oldest Test cricketer
| sign = '''[[User:Lugnuts|Lugnuts]]''' [[User talk:Lugnuts|Fire Walk with Me]] 14:05, 4 December 2021 (UTC) 
}}
*'''Support''' was coming by to nominate this myself. Article looks good enough for RD. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 16:44, 4 December 2021 (UTC)
*'''Support''' - the article is sourced and updated. What a long life! [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 18:28, 4 December 2021 (UTC)
*'''Posted'''. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 20:39, 4 December 2021 (UTC)
==== (Posted) 2021 Semeru eruption ====
{{ITN candidate
| article = 2021 Semeru eruption
| image =
| blurb = The Semeru volcano '''[[2021 Semeru eruption|erupts]]''' on the island of Java, killing at least 13 and injuring dozens.
|altblurb = [[Semeru|Mount Semeru]] in Indonesia '''[[2021 Semeru eruption|erupts]]''', killing 14 people and injuring hundreds of others.
| recent deaths = no
| ongoing = yes
| sources = [https://www.reuters.com/world/asia-pacific/ten-people-trapped-after-indonesias-semeru-volcano-erupts-evacuated-2021-12-05/ Reuters], [https://apnews.com/article/environment-and-nature-indonesia-java-4e7c30a65de39530c0ef53a086bb08ca AP], [https://www.aljazeera.com/news/2021/12/5/residents-in-shadow-of-indonesia-volcano-reckon-with-ruin AlJazeera], [https://www.bbc.com/news/world-asia-59532251 BBC], [https://www.dw.com/en/indonesia-rescuers-search-for-survivors-after-volcano-eruption/a-60024573 DW], [https://www.latimes.com/world-nation/story/2021-12-05/smoldering-debris-mud-hinder-indonesia-volcano-rescue LAT]

| updated = Dora the Axe-plorer
| nominator = Dora the Axe-plorer 
| creator = Dora the Axe-plorer
| updaters = Dora the Axe-plorer
| nom cmt = Still being updated, although most of the sources are not in English.
| sign = [[User:Dora the Axe-plorer|Dora the Axe-plorer]] ([[User talk:Dora the Axe-plorer|Nopen't]]) 05:49, 5 December 2021 (UTC) 
}}
:::Plenty of sources in English, including NYT, not listed above due to paywall. - [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 17:00, 5 December 2021 (UTC)
*'''Support''' - no referencing issues, long enough, death toll likely to increase. [[User:Mjroots|Mjroots]] ([[User talk:Mjroots|talk]]) 11:17, 5 December 2021 (UTC)
*'''Support''', death toll significant, article looks ok. [[User:Brandmeister|Brandmeister]][[User talk:Brandmeister|talk]] 11:38, 5 December 2021 (UTC)
*'''Support''' Significant incident and article is in good condition. --Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 14:43, 5 December 2021 (UTC)
*'''Posted''' --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 15:05, 5 December 2021 (UTC)
* '''Post-posting comment''' – Seems to me we should have waited a bit longer to see how much higher the toll gets. Thirty-five of 56 injuries termed severe, says ''AlJ''. "Hundreds" of injuries may be an exaggeration. AP said 57 were hospitalized, of which 16 were 'critical.' – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 15:12, 5 December 2021 (UTC)
**'''Post-response to comment by User:Sca''' – The source you were posting seems to be outdated. CNN Indonesia says that 102 were injured and most victims suffered burns. (added by me after a bunch of mess-ups) (link: https://www.cnnindonesia.com/nasional/20211205121045-20-730066/102-warga-lumajang-terluka-erupsi-semeru-sebagian-besar-luka-bakar) And I agree that "hundreds" of injuries may be somewhat exaggerating, given that "hundreds" is in the ballpark of 100-1000 (500 median or mean). [[User:Akmaie Ajam|ᐱᔌᕬᐱɭᕮ ᐱᒧᐱᕬ]] [[User talk:Akmaie Ajam|(Talk)]] 15:29, 5 December 2021 (UTC)
*::Personally, I'm not a big fan of CNN as a RS. The [https://www.bbc.com/news/world-asia-59532251 BBC] says "at least 56" injured. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 16:44, 5 December 2021 (UTC)
** Update 2: Nevermind - 45 people suffered burn marks according to Kompas (https://regional.kompas.com/read/2021/12/04/223240978/erupsi-semeru-dinkes-jatim-perkirakan-ada-warga-tertimbun-sulit-dievakuasi). Around 44% suffered burn marks. [[User:Akmaie Ajam|ᐱᔌᕬᐱɭᕮ ᐱᒧᐱᕬ]] [[User talk:Akmaie Ajam|(Talk)]] 15:41, 5 December 2021 (UTC)
:* No worries. The word "hundreds" (plural) did not make it to MainPage. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 16:12, 5 December 2021 (UTC)
:::Basically disagree. Current blurb says "injuring more than a hundred people." Most RS articles linked above don't support such a large number. Also, RS's put toll at 13-14, which one must acknowledge isn't really a huge number as natural disasters go. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 17:19, 5 December 2021 (UTC)
:::::''PS:'' FWIW, this event isn't currently listed in the French, Dutch, German, Swedish or Russian versions of ITN. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 17:22, 5 December 2021 (UTC)
::::::FWIW, it's Sunday afternoon in Europe.--[[Special:Contributions/65.94.214.139|65.94.214.139]] ([[User talk:65.94.214.139|talk]]) 19:50, 5 December 2021 (UTC)
* '''Comment''' {{re|PFHLai}}, you might use this image: [[:File:Ratusan rumah tertimbun abu vulkanik erupsi gunung Semeru.jpg]]. Shows the erupting mountain as well as the damages. --Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 15:48, 5 December 2021 (UTC)
:* Thanks for the suggestion, Jeromi Mikhael. Things look too tiny once the image is squished onto the ITN template. It would be better to put it in the article than on MainPage. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 16:10, 5 December 2021 (UTC)
:::The current blurb pic. of dormant Mt. Semeru on a blue-sky day, though fairly recent, doesn't do much to illustrate this news event, IMO. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 17:45, 5 December 2021 (UTC)
::::If you have a better pic, please proposed it here at ITN/C. If appropriate, a later screenshot from the same YouTube videoclip showing the damages may be useful. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 20:33, 5 December 2021 (UTC)
:::::{{u|PFHLai}}, how about [[:File:Ratusan rumah tertimbun abu vulkanik erupsi gunung Semeru (2).jpg|this]]? Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 23:54, 5 December 2021 (UTC)
[[File:Ratusan rumah tertimbun abu vulkanik erupsi gunung Semeru (2, cropped).jpg|right|175px]]
:::::: Thanks, [[User:Jeromi Mikhael|Jeromi Mikhael]]. That is pretty much what I had in mind. I have prepared a cropped and brightened version ''(shown on the right)'', hoping that viewers can see the damages caused by the eruption. Does this work? Need a caption? --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 11:38, 6 December 2021 (UTC)
::::::: {{re|PFHLai}} Thanks for the edit m8, I think you just need to add ''(damage pictured)'' or something like that to the blurb. Regards, [[User talk:Jeromi Mikhael|Jeromi Mikhael]] 12:15, 6 December 2021 (UTC)
:::::::: The pic is now '''posted'''. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 18:48, 6 December 2021 (UTC)
* '''Comment''' – Most RS reports put the number of injured at 56 (Reuters, BBC, AlJazeera, DW) or 57 (AP, LAT), substantially less than the "more than 100" in the current blurb. Suggest we replace "more than 100" with "more than 50," which would be accurate based on what's been reported so far. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 19:43, 5 December 2021 (UTC)
:*Please revise and update the article with reliable sources and afterwards let us know here at ITN. The blurb should match the article. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 20:33, 5 December 2021 (UTC)
:*And remember, a lot of people don't go to the hospital when they're hurt. I don't know how many. But AP (copied in LAT) says 57 "hospitalized". [[User:InedibleHulk|InedibleHulk]] ([[User talk:InedibleHulk|talk]]) 21:15, 5 December 2021 (UTC)
::*Updated after request from Sca, as Reuters (https://www.reuters.com/world/asia-pacific/ten-people-trapped-after-indonesias-semeru-volcano-erupts-evacuated-2021-12-05/) says that 56 were '''injured''', not hospitalized. [[User:Akmaie Ajam|ᐱᔌᕬᐱɭᕮ ᐱᒧᐱᕬ]] [[User talk:Akmaie Ajam|(Talk)]] 05:31, 6 December 2021 (UTC)
:::* The blurb on MainPage is now updated to "... killing fifteen people and injuring dozens <s>others</s>." --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 11:30, 6 December 2021 (UTC)

== December 3 ==
{{cot|[[Portal:Current events/2021 December 3]]}}
{{Portal:Current events/2021 December 3}}
{{cob}}
----
==== RD: Fortune FitzRoy, Duchess of Grafton ====
{{ITN candidate
| article = Fortune FitzRoy, Duchess of Grafton
| recent deaths = yes
| sources = https://www.thetimes.co.uk/article/fortune-duchess-of-grafton-obituary-8qvgnzq2w 
| updated = y 
| updaters = Richiepip , Jkaharper 
| nom cmt = Queen Elizabeth II's [[Mistress of the Robes]] since 1967. Wikibio looks clean enough and almost ready for RD consideration (needs a few more footnotes). 
| sign = -- [[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 00:03, 10 December 2021 (UTC) 
}}
'''Support''' Article looks good enough to be on RD, well done. [[User:Fakescientist8000|Fakescientist8000]] ([[User talk:Fakescientist8000|talk]]) 14:39, 10 December 2021 (UTC)

==== (Closed) 2021 Pakistan Sialkot lynching ====
{{atop|Stale. [[User:Stephen|Step]][[User talk:Stephen|hen]] 23:29, 9 December 2021 (UTC)}}
{{ITN candidate
| article = Lynching of Priyantha Kumara
| image =
| blurb = Islamist mob '''[[Lynching of Priyantha Kumara|lynch Sri Lankan worker]]''' for alleged blasphemy.
| recent deaths = no
| ongoing = no
| ITNR = no
| altblurb = A mob in Pakistan '''[[Lynching of Priyantha Kumara|lynches a foreign national]]''' for alleged [[Islam and blasphemy|blasphemy]].
| altblurb2 =
| sources = ''[https://www.theguardian.com/world/2021/dec/03/pakistan-sri-lankan-man-priyantha-diyawadana-tortured-killed-alleged-blasphemy-sialkot The Guardian]''
| updated = yes
| nominator = 27.7.182.218 
| creator =
| updaters =
| nom cmt = Highlights the growing Islamic radicalisation of Pakistan
| sign = [[Special:Contributions/27.7.182.218|27.7.182.218]] ([[User talk:27.7.182.218|talk]]) 07:52, 8 December 2021 (UTC) 
}}

*'''Comment''' Not sure at this stage whether this deserves listing, but I am sure the Nominator's comment above are inflammatory and unhelpful. I would prefer to see wording more like "An example of what some see as incidents of violence over alleged blasphemy in Pakistan." [[User:HiLo48|HiLo48]] ([[User talk:HiLo48|talk]]) 08:20, 8 December 2021 (UTC)
*'''{{s|Support}}''' Article is unusually well-referenced and composed. The very large number of people arrested for this, and the quick reactions from the international community, indicate that this is not a garden variety criminal proceeding. That this is directly related to Islamism and blasphemy is well-supported in sources. Altblurb added to conform to our general format.[[Special:Contributions/130.233.213.55|130.233.213.55]] ([[User talk:130.233.213.55|talk]]) 08:38, 8 December 2021 (UTC)
*Has this already been determined by a court of law to be a murder or lynching? [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 09:00, 8 December 2021 (UTC)
::The Pakistani PM calls it plainly "the Sialkot lynching". I'm unsure if Pakistan has a statutory definition of lynching (like the US), but it's the most concise and descriptive term available. Alternative could be "kills" but sources on all sides have gone beyond that.[[Special:Contributions/130.233.213.55|130.233.213.55]] ([[User talk:130.233.213.55|talk]]) 09:25, 8 December 2021 (UTC)
:::The PM is entitled to his opinion, but that is not a judgment by a court. (even just as a murder). [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 11:19, 8 December 2021 (UTC)
*'''Support''' Altblurb. Per 130.233.213.55, this is significant. [[User:Canadianerk|Canadianerk]] ([[User talk:Canadianerk|talk]]) 10:59, 8 December 2021 (UTC)
*'''Note''': Event occurred on December 3. The oldest current blurb is dated December 4. Story is stale. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 13:01, 8 December 2021 (UTC)
*:@[[User:Jayron32|Jayron32]] This is a major news. Was widely published in all countries. Please consider making an exception to post this. [[User:Venkat TL|Venkat TL]] ([[User talk:Venkat TL|talk]]) 13:03, 8 December 2021 (UTC)
*::We post stories chronologically. There's no where to put this story in the ITN box. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 13:07, 8 December 2021 (UTC)
*:::Understood. But [[WP:IAR]] should apply here. There is no date mentioned in the box anyway. So adding it will not break anything. Could be added on top or bottom. [[User:Venkat TL|Venkat TL]] ([[User talk:Venkat TL|talk]]) 13:15, 8 December 2021 (UTC)
*::::We simply don't do that. There could be arguments if this story had taken a while to propagate to major sources, but the Guardian piece is dated Dec 3, the day it happened. This simply missed the ITN window. --[[User:Masem|Masem]] ([[User Talk:Masem|t]]) 13:24, 8 December 2021 (UTC)
*::::{{ec}} You've asserted that IAR should apply. You've provided no particular evidence in support that it should. There are many widely published news stories. You've made no case that ''this one story'' is important enough to post even though you waited 5 extra days to nominate it. --[[User:Jayron32|Jayron]][[User talk:Jayron32|''32'']] 13:24, 8 December 2021 (UTC)
* '''Oppose''' – A mob riot that lacks general significance or wider impact. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:03, 8 December 2021 (UTC)
* '''Stale''' Significance is debatable, but not even in the ballpark of significance that would justify invoking IAR. Would also oppose as tabloid fodder; religious violence is somewhat common in the subcontinent. Inflating one story or another is inflammatory. ''[[User_talk:GreatCaesarsGhost|GreatCaesarsGhost]]'' 13:31, 8 December 2021 (UTC)
*'''Comment''' Support struck per stale, but just barely. An outside possibility might be to shorten either the Austrian or Nagaland blurbs to free up one line at ITN for this item, but I can't recall such happening before.[[Special:Contributions/130.233.213.55|130.233.213.55]] ([[User talk:130.233.213.55|talk]]) 13:35, 8 December 2021 (UTC)
{{abot}}

====(Posted) RD: Françoise Delord ====
{{ITN candidate
| article = Françoise Delord
| recent deaths = yes
| sources = [https://www.lepoint.fr/societe/la-fondatrice-du-zoo-de-beauval-francoise-delord-est-morte-03-12-2021-2455220_23.php Le Point]
| updated = Yes
| nominator = TJMSmith 
| creator = Jmanlucas
| updaters = KittenKlub
| nom cmt = French zookeeper and owner. Concise but well-sourced article. Hopefully it can be expanded a little more.
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 18:56, 4 December 2021 (UTC)
}}
*'''Support''' - Start-class and fully sourced.--[[User:Kacamata|Kacamata!]] [[User_talk:Kacamata|Dimmi!!!]] 21:49, 4 December 2021 (UTC)
* DYK check says there are only 1414 characters (235 words) of readable prose. Any more to add? This is a brand-new wikiarticle. Might as well expand it to 1500+ characters and qualify for a DYK nomination, too. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 00:01, 5 December 2021 (UTC)
*'''Comment:''' Needs 1 more fully fleshed out paragraph to meet ITN 3 paragraph minimum. '''[[User:Spencer|Spencer]]'''[[User talk:Spencer|T•]][[Special:Contributions/Spencer|C]] 02:49, 5 December 2021 (UTC)
::I added a few details with citations. I'm somewhat limited due to most sources being in French. [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 23:51, 5 December 2021 (UTC)
*'''Support'''. I’ve added some details to indicate the significance of her zoo. Seems to me it’s long enough now but just one editor’s opinion! [[User:Innisfree987|Innisfree987]] ([[User talk:Innisfree987|talk]]) 20:43, 6 December 2021 (UTC)
* With 2217 characters (379 words) of readable prose, this wikibio is long enough to not be considered a stub. There seems to be enough footnotes where they are expected. IMO, this wikibio is '''READY for RD'''. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 22:45, 6 December 2021 (UTC)
*'''Posted to RD'''. '''[[User:Spencer|Spencer]]'''[[User talk:Spencer|T•]][[Special:Contributions/Spencer|C]] 05:48, 7 December 2021 (UTC)

==== (Posted) RD: Charlotte Mailliard Shultz ====
{{ITN candidate
| article = Charlotte Mailliard Shultz
| recent deaths = yes
| sources = [https://www.sfchronicle.com/bayarea/article/Charlotte-Shultz-San-Francisco-s-longtime-16672961.php San Francisco Chronical]
| updated = 
| nominator = TJMSmith 
| creator = Plcs
| updaters = TJMSmith, Strattonsmith, Jkaharper
| nom cmt = American heiress, socialite, and philanthropist.
| sign = [[User:TJMSmith|TJMSmith]] ([[User talk:TJMSmith|talk]]) 01:07, 4 December 2021 (UTC)
}}
*'''Support''' Nicely done article. [[User:Fakescientist8000|Fakescientist8000]] ([[User talk:Fakescientist8000|talk]]) 13:06, 5 December 2021 (UTC)
*'''Posted'''. --[[User:PFHLai|PFHLai]] ([[User talk:PFHLai|talk]]) 14:20, 5 December 2021 (UTC)

==== RD: Antony Sher ====
{{ITN candidate
| article = Antony Sher
| recent deaths = yes
| sources = [https://www.bbc.co.uk/news/entertainment-arts-59520117] [https://theguardian.com/stage/2021/dec/03/antony-sher-celebrated-actor-on-stage-and-screen-dies-aged-72] [https://www.thetimes.co.uk/article/sir-antony-sher-obituary-actor-dies-of-cancer-q7q5nqjj9]
| updated = 
| nominator = Margueryllis 
| updaters = Margueryllis
| nom cmt = British actor and two time [[Laurence Olivier Award]] winner
| sign = [[User talk:Margueryllis|Margueryllis]] 16:50, 3 December 2021 (UTC) 
}}
*'''Comment''' needs lots more sourcing. [[User:Joseph2302|Joseph]][[User talk:Joseph2302|2302]] ([[User talk:Joseph2302|talk]]) 17:15, 3 December 2021 (UTC)
*'''Comment''' Joseph is being kind. Every item on Stage performances, Filmography and Awards and nominations has to be sourced (and current count is zero...) [[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 17:37, 3 December 2021 (UTC)

==== Laos-China Railway ====
{{ITN candidate
| article = Boten-Vientiane railway
| image = File:2021-12-03 China-Laos-Eisenbahn.jpg
| blurb = The [[Vientiane-Boten railway|Vientiane-Boten rail link]] opens as Laos' first [[High-speed rail in Laos|high-speed railway]] for passengers and freight services. 
| recent deaths = no 
| ongoing = no 
| ITNR = No 
| altblurb = The 1035 km [[Boten-Vientiane railway|Laos-China railway]], Laos' first [[high-speed rail in Laos|high-speed railway]], opens to passengers and freight services.
| altblurb2 = 
| sources = [https://www.vientianetimes.org.la/freeContent/FreeConten_FirstTrip_237_21.php Vientiane Times], [https://www.vientianetimes.org.la/freeContent/FreeConten_Religious_237_21.php Vientiane Times], [https://www.globaltimes.cn/page/202112/1240592.shtml Global Times], [https://news.cgtn.com/news/2021-12-03/China-Laos-high-speed-railway-goes-into-full-operation-15GviTLpCcU/index.html CGTN], [https://www.telesurenglish.net/news/Historic-China-Laos-Railway-Commences-Operations-20211203-0018.html teleSUR], [https://www.reuters.com/markets/deals/china-laos-open-6-billion-high-speed-rail-link-2021-12-03/ Reuters], [https://www.reuters.com/markets/deals/laos-gives-buddhist-blessings-its-new-high-speed-rail-line-2021-12-02/ Reuters], [https://www.france24.com/en/live-news/20211203-game-changer-laos-opens-chinese-built-railway-line France 24], [https://www.chinadaily.com.cn/a/202112/03/WS61a9d93ba310cdd39bc793ec.html China Daily],

| updated = no
| nominator = Viva Nicolás 
| creator = Pieceofmetalwork
| updaters = 
| nom cmt = This is the opening of the first high-speed railway in Laos. The railway included 167 tunnels and 301 bridges, which consisted over 900 km of its 1035 km length. Previously, the country only had 4 km of railway.
| sign = [[User:Viva Nicolás|Viva Nicolás]] ([[User talk:Viva Nicolás|talk]]) 11:53, 4 December 2021 (UTC) 
}}
*'''Oppose''' Very interesting article and development. Article is in good shape, however this is not high speed. "suitable for 160 km/h passenger" [[High-speed rail]] is defined as >= 250 km/h for new lines and >= 200 km/h for upgraded lines. [[User:KittenKlub|KittenKlub]] ([[User talk:KittenKlub|talk]]) 12:41, 4 December 2021 (UTC)
*Am I missing how this is ITNR? [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 13:05, 4 December 2021 (UTC)
*:Ditto. – [[User:Sca|Sca]] ([[User talk:Sca|talk]]) 13:22, 4 December 2021 (UTC)
*::Removed. [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 15:31, 4 December 2021 (UTC)
*''' Opposr per KittenKlub''' Even if it was high speed, I don't think we would nominate it unless it was the first on the continent or something.[[User:Scaramouche33|Scaramouche33]] ([[User talk:Scaramouche33|talk]]) 13:24, 4 December 2021 (UTC)
*'''Oppose''' per above. [[User:Alsoriano97|_-_Alsoriano97]] ([[User talk:Alsoriano97|talk]]) 16:34, 4 December 2021 (UTC)
*'''Support''' This is a significant link in China's [[Belt and Road Initiative]]. See [https://www.scmp.com/comment/opinion/article/3158184/how-chinas-belt-and-road-connecting-southeast-asia-political SCMP] for some analysis of its strategic significance. The current top blurb is days old and is a similar matter of South Asian development or lack of same. China is getting things done while India isn't. Either way, they are in the news. [[user:Andrew Davidson|Andrew]]🐉([[user talk:Andrew Davidson|talk]]) 23:53, 4 December 2021 (UTC)
*'''Oppose'''. This boils down to "railway opens". This isn't even technically a high speed railway. Countries spending money in other countries as diplomacy or part of a geopolitical struggle is common. [[User:331dot|331dot]] ([[User talk:331dot|talk]]) 13:57, 5 December 2021 (UTC)
*'''Comment''' - Laos' first high speed rail doesn't seem to be the key point of the news overall (especially when it should be considered [[higher-speed rail]] instead according to definition). This could be significant but the blurb needs to shift the focus. [[User:Sun8908|Sun8908]] [[User talk:Sun8908|Talk]] 18:24, 5 December 2021 (UTC)
*'''Oppose''' The broader impact of this beyond development aid/loan is not immediately discernible from the blurbs or the article. [[User:Gotitbro|Gotitbro]] ([[User talk:Gotitbro|talk]]) 07:01, 6 December 2021 (UTC)

==References==
Nominators often include links to external websites and other references in discussions on this page. It is usually best to provide such links using the [[WP:External Links#How to link|inline URL syntax]] <code><nowiki>[http://example.com]</nowiki></code> rather than using [[WP:CITE#How to format inline citations|<code><nowiki><ref></ref></nowiki></code> tags]], because that keeps all the relevant information in the same place as the nomination without having to jump to this section, and facilitates the archiving process.

For the times when <code><nowiki><ref></ref></nowiki></code> tags are being used, here are their contents:
{{Reflist}}{{NOINDEX}}
<noinclude>{{Main Page topics|state=collapsed}}
[[Category:Wikipedia In the news]]
[[Category:Main Page discussions]]
[[Category:Current events]]

Order topology

2021-10-27T12:40:00Z

IntegralPython: Nevermind I’m dumb

{{Distinguish|Order topology (functional analysis)}}

In [[mathematics]], an '''order topology''' is a certain [[topology]] that can be defined on any [[totally ordered set]]. It is a natural generalization of the topology of the [[real numbers]] to arbitrary totally ordered sets. 

If ''X'' is a totally ordered set, the '''order topology''' on ''X'' is generated by the [[subbase]] of "open rays"
:<math>\{ x \mid a < x\}</math>
:<math>\{x \mid x < b\}</math>
for all ''a, b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open [[interval (mathematics)|interval]]s
:<math>(a,b) = \{x \mid a < x < b\}</math>
together with the above rays form a [[base (topology)|base]] for the order topology. The open sets in ''X'' are the sets that are a [[union (set theory)|union]] of (possibly infinitely many) such open intervals and rays.

A [[topological space]] ''X'' is called '''orderable''' if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a [[completely normal space|completely normal]] [[Hausdorff space]].

The standard topologies on '''R''', '''Q''', '''Z''', and '''N''' are the order topologies.

== Induced order topology ==
If ''Y'' is a subset of ''X'', ''X'' a totally ordered set, then ''Y'' inherits a total order from ''X''. The set ''Y'' therefore has an order topology, the '''induced order topology'''. As a subset of ''X'', ''Y'' also has a [[subspace topology]]. The subspace topology is always at least as [[finer topology|fine]] as the induced order topology, but they are not in general the same.

For example, consider the subset ''Y'' = {–1} ∪ {1/''n''}''n''∈'''N''' in the [[rational numbers|rationals]]. Under the subspace topology, the singleton set {–1} is open in ''Y'', but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.

== An example of a subspace of a linearly ordered space whose topology is not an order topology ==
Though the subspace topology of ''Y'' = {–1} ∪ {1/''n''}''n''∈'''N''' in the section above is shown to be not generated by the induced order on ''Y'', it is nonetheless an order topology on ''Y''; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in ''Y'' for every y in ''Y''), so the subspace topology is the [[discrete topology]] on ''Y'' (the topology in which every subset of ''Y'' is an open set), and the discrete topology on any set is an order topology. To define a total order on ''Y'' that generates the discrete topology on ''Y'', simply modify the induced order on ''Y'' by defining -1 to be the greatest element of ''Y'' and otherwise keeping the same order for the other points, so that in this new order (call it say ''<''1) we have 1/''n'' ''<''1 –1 for all ''n'' ∈ '''N'''. Then, in the order topology on ''Y'' generated by ''<''1, every point of ''Y'' is isolated in ''Y''.

We wish to define here a subset ''Z'' of a linearly ordered topological space ''X'' such that no total order on ''Z'' generates the subspace topology on ''Z'', so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.

Let <math>Z = \{-1\}\cup (0,1) </math> in the real line. The same argument as before shows that the subspace topology on Z is not equal to the induced order topology on Z, but one can show that the subspace topology on Z cannot be equal to any order topology on Z.

An argument follows. Suppose by way of contradiction that there is some [[Totally ordered set#Strict total order|strict total order]] < on Z such that the order topology generated by < is equal to the subspace topology on Z (note that we are not assuming that < is the induced order on Z, but rather an arbitrarily given total order on Z that generates the subspace topology). In the following, interval notation should be interpreted relative to the < relation. Also, if ''A'' and ''B'' are sets, <math> A shall mean that <math> a for each ''a'' in ''A'' and ''b'' in ''B''.

Let ''M'' = ''Z'' \ {-1}, the unit interval. ''M'' is connected. If ''m'', ''n'' ∈ ''M'' and ''m'' < -1 < ''n'', then <math>(-\infty, -1)</math> and <math>(-1, \infty)</math> separate ''M'', a contradiction. Thus, ''M'' < {-1} or {-1} < ''M''. Assume without loss of generality that {-1} < ''M''. Since {-1} is open in ''Z'', there is some point ''p'' in ''M'' such that the interval (-1, ''p'') is empty. Since {-1} < ''M'', we know -1 is the only element of ''Z'' that is less than ''p'', so ''p'' is the minimum of ''M''. Then ''M'' \ {''p''} = ''A'' ∪ ''B'', where ''A'' and ''B'' are nonempty open and disjoint connected subsets of ''M'' (removing a point from an open interval yields two open intervals). By connectedness, no point of ''Z''\''B'' can lie between two points of ''B'', and no point of ''Z''\''A'' can lie between two points of A. Therefore, either ''A'' < ''B'' or ''B'' < ''A''. Assume without loss of generality that ''A'' < ''B''. If ''a'' is any point in ''A'', then ''p'' < ''a'' and (''p'',''a'')<math>\subseteq</math> ''A''. Then (-1,''a'')=[''p'',''a''), so [''p'',''a'') is open. {''p''}∪''A''=[''p'',''a'')∪''A'', so {''p''}∪''A'' is an open subset of ''M'' and hence ''M'' = ({''p''}∪''A'') ∪ ''B'' is the union of two disjoint open subsets of ''M'' so ''M'' is not connected, a contradiction.

==Left and right order topologies==
Several variants of the order topology can be given:

* The '''right order topology''' on ''X'' is the topology whose open sets consist of intervals of the form (''a'', ∞) (including (-∞, ∞)).<ref>Steen, [https://books.google.com/books?id=DkEuGkOtSrUC&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false p. 74].</ref>
* The '''left order topology''' on ''X'' is the topology whose open sets consist of intervals of the form (−∞, ''b'') (including (-∞, ∞)).

The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a [[compact space]] that is not Hausdorff.

The left order topology is the standard topology used for many [[set-theoretic]] purposes on a [[Boolean algebra (structure)|Boolean algebra]].{{Clarify|Boolean algebras are not totally ordered|date=April 2021}}

==Ordinal space==

For any [[ordinal number]] ''λ'' one can consider the spaces of ordinal numbers
:<math>[0,\lambda) = \{\alpha \mid \alpha < \lambda\}</math>
:<math>[0,\lambda] = \{\alpha \mid \alpha \le \lambda\}</math>
together with the natural order topology. These spaces are called '''ordinal spaces'''. (Note that in the usual set-theoretic construction of ordinal numbers we have ''λ'' = [0,''λ'') and ''λ'' + 1 = [0,''λ'']). Obviously, these spaces are mostly of interest when ''λ'' is an infinite ordinal; otherwise (for finite ordinals), the order topology is simply the [[discrete topology]].

When ''λ'' = ω (the first infinite ordinal), the space [0,ω) is just '''N''' with the usual (still discrete) topology, while [0,ω] is the [[Alexandroff_extension|one-point compactification]] of '''N'''.

Of particular interest is the case when ''λ'' = ω1, the set of all countable ordinals, and the [[first uncountable ordinal]]. The element ω1 is a [[limit point]] of the subset [0,ω1) even though no [[sequence (mathematics)|sequence]] of elements in [0,ω1) has the element ω1 as its limit. In particular, [0,ω1] is not [[First-countable space|first-countable]]. The subspace [0,ω1) is first-countable however, since the only point in [0,ω1] without a countable [[local base]] is ω1. Some further properties include
*neither [0,ω1) or [0,ω1] is [[separable space|separable]] or [[second-countable]]
*[0,ω1] is [[compact space|compact]], while [0,ω1) is [[Sequentially compact space|sequentially compact]] and [[Countably compact space|countably compact]], but not compact or [[paracompact]]

== Topology and ordinals ==

=== Ordinals as topological spaces ===
Any [[ordinal number]] can be made into a [[topological space]] by endowing it with the order topology (since, being well-ordered, an ordinal is in particular [[total order|totally ordered]]): in the absence of indication to the contrary, it is always that order topology that is meant when an ordinal is thought of as a topological space. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.)

The set of [[limit point]]s of an ordinal ''α'' is precisely the set of [[limit ordinal]]s less than ''α''. Successor ordinals (and zero) less than ''α'' are [[isolated point]]s in ''α''. In particular, the finite ordinals and ω are [[discrete space|discrete]] topological spaces, and no ordinal beyond that is discrete. The ordinal ''α'' is [[compact space|compact]] as a topological space if and only if ''α'' is a [[successor ordinal]].

The closed sets of a limit ordinal ''α'' are just the closed sets in the sense that we have [[#Closed unbounded sets and classes|already defined]], namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.

Any ordinal is, of course, an open subset of any further ordinal. We can also define the topology on the ordinals in the following inductive way: 0 is the empty topological space, ''α''+1 is obtained by taking the [[Compactification (mathematics)|one-point compactification]] of ''α'', and for ''δ'' a limit ordinal, ''δ'' is equipped with the [[direct limit|inductive limit]] topology. Note that if ''α'' is a successor ordinal, then ''α'' is compact, in which case its one-point compactification ''α''+1 is the disjoint union of ''α'' and a point.

As topological spaces, all the ordinals are [[Hausdorff space|Hausdorff]] and even [[normal space|normal]]. They are also [[totally disconnected space|totally disconnected]] (connected components are points), [[scattered space|scattered]] (every non-empty subspace has an isolated point; in this case, just take the smallest element), [[zero-dimensional space|zero-dimensional]] (the topology has a [[clopen]] [[basis (topology)|basis]]: here, write an open interval (''β'',''γ'') as the union of the clopen intervals (''β'',''γ''<nowiki>'</nowiki>+1)=<nowiki>[</nowiki>''β''+1,''γ''<nowiki>']</nowiki> for ''γ''<nowiki>'</nowiki><''γ''). However, they are not [[extremally disconnected space|extremally disconnected]] in general (there are open sets, for example the even numbers from ω, whose closure is not open).

The topological spaces ω1 and its successor ω1+1 are frequently used as text-book examples of non-countable topological spaces.
For example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no sequence of elements in ω1 has the element ω1 as its limit: an element in ω1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.

The space ω1 is [[first-countable space|first-countable]], but not [[second-countable space|second-countable]], and ω1+1 has neither of these two properties, despite being [[compact space|compact]]. It is also worthy of note that any continuous function from ω1 to '''R''' (the [[real line]]) is eventually constant: so the [[Compactification (mathematics)|Stone–Čech compactification]] of ω1 is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much ''larger'' than ω).

=== Ordinal-indexed sequences ===
If ''α'' is a limit ordinal and ''X'' is a set, an ''α''-indexed sequence of elements of ''X'' merely means a function from ''α'' to ''X''. This concept, a '''transfinite sequence''' or '''ordinal-indexed sequence''', is a generalization of the concept of a [[sequence]]. An ordinary sequence corresponds to the case ''α'' = ω.

If ''X'' is a topological space, we say that an ''α''-indexed sequence of elements of ''X'' ''converges'' to a limit ''x'' when it converges as a [[net (mathematics)|net]], in other words, when given any neighborhood ''U'' of ''x'' there is an ordinal ''β''<''α'' such that ''x''''ι'' is in ''U'' for all ''ι''≥''β''.

Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 ([[Ordinal number#Initial ordinal of a cardinal|omega-one]], the set of all countable ordinal numbers, and the smallest uncountable ordinal number), is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.

However, ordinal-indexed sequences are not powerful enough to replace nets (or [[filter (mathematics)|filters]]) in general: for example, on the [[Tychonoff plank]] (the product space <math>(\omega_1+1)\times(\omega+1)</math>), the corner point <math>(\omega_1,\omega)</math> is a limit point (it is in the closure) of the open subset <math>\omega_1\times\omega</math>, but it is not the limit of an ordinal-indexed sequence.

== See also ==

* [[List of topologies]]
* [[Lower limit topology]]
* [[Long line (topology)]]
* [[Linear continuum]]
* [[Order topology (functional analysis)]]
* [[Partially ordered space]]

==Notes==
<references/>

== References ==
* [[Lynn Arthur Steen|Steen, Lynn A.]] and [[J. Arthur Seebach, Jr.|Seebach, J. Arthur Jr.]]; ''[[Counterexamples in Topology]]'', Holt, Rinehart and Winston (1970). {{isbn|0-03-079485-4}}.
* Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
* {{PlanetMath attribution|id=1411|title=Order topology}}

[[Category:Order theory]]
[[Category:General topology]]
[[Category:Ordinal numbers]]
[[Category:Topological spaces]]

Order topology

2021-10-27T12:37:16Z

IntegralPython: /* An example of a subspace of a linearly ordered space whose topology is not an order topology */Feels like this section should be a subsection of the more general one above

{{Distinguish|Order topology (functional analysis)}}

In [[mathematics]], an '''order topology''' is a certain [[topology]] that can be defined on any [[totally ordered set]]. It is a natural generalization of the topology of the [[real numbers]] to arbitrary totally ordered sets. 

If ''X'' is a totally ordered set, the '''order topology''' on ''X'' is generated by the [[subbase]] of "open rays"
:<math>\{ x \mid a < x\}</math>
:<math>\{x \mid x < b\}</math>
for all ''a, b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open [[interval (mathematics)|interval]]s
:<math>(a,b) = \{x \mid a < x < b\}</math>
together with the above rays form a [[base (topology)|base]] for the order topology. The open sets in ''X'' are the sets that are a [[union (set theory)|union]] of (possibly infinitely many) such open intervals and rays.

A [[topological space]] ''X'' is called '''orderable''' if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a [[completely normal space|completely normal]] [[Hausdorff space]].

The standard topologies on '''R''', '''Q''', '''Z''', and '''N''' are the order topologies.

== Induced order topology ==
If ''Y'' is a subset of ''X'', ''X'' a totally ordered set, then ''Y'' inherits a total order from ''X''. The set ''Y'' therefore has an order topology, the '''induced order topology'''. As a subset of ''X'', ''Y'' also has a [[subspace topology]]. The subspace topology is always at least as [[finer topology|fine]] as the induced order topology, but they are not in general the same.

For example, consider the subset ''Y'' = {–1} ∪ {1/''n''}''n''∈'''N''' in the [[rational numbers|rationals]]. Under the subspace topology, the singleton set {–1} is open in ''Y'', but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.

=== An example of a subspace of a linearly ordered space whose topology is not an order topology ===
Though the subspace topology of ''Y'' = {–1} ∪ {1/''n''}''n''∈'''N''' in the section above is shown to be not generated by the induced order on ''Y'', it is nonetheless an order topology on ''Y''; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in ''Y'' for every y in ''Y''), so the subspace topology is the [[discrete topology]] on ''Y'' (the topology in which every subset of ''Y'' is an open set), and the discrete topology on any set is an order topology. To define a total order on ''Y'' that generates the discrete topology on ''Y'', simply modify the induced order on ''Y'' by defining -1 to be the greatest element of ''Y'' and otherwise keeping the same order for the other points, so that in this new order (call it say ''<''1) we have 1/''n'' ''<''1 –1 for all ''n'' ∈ '''N'''. Then, in the order topology on ''Y'' generated by ''<''1, every point of ''Y'' is isolated in ''Y''.

We wish to define here a subset ''Z'' of a linearly ordered topological space ''X'' such that no total order on ''Z'' generates the subspace topology on ''Z'', so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.

Let <math>Z = \{-1\}\cup (0,1) </math> in the real line. The same argument as before shows that the subspace topology on Z is not equal to the induced order topology on Z, but one can show that the subspace topology on Z cannot be equal to any order topology on Z.

An argument follows. Suppose by way of contradiction that there is some [[Totally ordered set#Strict total order|strict total order]] < on Z such that the order topology generated by < is equal to the subspace topology on Z (note that we are not assuming that < is the induced order on Z, but rather an arbitrarily given total order on Z that generates the subspace topology). In the following, interval notation should be interpreted relative to the < relation. Also, if ''A'' and ''B'' are sets, <math> A shall mean that <math> a for each ''a'' in ''A'' and ''b'' in ''B''.

Let ''M'' = ''Z'' \ {-1}, the unit interval. ''M'' is connected. If ''m'', ''n'' ∈ ''M'' and ''m'' < -1 < ''n'', then <math>(-\infty, -1)</math> and <math>(-1, \infty)</math> separate ''M'', a contradiction. Thus, ''M'' < {-1} or {-1} < ''M''. Assume without loss of generality that {-1} < ''M''. Since {-1} is open in ''Z'', there is some point ''p'' in ''M'' such that the interval (-1, ''p'') is empty. Since {-1} < ''M'', we know -1 is the only element of ''Z'' that is less than ''p'', so ''p'' is the minimum of ''M''. Then ''M'' \ {''p''} = ''A'' ∪ ''B'', where ''A'' and ''B'' are nonempty open and disjoint connected subsets of ''M'' (removing a point from an open interval yields two open intervals). By connectedness, no point of ''Z''\''B'' can lie between two points of ''B'', and no point of ''Z''\''A'' can lie between two points of A. Therefore, either ''A'' < ''B'' or ''B'' < ''A''. Assume without loss of generality that ''A'' < ''B''. If ''a'' is any point in ''A'', then ''p'' < ''a'' and (''p'',''a'')<math>\subseteq</math> ''A''. Then (-1,''a'')=[''p'',''a''), so [''p'',''a'') is open. {''p''}∪''A''=[''p'',''a'')∪''A'', so {''p''}∪''A'' is an open subset of ''M'' and hence ''M'' = ({''p''}∪''A'') ∪ ''B'' is the union of two disjoint open subsets of ''M'' so ''M'' is not connected, a contradiction.

==Left and right order topologies==
Several variants of the order topology can be given:

* The '''right order topology''' on ''X'' is the topology whose open sets consist of intervals of the form (''a'', ∞) (including (-∞, ∞)).<ref>Steen, [https://books.google.com/books?id=DkEuGkOtSrUC&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false p. 74].</ref>
* The '''left order topology''' on ''X'' is the topology whose open sets consist of intervals of the form (−∞, ''b'') (including (-∞, ∞)).

The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a [[compact space]] that is not Hausdorff.

The left order topology is the standard topology used for many [[set-theoretic]] purposes on a [[Boolean algebra (structure)|Boolean algebra]].{{Clarify|Boolean algebras are not totally ordered|date=April 2021}}

==Ordinal space==

For any [[ordinal number]] ''λ'' one can consider the spaces of ordinal numbers
:<math>[0,\lambda) = \{\alpha \mid \alpha < \lambda\}</math>
:<math>[0,\lambda] = \{\alpha \mid \alpha \le \lambda\}</math>
together with the natural order topology. These spaces are called '''ordinal spaces'''. (Note that in the usual set-theoretic construction of ordinal numbers we have ''λ'' = [0,''λ'') and ''λ'' + 1 = [0,''λ'']). Obviously, these spaces are mostly of interest when ''λ'' is an infinite ordinal; otherwise (for finite ordinals), the order topology is simply the [[discrete topology]].

When ''λ'' = ω (the first infinite ordinal), the space [0,ω) is just '''N''' with the usual (still discrete) topology, while [0,ω] is the [[Alexandroff_extension|one-point compactification]] of '''N'''.

Of particular interest is the case when ''λ'' = ω1, the set of all countable ordinals, and the [[first uncountable ordinal]]. The element ω1 is a [[limit point]] of the subset [0,ω1) even though no [[sequence (mathematics)|sequence]] of elements in [0,ω1) has the element ω1 as its limit. In particular, [0,ω1] is not [[First-countable space|first-countable]]. The subspace [0,ω1) is first-countable however, since the only point in [0,ω1] without a countable [[local base]] is ω1. Some further properties include
*neither [0,ω1) or [0,ω1] is [[separable space|separable]] or [[second-countable]]
*[0,ω1] is [[compact space|compact]], while [0,ω1) is [[Sequentially compact space|sequentially compact]] and [[Countably compact space|countably compact]], but not compact or [[paracompact]]

== Topology and ordinals ==

=== Ordinals as topological spaces ===
Any [[ordinal number]] can be made into a [[topological space]] by endowing it with the order topology (since, being well-ordered, an ordinal is in particular [[total order|totally ordered]]): in the absence of indication to the contrary, it is always that order topology that is meant when an ordinal is thought of as a topological space. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.)

The set of [[limit point]]s of an ordinal ''α'' is precisely the set of [[limit ordinal]]s less than ''α''. Successor ordinals (and zero) less than ''α'' are [[isolated point]]s in ''α''. In particular, the finite ordinals and ω are [[discrete space|discrete]] topological spaces, and no ordinal beyond that is discrete. The ordinal ''α'' is [[compact space|compact]] as a topological space if and only if ''α'' is a [[successor ordinal]].

The closed sets of a limit ordinal ''α'' are just the closed sets in the sense that we have [[#Closed unbounded sets and classes|already defined]], namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.

Any ordinal is, of course, an open subset of any further ordinal. We can also define the topology on the ordinals in the following inductive way: 0 is the empty topological space, ''α''+1 is obtained by taking the [[Compactification (mathematics)|one-point compactification]] of ''α'', and for ''δ'' a limit ordinal, ''δ'' is equipped with the [[direct limit|inductive limit]] topology. Note that if ''α'' is a successor ordinal, then ''α'' is compact, in which case its one-point compactification ''α''+1 is the disjoint union of ''α'' and a point.

As topological spaces, all the ordinals are [[Hausdorff space|Hausdorff]] and even [[normal space|normal]]. They are also [[totally disconnected space|totally disconnected]] (connected components are points), [[scattered space|scattered]] (every non-empty subspace has an isolated point; in this case, just take the smallest element), [[zero-dimensional space|zero-dimensional]] (the topology has a [[clopen]] [[basis (topology)|basis]]: here, write an open interval (''β'',''γ'') as the union of the clopen intervals (''β'',''γ''<nowiki>'</nowiki>+1)=<nowiki>[</nowiki>''β''+1,''γ''<nowiki>']</nowiki> for ''γ''<nowiki>'</nowiki><''γ''). However, they are not [[extremally disconnected space|extremally disconnected]] in general (there are open sets, for example the even numbers from ω, whose closure is not open).

The topological spaces ω1 and its successor ω1+1 are frequently used as text-book examples of non-countable topological spaces.
For example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no sequence of elements in ω1 has the element ω1 as its limit: an element in ω1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.

The space ω1 is [[first-countable space|first-countable]], but not [[second-countable space|second-countable]], and ω1+1 has neither of these two properties, despite being [[compact space|compact]]. It is also worthy of note that any continuous function from ω1 to '''R''' (the [[real line]]) is eventually constant: so the [[Compactification (mathematics)|Stone–Čech compactification]] of ω1 is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much ''larger'' than ω).

=== Ordinal-indexed sequences ===
If ''α'' is a limit ordinal and ''X'' is a set, an ''α''-indexed sequence of elements of ''X'' merely means a function from ''α'' to ''X''. This concept, a '''transfinite sequence''' or '''ordinal-indexed sequence''', is a generalization of the concept of a [[sequence]]. An ordinary sequence corresponds to the case ''α'' = ω.

If ''X'' is a topological space, we say that an ''α''-indexed sequence of elements of ''X'' ''converges'' to a limit ''x'' when it converges as a [[net (mathematics)|net]], in other words, when given any neighborhood ''U'' of ''x'' there is an ordinal ''β''<''α'' such that ''x''''ι'' is in ''U'' for all ''ι''≥''β''.

Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 ([[Ordinal number#Initial ordinal of a cardinal|omega-one]], the set of all countable ordinal numbers, and the smallest uncountable ordinal number), is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.

However, ordinal-indexed sequences are not powerful enough to replace nets (or [[filter (mathematics)|filters]]) in general: for example, on the [[Tychonoff plank]] (the product space <math>(\omega_1+1)\times(\omega+1)</math>), the corner point <math>(\omega_1,\omega)</math> is a limit point (it is in the closure) of the open subset <math>\omega_1\times\omega</math>, but it is not the limit of an ordinal-indexed sequence.

== See also ==

* [[List of topologies]]
* [[Lower limit topology]]
* [[Long line (topology)]]
* [[Linear continuum]]
* [[Order topology (functional analysis)]]
* [[Partially ordered space]]

==Notes==
<references/>

== References ==
* [[Lynn Arthur Steen|Steen, Lynn A.]] and [[J. Arthur Seebach, Jr.|Seebach, J. Arthur Jr.]]; ''[[Counterexamples in Topology]]'', Holt, Rinehart and Winston (1970). {{isbn|0-03-079485-4}}.
* Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
* {{PlanetMath attribution|id=1411|title=Order topology}}

[[Category:Order theory]]
[[Category:General topology]]
[[Category:Ordinal numbers]]
[[Category:Topological spaces]]

Africa

2021-09-14T15:21:30Z

IntegralPython: bluelink South Africa

{{Other uses}}
{{pp-move-indef}}
{{pp-semi-indef}}

{{Short description|Continent}}

{{Use dmy dates|date=August 2021}}
{{Use British English Oxford spelling|date=August 2016}}

{{Infobox Continent
|title = Africa
|image = {{Switcher|[[File:Africa (orthographic projection).svg|frameless]]|Show national borders|[[File:Africa (orthographic projection) blank.svg|frameless]]|Hide national borders|default=1}}
|area = {{convert|30370000|km2|sqmi|abbr=on}}  ([[List of continents by area|2nd]])
|population = {{UN_Population|Africa}}{{UN_Population|ref}} ({{UN_Population|Year}}; [[List of continents by population|2nd]])
|density = {{pop density|1100000000|30221532|km2|sqmi}}
|religions = {{unbulleted list
| [[Christianity in Africa|Christianity]] (51%)
| [[Islam in Africa|Islam]] (40%)
| [[African traditional religion|Traditional faiths]] (6%)
| [[Irreligion|No religion]] (2%)
| [[Religion in Africa|Others]] (1%)<ref>[https://www.gordonconwell.edu/blog/african-christianity-101/?__cf_chl_jschl_tk__=a1d4c1d931e6c38110d5c3f059ae64bb66bafafa-1590463375-0-AezHrWRbV9jUadPbMq1KCYOzXRnMTcuigdG5X7oahVbSoI1-HbOZFVzICpNQM3DD6h-V4OowV97KMQvA_Z5xrEIueURh3cAjh_JOwgzb_0xJ8ApebiYm1YKfWINm1tpYbvki0LdD6UCp1tdLlxQ9SRwtdKFDMRidCaiTEuKpAgqahxqDYDT9efnF_jaiIEUQu0uIx-pJ0jUDCQtArMqdHTN8eI_S59hxJlvlxrSqBFOFsKFbiRy66EYOzblYbhaniwzQPIxiovSOAM7Yj6fu-5jMYVAPJtBJplpKoRDBlTtl44pnDC6wJInEyJbLw46dPuXcViyFEB57ebEfmUnpcYoJDlysExw35Ay28x7nvUDx3aIEa6ZhJsxwn62dv-R57g Gordon Conwell Theological Seminary, African Christianity, 2020]</ref>}}
|GDP_PPP = {{nowrap|$6.84 trillion (2021 est; 4th)<ref>{{Cite web|url=https://www.imf.org/external/datamapper/PPPGDP@WEO/OEMDC/ADVEC/WEOWORLD|title=GDP PPP, current prices|publisher=International Monetary Fund|date=2021|access-date=16 January 2021|archive-date=22 January 2021|archive-url=https://web.archive.org/web/20210122001107/https://www.imf.org/external/datamapper/PPPGDP@WEO/OEMDC/ADVEC/WEOWORLD|url-status=live}}</ref>}}
|GDP_nominal = $2.49 trillion (2021 est; [[List of continents by GDP (nominal)|5th]])<ref>{{cite web|title=GDP Nominal, current prices|url=https://www.imf.org/external/datamapper/NGDPD@WEO/OEMDC/ADVEC/WEOWORLD|publisher=International Monetary Fund|date=2021|access-date=16 January 2021|archive-date=25 February 2017|archive-url=https://web.archive.org/web/20170225211431/https://www.imf.org/external/datamapper/NGDPD@WEO/OEMDC/ADVEC/WEOWORLD|url-status=live}}</ref>
|GDP_per_capita = $1,860 (2021 est; [[List of continents by GDP (nominal)#GDP per capita (nominal) by continents|6th]])<ref>{{Cite web|url=https://www.imf.org/external/datamapper/NGDPDPC@WEO/OEMDC/ADVEC/WEOWORLD|title=Nominal GDP per capita|publisher=International Monetary Fund|date=2021|access-date=16 January 2021|archive-date=11 January 2020|archive-url=https://web.archive.org/web/20200111084550/https://www.imf.org/external/datamapper/NGDPDPC@WEO/OEMDC/ADVEC/WEOWORLD|url-status=live}}</ref>
|demonym = [[List of ethnic groups of Africa|African]]
|countries = 54+2*+4** (*disputed) (**territories)
|list_countries = List of sovereign states and dependent territories in Africa
|dependencies = {{Collapsible list
|list_style = text-align:left;
|title = [[List of African dependencies#External territories|External]] (3)
| 1 = {{Flag|British Indian Ocean Territory}} | 2 = {{flag|French Southern and Antarctic Lands|name=French Southern Territories}} | 3 = {{flag|Saint Helena, Ascension and Tristan da Cunha}}
}}
{{Collapsible list
|list_style = text-align:left;
|title = [[List of African dependencies#Internal territories|Internal]] (9+1 disputed)
| 1 = {{flag|Azores}} | 2 = {{flag|Canary Islands}} | 3 = {{flag|Ceuta}} | 4 = {{flag|Madeira}} | 5 = {{flag|Mayotte|local}} | 6 = {{flag|Melilla}} | 7 = {{flagdeco|Spain}} [[Plazas de soberanía]] | 8 = {{flagdeco|South Africa}} [[Prince Edward Islands]] | 9 = {{flag|Réunion}} | 10 = ''{{flagdeco|Morocco}} [[Southern Provinces]]''
}}
|languages = [[Languages of Africa|1250–3000 native languages]]
|time = [[UTC-1]] to [[UTC+4]]
|cities = [[List of urban agglomerations in Africa|Largest urban areas]]:{{hlist
|item_style=white-space:break;
|[[Cairo]]
|[[Lagos]]
|[[Kinshasa]]
|[[Johannesburg]]
|[[Luanda]]
|[[Khartoum]]
|[[Dar es Salaam]]
|[[Abidjan]]
|[[Alexandria]]
|[[Kigali]]
|[[Nairobi]]
|[[Algiers]]}}
|[[Cape Town]]
|[[Kano (city)|Kano]]
|[[Dakar]]
|[[Casablanca]]
|[[Addis Ababa]]}}

'''Africa''' is the world's second-largest and second-most populous [[continent]], after Asia in both cases. At about 30.3 million km2 (11.7 million square miles) including adjacent islands, it covers 6% of Earth's total surface area and 20% of its land area.<ref name=Sayre>Sayre, April Pulley (1999), ''Africa'', Twenty-First Century Books. {{ISBN|0-7613-1367-2}}.</ref> With {{#expr:{{formatnum:{{UN_Population|Africa}}|R}}/1e9 round 1}} billion people{{UN_Population|ref}} as of {{UN_Population|Year}}, it accounts for about 16% of the world's [[human population]]. Africa's population is the youngest amongst all the continents;<ref>{{cite web|title=5 ways the world will look dramatically different in 2100|url=https://www.washingtonpost.com/news/wonk/wp/2015/08/17/5-ways-the-world-will-look-dramatically-different-in-2100/|last=Swanson|first=Ana|date=17 August 2015|work=[[The Washington Post]]|access-date=26 September 2017|archive-date=26 September 2017|archive-url=https://web.archive.org/web/20170926194109/https://www.washingtonpost.com/news/wonk/wp/2015/08/17/5-ways-the-world-will-look-dramatically-different-in-2100/|url-status=live}}</ref><ref>{{cite news|last=[[Njideka Harry|Harry]]|first=Njideka U.|date=11 September 2013|title=African Youth, Innovation and the Changing Society|work=Huffington Post|url=http://www.huffingtonpost.com/njideka-u-harry/african-youth-innovation-_b_3904408.html|access-date=27 September 2013|archive-date=20 September 2013|archive-url=https://web.archive.org/web/20130920184934/http://www.huffingtonpost.com/njideka-u-harry/african-youth-innovation-_b_3904408.html|url-status=live}}</ref> the [[median]] age in 2012 was 19.7, when the worldwide median age was 30.4.<ref>{{cite web|title=item,4 of the provisional agenda – General debate on national experience in population matters: adolescents and youth|url=https://www.un.org/esa/population/cpd/cpd2012/Agenda%20item%204/UN%20system%20statements/ECA_Item4.pdf|author=Janneh, Abdoulie|date=April 2012|work=United Nations Economic Commission for Africa|access-date=15 December 2015|archive-date=10 November 2013|archive-url=https://web.archive.org/web/20131110111359/http://www.un.org/esa/population/cpd/cpd2012/Agenda%20item%204/UN%20system%20statements/ECA_Item4.pdf|url-status=live}}</ref> Despite a wide range of [[natural resource]]s, Africa is the least wealthy continent per capita, in part due to geographic impediments,<ref name=":1" /> legacies of [[Scramble for Africa|European colonization in Africa]] and the [[Cold War]],<ref>{{Cite book|last=Fwatshak|first=S. U.|title=Contemporary Africa|publisher=Springer|year=2014|isbn=978-1-349-49413-2|pages=89–125|chapter=The Cold War and the Emergence of Economic Divergences: Africa and Asia Compared|doi=10.1057/9781137444134_5}}</ref><ref>{{Cite journal|last=Austin|first=Gareth|date=1 March 2010|title=African Economic Development, and Colonial Legacies|url=http://journals.openedition.org/poldev/78|journal=International Development Policy {{!}} Revue internationale de politique de développement|language=en|issue=1|pages=11–32|doi=10.4000/poldev.78|issn=1663-9375|doi-access=free|access-date=12 November 2020|archive-date=21 January 2021|archive-url=https://web.archive.org/web/20210121082142/https://journals.openedition.org/poldev/78|url-status=live}}</ref><ref>{{Cite journal|last=Dunning|first=Thad|date=2004|title=Conditioning the Effects of Aid: Cold War Politics, Donor Credibility, and Democracy in Africa|url=https://www.jstor.org/stable/3877863|journal=International Organization|volume=58|issue=2|pages=409–423|doi=10.1017/S0020818304582073|jstor=3877863|s2cid=154368924|issn=0020-8183|access-date=12 November 2020|archive-date=12 November 2020|archive-url=https://web.archive.org/web/20201112223528/https://www.jstor.org/stable/3877863|url-status=live}}</ref><ref>{{Cite journal|last=Alemazung|first=J.|date=2010|title=Post-Colonial Colonialism: An Analysis of International Factors and Actors Marring African Socio-Economic and Political Development|url=https://www.semanticscholar.org/paper/Post-Colonial-Colonialism%3A-An-Analysis-of-Factors-Alemazung/3c7b2be6a73e33f366c835ce06ff8a6dd6b2cbf3|access-date=12 November 2020|website=undefined|s2cid=140806396|language=en|archive-date=28 January 2021|archive-url=https://web.archive.org/web/20210128073443/https://www.semanticscholar.org/paper/Post-Colonial-Colonialism%3A-An-Analysis-of-Factors-Alemazung/3c7b2be6a73e33f366c835ce06ff8a6dd6b2cbf3|url-status=live}}</ref><ref>{{Cite journal|last=Bayeh|first=E.|date=2015|title=THE POLITICAL AND ECONOMIC LEGACY OF COLONIALISM IN THE POST-INDEPENDENCE AFRICAN STATES|url=https://www.semanticscholar.org/paper/THE-POLITICAL-AND-ECONOMIC-LEGACY-OF-COLONIALISM-IN-Bayeh/82148a1b7703edad0ca74394e236163eb17c82f5|access-date=12 November 2020|website=www.semanticscholar.org|s2cid=198939744|language=en|archive-date=21 January 2021|archive-url=https://web.archive.org/web/20210121000339/https://www.semanticscholar.org/paper/THE-POLITICAL-AND-ECONOMIC-LEGACY-OF-COLONIALISM-IN-Bayeh/82148a1b7703edad0ca74394e236163eb17c82f5|url-status=live}}</ref> predatory/neo-colonialistic activities by Western nations and China, and undemocratic rule and deleterious policies.<ref name=":1">{{Cite journal|last1=Collier|first1=Paul|last2=Gunning|first2=Jan Willem|date=1 August 1999|title=Why Has Africa Grown Slowly?|url=https://pubs.aeaweb.org/doi/10.1257/jep.13.3.3|journal=Journal of Economic Perspectives|language=en|volume=13|issue=3|pages=3–22|doi=10.1257/jep.13.3.3|issn=0895-3309|access-date=12 November 2020|archive-date=30 March 2021|archive-url=https://web.archive.org/web/20210330032528/https://www.aeaweb.org/articles?id=10.1257%2Fjep.13.3.3|url-status=live}}</ref> Despite this low concentration of wealth, recent economic expansion and the large and young population make Africa an important economic market in the broader global context.

The continent is surrounded by the [[Mediterranean Sea]] to the north, the [[Isthmus of Suez]] and the [[Red Sea]] to the northeast, the Indian Ocean to the southeast and the Atlantic Ocean to the west. The continent includes [[Madagascar]] and various [[archipelago]]s. It contains [[List of sovereign states and dependent territories in Africa|54]] [[diplomatic recognition|fully recognised]] [[sovereign state]]s ([[country|countries]]), eight [[territory|territories]] and two ''de facto'' independent [[List of states with limited recognition|states with limited or no recognition]]. [[Algeria]] is Africa's largest country by area, and [[Nigeria]] is its largest by population. African nations cooperate through the establishment of the [[African Union]], which is headquartered in [[Addis Ababa]].

Africa straddles the [[equator]] and the [[prime meridian]] making it the only continent in the world to be situated in all four [[Points of the compass|cardinal]] [[Hemispheres of Earth|hemispheres]]. It is the only continent to stretch from the northern [[temperate]] to southern temperate zones.<ref>{{cite web|title=Africa. General info|url=http://www.visualgeography.com/continents/africa.html|publisher=Visual Geography|url-status=dead|archive-url=https://web.archive.org/web/20110424072430/http://www.visualgeography.com/continents/africa.html|archive-date=24 April 2011|access-date=24 November 2007}}</ref> The majority of the continent and its countries are in the [[Northern Hemisphere]], with a substantial portion and number of countries in the [[Southern Hemisphere]]. Most of the continent lies in the tropics, except for a large part of [[Western Sahara]], [[Algeria]], [[Libya]] and [[Egypt]], the northern tip of [[Mauritania]], the entire territories of [[Morocco]], [[Ceuta]], [[Melilla]] and [[Tunisia]] which in turn are located above the [[tropic of Cancer]], in the [[temperate zone|northern temperate zone]]. In the other extreme of the continent, southern [[Namibia]], southern [[Botswana]], great parts of [[South Africa]], the entire territories of [[Lesotho]] and [[Eswatini]] and the southern tips of [[Mozambique]] and [[Madagascar]] are located below the [[tropic of Capricorn]], in the [[temperate zone|southern temperate zone]].

Africa is home to much biodiversity; it is the continent with the largest number of [[megafauna]] species, as it was least affected by the [[Quaternary extinction event#The Pleistocene or Ice Age extinction event|extinction of the Pleistocene megafauna]]. However, Africa also is [[Environmental issues in Africa|heavily affected by a wide range of environmental issues]], including desertification, deforestation, [[water scarcity]], and other issues. These entrenched environmental concerns are expected to worsen as [[Climate change in Africa|climate change impacts Africa]]. The UN [[Intergovernmental Panel on Climate Change]] has identified Africa as the continent most [[Climate change vulnerability|vulnerable to climate change]].<ref>{{cite book|author=Schneider, S.H.|url=http://www.ipcc.ch/publications_and_data/ar4/wg2/en/ch19s19-3-3.html|title=Chapter 19: Assessing Key Vulnerabilities and the Risk from Climate Change|publisher=Print version: CUP. This version: IPCC website|year=2007|isbn=978-0-521-88010-7|editor=Parry, M.L.|series=Climate change 2007: impacts, adaptation, and vulnerability: contribution of Working Group II to the fourth assessment report of the Intergovernmental Panel on Climate Change (IPCC)|location=Cambridge University Press (CUP): Cambridge, UK|contribution=19.3.3 Regional vulnerabilities|display-authors=etal|access-date=15 September 2011|display-editors=etal|archive-url=https://web.archive.org/web/20130312104158/http://www.ipcc.ch/publications_and_data/ar4/wg2/en/ch19s19-3-3.html|archive-date=12 March 2013|url-status=dead}}</ref><ref name=":10Africa">Niang, I., O.C. Ruppel, M.A. Abdrabo, A. Essel, C. Lennard, J. Padgham, and P. Urquhart, 2014: Africa. In: Climate Change 2014: Impacts, Adaptation, and Vulnerability. Part B: Regional Aspects. Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change [Barros, V.R., C.B. Field, D.J. Dokken et al. (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, pp. 1199–1265. https://www.ipcc.ch/site/assets/uploads/2018/02/WGIIAR5-Chap22_FINAL.pdf {{Webarchive|url=https://web.archive.org/web/20200619170833/https://www.ipcc.ch/site/assets/uploads/2018/02/WGIIAR5-Chap22_FINAL.pdf |date=19 June 2020 }}</ref>

The [[history of Africa]] is long, complex, and has often been under-appreciated by the global [[African historiography|historical community]].<ref>{{Cite news|date=1 July 2017|title=One of Africa's best kept secrets – its history|language=en-GB|work=BBC News|url=https://www.bbc.com/news/world-africa-40420910|access-date=29 July 2021}}</ref> Africa, particularly [[East Africa|Eastern Africa]], is widely accepted as the place of origin of humans and the [[Hominidae]] [[clade]] ([[great ape]]s). The earliest [[hominids]] and their ancestors have been dated to around 7 million years ago, including ''[[Sahelanthropus tchadensis]]'', ''[[Australopithecus africanus]]'', ''[[Australopithecus afarensis|A. afarensis]]'', ''[[Homo erectus]]'', ''[[Homo habilis|H. habilis]]'' and ''[[Homo ergaster|H. ergaster]]''— the earliest ''[[Homo sapiens]]'' (modern human) remains, found in [[Omo remains|Ethiopia]], [[Florisbad Skull|South Africa]], and [[Jebel Irhoud|Morocco]], date to circa 200,000, 259,000, and 300,000 years ago respectively, and ''Homo sapiens'' is believed to have originated in Africa around 350,000–260,000 years ago.<ref>{{Cite web|url=http://web.utah.edu/unews/releases/05/feb/homosapiens.html|archive-url=https://web.archive.org/web/20071024234234/http://web.utah.edu/unews/releases/05/feb/homosapiens.html|url-status=dead|title=Homo sapiens: University of Utah News Release: 16 February 2005|archive-date=24 October 2007}}</ref><ref name="Schlebusch2017">{{cite journal |doi=10.1126/science.aao6266 |pmid=28971970 |title=Southern African ancient genomes estimate modern human divergence to 350,000 to 260,000 years ago |journal=Science |volume=358 |issue=6363 |pages=652–655 |year=2017 |last1=Schlebusch |first1=Carina M |last2=Malmström |first2=Helena |last3=Günther |first3=Torsten |last4=Sjödin |first4=Per |last5=Coutinho |first5=Alexandra |last6=Edlund |first6=Hanna |last7=Munters |first7=Arielle R |last8=Vicente |first8=Mário |last9=Steyn |first9=Maryna |last10=Soodyall |first10=Himla |last11=Lombard |first11=Marlize |last12=Jakobsson |first12=Mattias |bibcode=2017Sci...358..652S |doi-access=free }}</ref><ref name="Guardian">{{cite news|url=https://www.theguardian.com/science/2017/jun/07/oldest-homo-sapiens-bones-ever-found-shake-foundations-of-the-human-story|title=Oldest ''Homo sapiens'' bones ever found shake foundations of the human story|last=Sample|first=Ian|work=The Guardian|date=7 June 2017|access-date=7 June 2017|archive-date=31 October 2019|archive-url=https://web.archive.org/web/20191031005024/https://www.theguardian.com/science/2017/jun/07/oldest-homo-sapiens-bones-ever-found-shake-foundations-of-the-human-story|url-status=live}}</ref><ref name="NYT-20190910">{{cite news |last=Zimmer |first=Carl |author-link=Carl Zimmer |title=Scientists Find the Skull of Humanity's Ancestor — on a Computer – By comparing fossils and CT scans, researchers say they have reconstructed the skull of the last common forebear of modern humans. |url=https://www.nytimes.com/2019/09/10/science/human-ancestor-skull-computer.html |date=10 September 2019 |work=[[The New York Times]] |access-date=10 September 2019 |archive-date=31 December 2019 |archive-url=https://web.archive.org/web/20191231125331/https://www.nytimes.com/2019/09/10/science/human-ancestor-skull-computer.html |url-status=live }}</ref><ref name="NAT-20190910">{{cite journal |last1=Mounier |first1=Aurélien |last2=Lahr |first2=Marta |title=Deciphering African late middle Pleistocene hominin diversity and the origin of our species |journal=[[Nature Communications]] |volume=10 |issue=1 |page=3406 |doi=10.1038/s41467-019-11213-w |pmid=31506422 |pmc=6736881 |year=2019 |bibcode=2019NatCo..10.3406M }}</ref>

Early human civilizations, such as [[Ancient Egypt]] and [[Carthage]] emerged in [[History of North Africa|North Africa]]. Following a subsequent long and complex history of civilizations, migration and trade, Africa hosts a large diversity of [[List of ethnic groups of Africa|ethnicities]], [[Culture of Africa|cultures]] and [[Languages of Africa|languages]]. The last 400 years have witnessed an increasing European influence on the continent. Starting in the 16th century, this was driven by trade, including the [[Slavery in Africa|Trans-Atlantic slave trade]], which created large [[African diaspora]] populations in the Americas. In the [[New Imperialism|late 19th century]], European countries [[Scramble for Africa|colonized almost all of Africa]], extracting resources from the continent and exploiting local communities; most present states in Africa emerged from a process of [[Decolonisation of Africa|decolonisation]] in the 20th century.

==Etymology==
[[File:Palazzo Ferreria statue 2.jpeg|thumb|upright|Statue representing Africa at [[Palazzo Ferreria]], in [[Valletta]], [[Malta]]]]
''[[Afri]]'' was a [[Latin]] name used to refer to the inhabitants of then-known northern Africa to the west of the [[Nile]] river, and in its widest sense referred to all lands south of the Mediterranean ([[Ancient Libya]]).<ref>{{cite encyclopedia|last1=Georges|first1=Karl Ernst|editor1-last=Georges|editor1-first=Heinrich|encyclopedia=Ausführliches lateinisch-deutsches Handwörterbuch|date=1913–1918|location=Hannover|edition=8th|url=http://latin_german.deacademic.com/1644|access-date=20 September 2015|language=de|title=Afri|archive-url=https://web.archive.org/web/20160116044500/http://latin_german.deacademic.com/1644|archive-date=16 January 2016|url-status=dead}}</ref><ref>{{cite encyclopedia|last1=Lewis|first1=Charlton T.|last2=Short|first2=Charles|encyclopedia=A Latin Dictionary|date=1879|publisher=Clarendon Press|location=Oxford|url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3DAfer|access-date=20 September 2015|title=Afer|archive-date=16 January 2016|archive-url=https://web.archive.org/web/20160116044500/http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3DAfer|url-status=live}}</ref> This name seems to have originally referred to a [[Ancient Libya|native Libyan]] tribe, an ancestor of modern [[Berbers]]; see [[Terence#Biography|Terence]] for discussion. The name had usually been connected with the [[Phoenician language|Phoenician]] word ''{{lang|phn|ʿafar}}'' meaning "dust",<ref>Venter & Neuland, ''NEPAD and the African Renaissance'' (2005), p. 16</ref> but a 1981 hypothesis<ref>{{cite web |url=http://michel-desfayes.org/namesofcountries.html |title=The Names of Countries |last=Desfayes |first=Michel |website=michel-desfayes.org |quote=Africa. From the name of an ancient tribe in Tunisia, the ''Afri'' (adjective: ''Afer''). The name is still extant today as ''Ifira'' and ''Ifri-n-Dellal'' in Greater Kabylia (Algeria). A Berber tribe was called ''Beni-Ifren'' in the Middle Ages and ''Ifurace'' was the name of a Tripolitan people in the 6th century. The name is from the Berber language ''ifri'' 'cave'. Troglodytism was frequent in northern Africa and still occurs today in southern Tunisia. Herodote wrote that the Garamantes, a North African people, used to live in caves. The Ancient Greek called ''troglodytēs'' an African people who lived in caves. ''Africa'' was coined by the Romans and {{'}}''Ifriqiyeh''{{'}} is the arabized Latin name. (Most details from Decret & Fantar, 1981). |date=25 January 2011 |access-date=9 April 2019 |archive-url=https://web.archive.org/web/20190627022921/http://michel-desfayes.org/namesofcountries.html |archive-date=27 June 2019 |url-status=dead }}</ref> has asserted that it stems from the [[Berber languages|Berber]] word ''ifri'' (plural ''ifran'') meaning "cave", in reference to cave dwellers.<ref name="Michell">{{Cite journal|jstor = 714549|title = The Berbers|journal = Journal of the Royal African Society|volume = 2|issue = 6|pages = 161–194|last1 = Babington Michell|first1 = Geo|year = 1903|doi = 10.1093/oxfordjournals.afraf.a093193|url = https://zenodo.org/record/1782363|access-date = 30 August 2020|archive-date = 30 December 2020|archive-url = https://web.archive.org/web/20201230012624/https://zenodo.org/record/1782363|url-status = live}}</ref> The same word<ref name="Michell" /> may be found in the name of the [[Banu Ifran]] from [[Algeria]] and [[Tripolitania]], a Berber tribe originally from [[Yafran]] (also known as ''Ifrane'') in northwestern Libya,<ref>[https://books.google.com/books?id=SLSzNfdcqfoC&pg=PA200 Edward Lipinski, ''Itineraria Phoenicia''] {{Webarchive|url=https://web.archive.org/web/20160116044459/https://books.google.com/books?id=SLSzNfdcqfoC&pg=PA200 |date=16 January 2016 }}, Peeters Publishers, 2004, p. 200. {{ISBN|90-429-1344-4}}</ref> as well as the city of [[Ifrane]] in Morocco.

Under [[Roman Empire|Roman]] rule, [[Carthage]] became the capital of the province it then named ''[[Africa Proconsularis]]'', following its defeat of the [[Ancient Carthage|Carthaginians]] in the [[Third Punic War]] in 146 BC, which also included the coastal part of modern [[Libya]].<ref>{{cite web |url=http://www.consultsos.com/pandora/africa.htm |title=Africa African Africanus Africus |publisher=Consultos.com |access-date=14 November 2006 |archive-date=29 January 2009 |archive-url=https://web.archive.org/web/20090129111458/http://www.consultsos.com/pandora/africa.htm |url-status=live }}</ref> The Latin suffix ''[[wikt:-ica#Latin|-ica]]'' can sometimes be used to denote a land (e.g., in ''[[Gallia Celtica|Celtica]]'' from ''[[Celts|Celtae]]'', as used by [[Julius Caesar]]). The later Muslim region of [[Ifriqiya]], following its conquest of the [[Byzantine Empire|Byzantine (Eastern Roman)]] Empire's ''[[Exarchate of Africa|Exarchatus Africae]]'', also preserved a form of the name.

According to the Romans, Africa lies to the west of Egypt, while "Asia" was used to refer to [[Anatolia]] and lands to the east. A definite line was drawn between the two continents by the geographer [[Ptolemy]] (85–165 AD), indicating [[Alexandria]] along the [[Prime Meridian]] and making the isthmus of Suez and the [[Red Sea]] the boundary between Asia and Africa. As Europeans came to understand the real extent of the continent, the idea of "Africa" expanded with their knowledge.

Other etymological hypotheses have been postulated for the ancient name "Africa":
* The 1st-century Jewish historian [[Flavius Josephus]] (''Ant. 1.15'') asserted that it was named for [[Epher]] ('calf'), grandson of [[Abraham]] according to Gen. 25:4, whose descendants, he claimed, had invaded Libya.
* [[Isidore of Seville]] in his 7th-century ''[[Etymologiae]]'' XIV.5.2. suggests "Africa comes from the [[Latin]] ''aprica'', meaning "sunny".
* Massey, in 1881, stated that Africa is derived from the Egyptian ''af-rui-ka'', meaning "to turn toward the opening of the Ka." The [[Egyptian soul#Ka|Ka]] is the energetic double of every person and the "opening of the Ka" refers to a womb or birthplace. Africa would be, for the Egyptians, "the birthplace."<ref>{{cite web|url=http://gerald-massey.org.uk/massey/cmc_nile_genesis.htm|title=Nile Genesis: the opus of Gerald Massey|publisher=Gerald-massey.org.uk|date=29 October 1907|access-date=18 May 2010|archive-url=https://web.archive.org/web/20100130200159/http://gerald-massey.org.uk/massey/cmc_nile_genesis.htm|archive-date=30 January 2010|url-status=dead}}</ref>
* Michèle Fruyt in 1976 proposed<ref>{{cite journal|author=Fruyt, M. |title= D'Africus ventus a Africa terrain |journal=Revue de Philologie|volume= 50|year= 1976|pages= 221–38}}</ref> linking the Latin word with ''africus'' "south wind", which would be of Umbrian origin and mean originally "rainy wind".
* Robert R. Stieglitz of [[Rutgers University]] in 1984 proposed: "The name Africa, derived from the Latin *Aphir-ic-a, is cognate to Hebrew [[Ophir]] ['rich']."<ref>{{cite journal|doi=10.2307/3209914|jstor=3209914|title=Long-Distance Seafaring in the Ancient Near East|journal=The Biblical Archaeologist|volume=47|issue=3|pages=134–142|year=1984|last1=Stieglitz|first1=Robert R.|s2cid=130072563}}</ref>
* [[Ibn Khallikan]] and some other historians claim that the name of Africa came from a [[Himyarite Kingdom|Himyarite]] king called Afrikin ibn Kais ibn Saifi also called "Afrikus son of Abraham" who subdued Ifriqiya.<ref>{{Cite book|url=https://books.google.com/books?id=3cdLAAAAcAAJ&pg=PA35|title=Kitab Wafayat Ala'yan. Ibn Khallikan's Biographical Dictionary Transl. by (Guillaume) B(aro)n Mac-Guckin de Slane|last=Hallikan|first='Abu-l-'Abbas Sams-al-din 'Ahmad ibn Muhammad Ibn|date=1842|publisher=Benjamin Duprat|language=en|access-date=30 July 2018|archive-date=24 September 2019|archive-url=https://web.archive.org/web/20190924231636/https://books.google.com/books?id=3cdLAAAAcAAJ&pg=PA35|url-status=live}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=mcN7AAAAQBAJ&pg=PA38|title=Science in the Medieval World|last=al-Andalusi|first=Sa'id|year=2010|publisher=University of Texas Press|isbn=9780292792319|language=en|access-date=30 July 2018|archive-date=24 September 2019|archive-url=https://web.archive.org/web/20190924231632/https://books.google.com/books?id=mcN7AAAAQBAJ&pg=PA38|url-status=live}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=pbo5AQAAMAAJ&pg=PA113|title=Travels in the Arabian Desert: With Special Reference to the Arabian Horse and Its Pedigree|last=Upton|first=Roger D.|date=1881|publisher=C.K. Paul & Company|language=en|access-date=30 July 2018|archive-date=24 September 2019|archive-url=https://web.archive.org/web/20190924231639/https://books.google.com/books?id=pbo5AQAAMAAJ&pg=PA113|url-status=live}}</ref>
* Arabic ''afrīqā'' (feminine noun) and ''ifrīqiyā'', now usually pronounced ''afrīqiyā'' (feminine) ‘Africa’, from ''‘afara'' [‘ = ''‘ain'', not ''’alif''] ’to be dusty’ from ''‘afar'' ‘dust, powder’ and ''‘afir'' ‘dried, dried up by the sun, withered’ and ''‘affara'' ‘to dry in the sun on hot sand’ or ‘to sprinkle with dust’.<ref>Modified from Wilhelm Sturmfels and Heinz Bischof: ''Unsere Ortsnamen im ABC erklärt nach Herkunft und Bedeutung'', Bonn, 1961, Ferdinand Dümmlers Verlag.</ref>
* Possibly Phoenician ''faraqa'' in the sense of ‘colony, separation’.<ref>Serge Losique: ''Dictionnaire étymologique des noms de pays et de peuples'', Paris, 1971, Éditions Klincksieck.</ref>

==History==
{{Main|History of Africa}}
{{Further|History of North Africa|History of West Africa|History of Central Africa|History of East Africa|History of Southern Africa}}

===Prehistory===
{{Main|Recent African origin of modern humans}}
[[File:Lucy blackbg.jpg|thumb|upright=0.7|[[Lucy (Australopithecus)|Lucy]], an ''[[Australopithecus afarensis]]'' skeleton discovered 24 November 1974 in the [[Awash Valley]] of [[Ethiopia]]'s [[Afar Depression]] ]]

Africa is considered by most [[paleoanthropology|paleoanthropologists]] to be the [[cradle of Humankind|oldest inhabited territory]] on Earth, with the Human species originating from the continent.<ref name="HerreraGarcia-Bertrand2018">{{cite book|author1=Rene J. Herrera|author2=Ralph Garcia-Bertrand|title=Ancestral DNA, Human Origins, and Migrations|url=https://books.google.com/books?id=ZF1gDwAAQBAJ&pg=PA61|year=2018|publisher=Elsevier Science|isbn=978-0-12-804128-4|pages=61–|access-date=18 October 2020|archive-date=30 March 2021|archive-url=https://web.archive.org/web/20210330032459/https://books.google.com/books?id=ZF1gDwAAQBAJ&pg=PA61|url-status=live}}</ref> During the mid-20th century, [[Anthropology|anthropologists]] discovered many [[fossil]]s and evidence of human occupation perhaps as early as 7 million years ago (BP=before present). Fossil remains of several species of early apelike humans thought to have [[Evolution|evolved]] into modern man, such as ''[[Australopithecus afarensis]]'' ([[Radiometric dating|radiometrically dated]] to approximately 3.9–3.0 million years BP,<ref>Kimbel, William H. and Yoel Rak and Donald C. Johanson. (2004) ''The Skull of Australopithecus Afarensis'', Oxford University Press US. {{ISBN|0-19-515706-0}}</ref> ''[[Paranthropus boisei]]'' (c. 2.3–1.4 million years BP)<ref>Tudge, Colin. (2002) ''The Variety of Life.'', Oxford University Press. {{ISBN|0-19-860426-2}}</ref> and ''[[Homo ergaster]]'' (c. 1.9 million–600,000 years BP) have been discovered.<ref name=Sayre/>

After the evolution of ''[[Homo sapiens]]'' approximately 350,000 to 260,000 years BP in Africa,<ref name="Schlebusch2017"/><ref name="Guardian"/><ref name="NYT-20190910"/><ref name="NAT-20190910"/> the continent was mainly populated by groups of [[hunter-gatherer]]s.<ref>[[Ivan van Sertima|van Sertima, Ivan]]. (1995) ''Egypt: Child of Africa/S V12 (Ppr)'', Transaction Publishers. pp. 324–25. {{ISBN|1-56000-792-3}}</ref><ref>Mokhtar, G. (1990) ''UNESCO [[General History of Africa]], Vol. II, Abridged Edition: Ancient Africa'', University of California Press. {{ISBN|0-85255-092-8}}</ref><ref>Eyma, A.K. and C.J. Bennett. (2003) ''Delts-Man in Yebu: Occasional Volume of the Egyptologists' Electronic Forum No. 1'', Universal Publishers. p. 210. {{ISBN|1-58112-564-X}}</ref> These first modern humans left Africa and populated the rest of the globe during the [[Recent African origin of modern humans|Out of Africa II]] migration dated to approximately 50,000 years BP, exiting the continent either across [[Bab-el-Mandeb]] over the [[Red Sea]],<ref>Wells, Spencer (December 2002) [http://news.nationalgeographic.com/news/2002/12/1212_021213_journeyofman.html The Journey of Man] {{Webarchive|url=https://web.archive.org/web/20110427020944/http://news.nationalgeographic.com/news/2002/12/1212_021213_journeyofman.html |date=27 April 2011 }}. ''National Geographic''</ref><ref>Oppenheimer, Stephen. [http://www.bradshawfoundation.com/journey/gates2.html The Gates of Grief] {{Webarchive|url=https://web.archive.org/web/20140530001241/http://www.bradshawfoundation.com/journey/gates2.html |date=30 May 2014 }}. bradshawfoundation.com</ref> the [[Strait of Gibraltar]] in Morocco,<ref>{{Cite web|title=15. Strait of Gibraltar, Atlantic Ocean/Mediterranean Sea|url=https://www.lpi.usra.edu/publications/slidesets/humanimprints/slide_15.html|website=www.lpi.usra.edu|access-date=13 May 2020|archive-date=26 January 2021|archive-url=https://web.archive.org/web/20210126205023/https://www.lpi.usra.edu/publications/slidesets/humanimprints/slide_15.html|url-status=live}}</ref><ref>{{Cite journal|last1=Fregel|first1=Rosa|last2=Méndez|first2=Fernando L.|last3=Bokbot|first3=Youssef|last4=Martín-Socas|first4=Dimas|last5=Camalich-Massieu|first5=María D.|last6=Santana|first6=Jonathan|last7=Morales|first7=Jacob|last8=Ávila-Arcos|first8=María C.|last9=Underhill|first9=Peter A.|last10=Shapiro|first10=Beth|last11=Wojcik|first11=Genevieve|date=26 June 2018|title=Ancient genomes from North Africa evidence prehistoric migrations to the Maghreb from both the Levant and Europe|journal=Proceedings of the National Academy of Sciences|language=en|volume=115|issue=26|pages=6774–6779|doi=10.1073/pnas.1800851115|issn=0027-8424|pmid=29895688|pmc=6042094|doi-access=free}}</ref> or the [[Isthmus of Suez]] in Egypt.<ref>{{cite journal|url=http://www.ffzg.unizg.hr/arheo/ska/tekstovi/out_of_africa.pdf|doi=10.1007/s10963-006-9002-z|title=Getting "Out of Africa": Sea Crossings, Land Crossings and Culture in the Hominin Migrations|journal=Journal of World Prehistory|volume=19|issue=2|pages=119–132|year=2005|last1=Derricourt|first1=Robin|s2cid=28059849|access-date=26 December 2013|archive-date=22 February 2012|archive-url=https://web.archive.org/web/20120222031934/http://www.ffzg.unizg.hr/arheo/ska/tekstovi/out_of_africa.pdf|url-status=live}}</ref>

Other migrations of modern humans within the African continent have been dated to that time, with evidence of early human settlement found in Southern Africa, Southeast Africa, North Africa, and the [[Sahara]].<ref>{{cite book|author1=Goucher, Candice|author2=Walton, Linda|title=World History: Journeys from Past to Present|url=https://books.google.com/books?id=gY7cAAAAQBAJ|year=2013|publisher=Routledge|isbn=978-1-134-72354-6|pages=2–20|access-date=5 February 2018|archive-date=11 June 2020|archive-url=https://web.archive.org/web/20200611044204/https://books.google.com/books?id=gY7cAAAAQBAJ|url-status=live}}</ref>

=== Emergence of civilization ===
{{see|Cradle of civilization#Ancient Egypt}}
The size of the Sahara has historically been extremely variable, with its area rapidly fluctuating and at times disappearing depending on global climatic conditions.<ref>{{cite book|author=Keenan, Jeremy|title=The Sahara: Past, Present and Future|url=https://books.google.com/books?id=KUKPAQAAQBAJ|year=2013|publisher=Routledge|isbn=978-1-317-97001-9|access-date=5 February 2018|archive-date=28 February 2017|archive-url=https://web.archive.org/web/20170228175639/https://books.google.com/books?id=KUKPAQAAQBAJ|url-status=live}}</ref> At the end of the [[Ice age]]s, estimated to have been around 10,500 BC, the Sahara had again become a green fertile valley, and its African populations returned from the interior and coastal highlands in [[Sub-Saharan Africa]], with [[Saharan rock art|rock art paintings]] depicting a fertile Sahara and large populations discovered in [[Tassili n'Ajjer]] dating back perhaps 10 millennia.<ref>{{cite journal|last1=Mercier|first1=Norbert|display-authors=etal|date=2012|title=OSL dating of quaternary deposits associated with the parietal art of the Tassili-n-Ajjer plateau (Central Sahara)|journal=Quaternary Geochronology|volume=10|pages=367–73|doi=10.1016/j.quageo.2011.11.010}}</ref> However, the warming and drying climate meant that by 5000 BC, the Sahara region was becoming increasingly dry and hostile. Around 3500 BC, due to a tilt in the earth's orbit, the Sahara experienced a period of rapid desertification.<ref>[https://www.sciencedaily.com/releases/1999/07/990712080500.htm "Sahara's Abrupt Desertification Started by Changes in Earth's Orbit, Accelerated by Atmospheric and Vegetation Feedbacks"] {{webarchive|url=https://web.archive.org/web/20140307060153/https://www.sciencedaily.com/releases/1999/07/990712080500.htm|date=7 March 2014 }}, ''Science Daily''</ref> The population trekked out of the Sahara region towards the Nile Valley below the [[Cataracts of the Nile|Second Cataract]] where they made permanent or semi-permanent settlements. A major climatic recession occurred, lessening the heavy and persistent rains in Central and [[East Africa|Eastern Africa]]. Since this time, dry conditions have prevailed in Eastern Africa and, increasingly during the last 200 years, in [[Ethiopia]].

The domestication of cattle in Africa preceded agriculture and seems to have existed alongside hunter-gatherer cultures. It is speculated that by 6000 BC, cattle were domesticated in North Africa.<ref>Diamond, Jared. (1999) ''Guns, Germs and Steel: The Fates of Human Societies''. New York: Norton, p. 167. {{ISBN|978-0813498027}}</ref> In the Sahara-Nile complex, people domesticated many animals, including the donkey and a small screw-horned goat which was common from [[Algeria]] to [[Nubia]].

Between the 10,000–9,000 BC, pottery was independently invented in the region of Mali in the savannah of West Africa.<ref name="Pottery">{{cite journal |last1=Jesse |first1=Friederike |title=Early Pottery in Northern Africa – An Overview |issue=2 |pages=219–238 |journal=[[Journal of African Archaeology]]|volume=8 |jstor=43135518 |year=2010 |doi=10.3213/1612-1651-10171 }}</ref><ref name=swissinfo>[http://www.swissinfo.ch/eng/Home/Archive/Swiss_archaeologist_digs_up_West_Africas_past.html?cid=5675736 Simon Bradley, ''A Swiss-led team of archaeologists has discovered pieces of the oldest African pottery in central Mali, dating back to at least 9,400BC''] {{webarchive|url=https://web.archive.org/web/20120306002155/http://www.swissinfo.ch/eng/Home/Archive/Swiss_archaeologist_digs_up_West_Africas_past.html?cid=5675736 |date=6 March 2012 }}, SWI swissinfo.ch – the international service of the Swiss Broadcasting Corporation (SBC), 18 January 2007</ref>
[[File:Mathendous giraffes.jpg|thumb|[[Saharan rock art]] in the [[Fezzan]]]]
In the [[steppe]]s and [[savanna]]hs of the [[Sahara]] and [[Sahel]] in Northern West Africa, the [[Nilo-Saharan languages|Nilo-Saharan speakers]] and [[Mandé peoples]] started to collect and domesticate wild millet, [[African rice]] and [[sorghum]] between 8,000 and 6,000 BC. Later, [[gourd]]s, [[watermelon]]s, [[castor bean]]s, and cotton were also collected and domesticated.<ref>Ehret (2002), pp. 64–75.</ref> They also started making [[pottery]] and built stone settlements (e.g., [[Tichitt]], [[Oualata]]). Fishing, using bone-tipped [[harpoon]]s, became a major activity in the numerous streams and lakes formed from the increased rains.<ref>{{Cite web|url=http://humanorigins.si.edu/evidence/behavior/getting-food/katanda-bone-harpoon-point|title=Katanda Bone Harpoon Point|date=22 January 2010|website=The Smithsonian Institution's Human Origins Program|language=en|access-date=19 February 2019|archive-date=14 August 2020|archive-url=https://web.archive.org/web/20200814055506/https://humanorigins.si.edu/evidence/behavior/getting-food/katanda-bone-harpoon-point|url-status=live}}</ref> Mande peoples have been credited with the independent development of agriculture by about 3,000–4,000 BC.<ref>{{Cite web | url=https://www.britannica.com/topic/Mande | title=Mande | people | access-date=22 August 2020 | archive-date=16 August 2020 | archive-url=https://web.archive.org/web/20200816032310/https://www.britannica.com/topic/Mande | url-status=live }}</ref> In West Africa, the wet phase ushered in an expanding [[rainforest]] and wooded savanna from [[Senegal]] to [[Cameroon]]. Between 9,000 and 5,000 BC, [[Niger–Congo languages|Niger–Congo speakers]] domesticated the [[Elaeis guineensis|oil palm]] and [[raffia palm]]. [[Black-eyed pea]]s and [[voandzeia]] (African groundnuts), were domesticated, followed by [[okra]] and [[kola nut]]s. Since most of the plants grew in the forest, the Niger–Congo speakers invented polished stone axes for clearing forest.<ref>Ehret (2002), pp. 82–84.</ref>

Around 4000 BC, the Saharan climate started to become drier at an exceedingly fast pace.<ref name="O'Brien">O'Brien, Patrick K. ed. (2005) ''Oxford Atlas of World History''. New York: Oxford University Press. pp. 22–23. {{ISBN|9780199746538}}</ref> This climate change caused lakes and rivers to shrink significantly and caused increasing [[desertification]]. This, in turn, decreased the amount of land conducive to settlements and helped to cause migrations of farming communities to the more tropical climate of West Africa.<ref name="O'Brien"/>

By the first millennium BC, [[Ferrous metallurgy|ironworking]] had been introduced in Northern Africa. Around that time it also became established in parts of sub-Saharan Africa, either through independent invention there or diffusion from the north<ref>[http://princetonol.com/groups/iad/lessons/middle/history1.htm#Irontechnology Martin and O'Meara, "Africa, 3rd Ed."] {{webarchive|url=https://web.archive.org/web/20071011083356/http://princetonol.com/groups/iad/lessons/middle/history1.htm |date=11 October 2007 }} Indiana: Indiana University Press, 1995</ref><ref name="PB 2014">Breunig, Peter. 2014. Nok: African Sculpture in Archaeological Context: p. 21.</ref> and vanished under unknown circumstances around 500 AD, having lasted approximately 2,000 years.<ref name="FB 1969">Fagg, Bernard. 1969. Recent work in west Africa: New light on the Nok culture. World Archaeology 1(1): 41–50.</ref> and by 500 BC, metalworking began to become commonplace in West Africa. Ironworking was fully established by roughly 500 BC in many areas of East and West Africa, although other regions didn't begin ironworking until the early centuries AD. Copper objects from [[Egypt]], North Africa, Nubia, and Ethiopia dating from around 500 BC have been excavated in West Africa, suggesting that [[Trans-Saharan trade]] networks had been established by this date.<ref name="O'Brien"/>

===Early civilizations===
{{Main|Ancient African history}}
[[File:African-civilizations-map-pre-colonial.svg|thumb|upright=1.2|Diachronic map showing [[African empires]] spanning roughly 500 BCE to 1500 CE]]

At about 3300 BC, the historical record opens in Northern Africa with the rise of literacy in the [[Pharaoh|Pharaonic]] civilization of [[Ancient Egypt]].<ref>[http://news.bbc.co.uk/1/hi/sci/tech/235724.stm Were Egyptians the first scribes?] {{Webarchive|url=https://web.archive.org/web/20060626190345/http://news.bbc.co.uk/1/hi/sci/tech/235724.stm |date=26 June 2006 }} BBC News (15 December 1998)</ref> One of the world's earliest and longest-lasting civilizations, the Egyptian state continued, with varying levels of influence over other areas, until 343 BC.<ref>Hassan, Fekri A. (2002) ''Droughts, Food and Culture'', Springer. p. 17. {{ISBN|0-306-46755-0}}</ref><ref>McGrail, Sean. (2004) ''Boats of the World'', Oxford University Press. p. 48. {{ISBN|0-19-927186-0}}</ref> Egyptian influence reached deep into modern-day [[Ancient Libya|Libya]] and [[Nubia]], and, according to Martin Bernal, as far north as Crete.<ref>{{cite book|title=History in Black: African-Americans in Search of an Ancient Past|first1=Jacob|last1=Shavit|first2=Yaacov|last2=Shavit|publisher=Taylor & Francis|date=2001|isbn=978-0-7146-8216-7|url=https://books.google.com/books?id=VlNkzTO6IecC&pg=PA77|page=77|access-date=30 August 2020|archive-date=5 April 2015|archive-url=https://web.archive.org/web/20150405133055/http://books.google.com/books?id=VlNkzTO6IecC&pg=PA77|url-status=live}}</ref>

An independent centre of [[civilization]] with trading links to [[Phoenicia]] was established by [[Phoenicia]]ns from [[Tyre, Lebanon|Tyre]] on the north-west African coast at [[Carthage]].<ref>Fage, J.D. (1979), ''The Cambridge History of Africa'', Cambridge University Press. {{ISBN|0-521-21592-7}}</ref><ref>Fage, J.D., et al. (1986), [https://www.sahistory.org.za/sites/default/files/file%20uploads%20/j._d._fage_the_cambridge_history_of_africa_volubook4you.pdf ''The Cambridge History of Africa''] {{Webarchive|url=https://web.archive.org/web/20180228041419/http://www.sahistory.org.za/sites/default/files/file%20uploads%20/j._d._fage_the_cambridge_history_of_africa_volubook4you.pdf |date=28 February 2018 }}, Cambridge University Press. Vol. 2, p. 118. {{DOI|10.1017/CHOL9780521215923.004}}. {{ISBN|9781139054560}}</ref><ref>Oliver, Roland and Anthony Atmore (1994), ''Africa Since 1800'', Cambridge University Press. {{ISBN|0-521-42970-6}}</ref>

[[European exploration of Africa]] began with [[Ancient Greeks]] and [[Ancient Rome|Romans]].<ref>{{Cite web|title=The Berlin Conference {{!}} Western Civilization II (HIS 104) – Biel|url=https://courses.lumenlearning.com/suny-fmcc-worldcivilization2-1/chapter/the-berlin-conference/|website=courses.lumenlearning.com|access-date=13 May 2020|archive-date=10 June 2020|archive-url=https://web.archive.org/web/20200610192536/https://courses.lumenlearning.com/suny-fmcc-worldcivilization2-1/chapter/the-berlin-conference/|url-status=live}}</ref><ref>{{Cite web|title=Greeks, Romans and Barbarians|url=https://www.britannica.com/topic/history-of-Europe/Greeks-Romans-and-barbarians|access-date=30 May 2020|archive-date=4 May 2020|archive-url=https://web.archive.org/web/20200504023034/https://www.britannica.com/topic/history-of-Europe/Greeks-Romans-and-barbarians|url-status=live}}</ref> In 332 BC, [[Alexander the Great]] was welcomed as a liberator in [[History of Ptolemaic Egypt|Persian-occupied Egypt]]. He founded [[Alexandria]] in Egypt, which would become the prosperous capital of the [[Ptolemaic dynasty]] after his death.<ref>{{cite web|url=http://www.wsu.edu/~dee/EGYPT/PTOLEMY.HTM |title=Ptolemaic and Roman Egypt: 332 BC – 395 AD |publisher=Wsu.edu |date=6 June 1999 |access-date=18 May 2010 |archive-url=https://web.archive.org/web/20100528152425/http://www.wsu.edu/~dee/EGYPT/PTOLEMY.HTM |archive-date=28 May 2010 |url-status=dead}}</ref>

Following the conquest of North Africa's Mediterranean coastline by the [[Roman Empire]], the area was integrated economically and culturally into the Roman system. [[Africa Province|Roman settlement]] occurred in modern Tunisia and elsewhere along the coast. The first [[Roman emperor]] native to North Africa was [[Septimius Severus]], born in [[Leptis Magna]] in present-day Libya—his mother was Italian Roman and his father was [[Punics|Punic]].<ref>{{cite news|title=New exhibition about Roman Emperor Septimius Severus at the Yorkshire Museum|url=http://www.yorkpress.co.uk/features/features/8826893.New_exhibition_about_Roman_Emperor_Septimius_Severus_at_the_Yorkshire_Museum/|access-date=15 December 2013|newspaper=The Press|date=2 February 2011|archive-date=15 December 2013|archive-url=https://web.archive.org/web/20131215152114/http://www.yorkpress.co.uk/features/features/8826893.New_exhibition_about_Roman_Emperor_Septimius_Severus_at_the_Yorkshire_Museum/|url-status=live}}</ref>
[[File:Aksum, iscrizione di re ezana, in greco, sabeo e ge'ez, 330-350 dc ca. 10.jpg|right|thumb|The [[Ezana Stone]] records King Ezana's conversion to Christianity and his subjugation of various neighboring peoples, including [[Meroë]].]]
Christianity spread across these areas at an early date, from Judaea via Egypt and beyond the borders of the Roman world into Nubia;<ref>{{cite web|title=The Story of Africa – Christianity|url=https://www.bbc.co.uk/worldservice/africa/features/storyofafrica/index_section8.shtml|work=BBC World Service|publisher=BBC|access-date=15 December 2013|archive-date=9 July 2013|archive-url=https://web.archive.org/web/20130709142011/http://www.bbc.co.uk/worldservice/africa/features/storyofafrica/index_section8.shtml|url-status=live}}</ref> by AD 340 at the latest, it had become the [[state religion]] of the [[Aksumite Empire]]. [[Frumentius|Syro-Greek missionaries]], who arrived by way of the Red Sea, were responsible for this theological development.<ref>{{cite book|author=Tesfagiorgis, Mussie|title=Eritrea|url=https://books.google.com/books?id=f0R7iHoaykoC&pg=PA153|year=2010|publisher=ABC-CLIO|isbn=978-1-59884-232-6|page=153|access-date=14 October 2015|archive-date=11 June 2020|archive-url=https://web.archive.org/web/20200611044525/https://books.google.com/books?id=f0R7iHoaykoC&pg=PA153|url-status=live}}</ref>

In the early 7th century, the newly formed Arabian Islamic [[Caliphate]] expanded into Egypt, and then into North Africa. In a short while, the local Berber elite had been integrated into Muslim Arab tribes. When the Umayyad capital Damascus fell in the 8th century, the Islamic centre of the Mediterranean shifted from Syria to [[Qayrawan]] in North Africa. Islamic North Africa had become diverse, and a hub for mystics, scholars, jurists, and philosophers. During the above-mentioned period, Islam spread to sub-Saharan Africa, mainly through trade routes and migration.<ref name =Ayoub>{{cite book|last=Ayoub|first=Mahmoud M.|author-link=Mahmoud M. Ayoub|title=Islam: Faith and History|publisher=Oneworld|date=2004|location=Oxford|pages=76, 92–93, 96–97}}</ref>

In West Africa, [[Dhar Tichitt]] and [[Oualata]] in present-day [[Mauritania]] figure prominently among the early urban centers, dated to 2,000 BC. About 500 stone settlements litter the region in the former savannah of the Sahara. Its inhabitants fished and grew millet. It has been found by Augustin Holl that the [[Soninke people|Soninke]] of the [[Mandé peoples]] were likely responsible for constructing such settlements. Around 300 BC the region became more desiccated and the settlements began to decline, most likely relocating to [[Koumbi Saleh]].<ref name="HollA1985">{{cite journal | vauthors = Holl, Augustin | title = Background to the Ghana Empire: archaeological investigations on the transition to statehood in the Dhar Tichitt region (Mauritania) | journal = Journal of Anthropological Archaeology | volume = 4 | issue = 2 | pages = 73–115 | date = 1985 | url = https://www.academia.edu/2558381 | doi = 10.1016/0278-4165(85)90005-4 | access-date = 15 October 2019 | archive-date = 30 March 2021 | archive-url = https://web.archive.org/web/20210330032526/https://www.academia.edu/2558381/Background_to_the_Ghana_Empire_archaeological_investigations_on_the_transition_to_statehood_in_the_Dhar_Tichitt_region_Mauritania_ | url-status = live }}</ref> Architectural evidence and the comparison of pottery styles suggest that Dhar Tichitt was related to the subsequent [[Ghana Empire]]. [[Djenné-Djenno]] (in present-day [[Mali]]) was settled around 300 BC, and the town grew to house a sizable [[Iron Age]] population, as evidenced by crowded cemeteries. Living structures were made of sun-dried mud. By 250 BC [[Djenné-Djenno]] had become a large, thriving market town.<ref>Iliffe, John (2007). pp. 49–50</ref><ref>Collins and Burns (2007), p. 78.</ref>

Farther south, in central [[Nigeria]], around 1,500 BC, the [[Nok culture]] developed on the [[Jos Plateau]]. It was a highly centralized community. The Nok people produced lifelike representations in [[terracotta]], including human heads and human figures, elephants, and other animals. By 500 BC, and possibly earlier, they were smelting iron. By 200 AD the Nok culture had vanished.<ref name="PB 2014"/> and vanished under unknown circumstances around 500 AD, having lasted approximately 2,000 years. Based on stylistic similarities with the Nok terracottas, the bronze figurines of the [[Yoruba people|Yoruba]] kingdom of [[Ife]] and those of the [[Bini people|Bini]] kingdom of [[Benin]] are suggested to be continuations of the traditions of the earlier Nok culture.<ref>Shillington, Kevin (2005), p. 39.</ref><ref name="FB 1969"/>

===Ninth to eighteenth centuries===

[[File:Bronze ornamental staff head, 9th century, Igbo-Ukwu.JPG|thumb|left|upright|The intricate 9th-century bronzes from [[Archaeology of Igbo-Ukwu|Igbo-Ukwu]], in [[Nigeria]] displayed a level of technical accomplishment that was notably more advanced than European bronze casting of the same period.<ref name=Honour-2005>{{cite book |last1=Honour |first1=Hugh|last2=Fleming|first2=John|title=A world history of art|date=2005|publisher=Laurence King |location=London |isbn=9781856694513 |edition=7th}}</ref>]]
Pre-colonial Africa possessed perhaps as many as 10,000 different states and polities<ref>{{cite news|url=https://www.washingtonpost.com/wp-dyn/content/discussion/2006/01/11/DI2006011101372.html|title=The Fate of Africa – A Survey of Fifty Years of Independence|access-date=23 July 2007|work=[[The Washington Post]]|first=Martin|last=Meredith|date=20 January 2006|archive-date=2 May 2019|archive-url=https://web.archive.org/web/20190502070029/http://www.washingtonpost.com/wp-dyn/content/discussion/2006/01/11/DI2006011101372.html|url-status=live}}</ref> characterized by many different sorts of political organization and rule. These included small family groups of hunter-gatherers such as the [[San people]] of southern Africa; larger, more structured groups such as the family clan groupings of the [[Bantu languages|Bantu-speaking]] [[Bantu peoples|peoples]] of central, southern, and eastern Africa; heavily structured clan groups in the [[Horn of Africa]]; the large [[Sahelian kingdoms]]; and autonomous city-states and kingdoms such as those of the [[Akan people|Akan]]; [[Kingdom of Benin|Edo]], [[Yoruba people|Yoruba]], and [[Igbo people]] in West Africa; and the [[Swahili people|Swahili]] coastal trading towns of Southeast Africa.

By the ninth century AD, a string of dynastic states, including the earliest [[Hausa Kingdoms|Hausa]] states, stretched across the sub-Saharan savannah from the western regions to central Sudan. The most powerful of these states were [[Ghana Empire|Ghana]], [[Gao Region|Gao]], and the [[Kanem Empire|Kanem-Bornu Empire]]. Ghana declined in the eleventh century, but was succeeded by the [[Mali Empire]] which consolidated much of western Sudan in the thirteenth century. Kanem accepted Islam in the eleventh century.

In the forested regions of the West African coast, independent kingdoms grew with little influence from the [[Muslim]] north. The [[Kingdom of Nri]] was established around the ninth century and was one of the first. It is also one of the oldest kingdoms in present-day [[Nigeria]] and was ruled by the [[Eze Nri]]. The Nri kingdom is famous for its elaborate [[Igbo-Ukwu#Bronzes|bronzes]], found at the town of [[Igbo-Ukwu]]. The bronzes have been dated from as far back as the ninth century.<ref>{{cite web|url=http://www.metmuseum.org/toah/hd/igbo/hd_igbo.htm|title=Igbo-Ukwu (c. 9th century) | Thematic Essay | Heilbrunn Timeline of Art History | The Metropolitan Museum of Art|publisher=Metmuseum.org|access-date=18 May 2010|archive-date=4 December 2008|archive-url=https://web.archive.org/web/20081204053356/http://www.metmuseum.org/toah/hd/igbo/hd_igbo.htm|url-status=live}}</ref>

The [[Ifẹ|Kingdom of Ife]], historically the first of these Yoruba city-states or kingdoms, established government under a priestly [[oba (ruler)|oba]] ('king' or 'ruler' in the [[Yoruba language]]), called the ''Ooni of Ife''. Ife was noted as a major religious and cultural centre in West Africa, and for its unique naturalistic tradition of bronze sculpture. The Ife model of government was adapted at the [[Oyo Empire]], where its obas or kings, called the ''Alaafins of Oyo'', once controlled a large number of other Yoruba and non-Yoruba city-states and kingdoms; the [[Fon people|Fon]] ''Kingdom of [[Dahomey]]'' was one of the non-Yoruba domains under Oyo control.
[[File:Great Zimbabwe Closeup.jpg|thumb|left|Ruins of [[Great Zimbabwe]] (flourished eleventh to fifteenth centuries)]]
The [[Almoravid dynasty|Almoravids]] were a [[Berber people|Berber]] dynasty from the [[Sahara]] that spread over a wide area of northwestern Africa and the Iberian peninsula during the eleventh century.<ref>Glick, Thomas F. (2005) ''Islamic And Christian Spain in the Early Middle Ages''. Brill Academic Publishers, p. 37. {{ISBN|9789004147713}}</ref> The [[Banu Hilal]] and [[Maqil|Banu Ma'qil]] were a collection of [[Arab]] [[Bedouin]] tribes from the [[Arabian Peninsula]] who migrated westwards via Egypt between the eleventh and thirteenth centuries. Their [[Human migration|migration]] resulted in the fusion of the Arabs and Berbers, where the locals were [[Arabization|Arabized]],<ref>{{cite web|url=http://countrystudies.us/mauritania/8.htm|title=Mauritania – Arab Invasions|website=countrystudies.us|access-date=25 April 2010|archive-date=23 June 2011|archive-url=https://web.archive.org/web/20110623125418/http://countrystudies.us/mauritania/8.htm|url-status=live}}</ref> and Arab culture absorbed elements of the local culture, under the unifying framework of Islam.<ref>{{cite journal|title=Genetic Evidence for the Expansion of Arabian Tribes into the Southern Levant and North Africa |date=1 April 2010 |pmc=379148 |volume=70|issue=6|pmid=11992266|last1=Nebel|first1=A|display-authors=etal|pages=1594–96 |doi=10.1086/340669 |journal=American Journal of Human Genetics}}</ref>

Following the breakup of Mali, a local leader named [[Sonni Ali]] (1464–1492) founded the [[Songhai Empire]] in the region of middle [[Niger]] and the western [[Sudan (region)|Sudan]] and took control of the trans-Saharan trade. Sonni Ali seized [[Timbuktu]] in 1468 and [[Djenné|Jenne]] in 1473, building his regime on trade revenues and the cooperation of Muslim merchants. His successor [[Askia Mohammad I]] (1493–1528) made Islam the official religion, built mosques, and brought to Gao Muslim scholars, including al-Maghili (d.1504), the founder of an important tradition of Sudanic African Muslim scholarship.<ref name="multiple">Lapidus, Ira M. (1988) ''A History of Islamic Societies'', Cambridge.</ref> By the eleventh century, some [[Hausa Kingdoms|Hausa]] states – such as [[Kano (city)|Kano]], [[jigawa]], [[Katsina]], and [[Gobir]] – had developed into walled towns engaging in trade, servicing [[camel train|caravans]], and the manufacture of goods. Until the fifteenth century, these small states were on the periphery of the major Sudanic empires of the era, paying tribute to Songhai to the west and Kanem-Borno to the east.

===Height of the slave trade===
{{See also|Trans-Saharan slave trade|Atlantic slave trade|Indian Ocean slave trade}}
[[File:Africa slave Regions.svg|thumb|upright=1.2|right|Major slave trading regions of Africa, 15th–19th centuries.]]
[[Slavery]] had long been practiced in Africa.<ref>[http://www.britannica.com/blackhistory/article-24157 Historical survey: Slave societies] {{Webarchive|url=https://web.archive.org/web/20071230184609/http://www.britannica.com/blackhistory/article-24157 |date=30 December 2007 }}, ''Encyclopædia Britannica''</ref><ref>[http://www7.nationalgeographic.com/ngm/data/2001/10/01/html/ft_20011001.6.html Swahili Coast] {{Webarchive|url=https://web.archive.org/web/20071206102932/http://www7.nationalgeographic.com/ngm/data/2001/10/01/html/ft_20011001.6.html |date=6 December 2007 }}, National Geographic</ref> Between the 15th and the 19th centuries, the Atlantic slave trade took an estimated 7–12 million slaves to the New World.<ref>[http://www.britannica.com/blackhistory/article-24156 Welcome to Encyclopædia Britannica's Guide to Black History] {{Webarchive|url=https://web.archive.org/web/20070223090720/http://www.britannica.com/blackhistory/article-24156 |date=23 February 2007 }}, ''[[Encyclopædia Britannica]]''</ref><ref>{{Cite news|url=http://news.bbc.co.uk/2/hi/africa/1523100.stm|title=Focus on the slave trade|publisher=BBC News – Africa|work=bbc.co.uk|date=3 September 2001|access-date=28 February 2008|archive-date=28 July 2011|archive-url=https://web.archive.org/web/20110728134034/http://news.bbc.co.uk/2/hi/africa/1523100.stm|url-status=live}}</ref><ref>{{cite book|author=Lovejoy, Paul E. |title=Transformations in Slavery: A History of Slavery in Africa|url=https://archive.org/details/transformationsi0000love|url-access=registration |year=2000|publisher=Cambridge University Press|isbn=978-0-521-78430-6|page=[https://archive.org/details/transformationsi0000love/page/25 25]}}</ref> In addition, more than 1 million Europeans were captured by [[Barbary pirates]] and sold as slaves in North Africa between the 16th and 19th centuries.<ref>Rees Davies, [https://www.bbc.co.uk/history/british/empire_seapower/white_slaves_01.shtml "British Slaves on the Barbary Coast"] {{Webarchive|url=https://web.archive.org/web/20110425235016/http://www.bbc.co.uk/history/british/empire_seapower/white_slaves_01.shtml |date=25 April 2011 }}, [[BBC]], 1 July 2003</ref>

In West Africa, the decline of the Atlantic slave trade in the 1820s caused dramatic economic shifts in local polities. The gradual decline of slave-trading, prompted by a lack of demand for slaves in the [[New World]], increasing [[abolitionism|anti-slavery]] legislation in Europe and America, and the [[Royal Navy|British Royal Navy's]] increasing presence off the West African coast, obliged African states to adopt new economies. Between 1808 and 1860, the British [[West Africa Squadron]] seized approximately 1,600 slave ships and freed 150,000 Africans who were aboard.<ref>[https://www.bbc.co.uk/devon/content/articles/2007/03/20/abolition_navy_feature.shtml Jo Loosemore, Sailing against slavery] {{Webarchive|url=https://web.archive.org/web/20081103004954/https://www.bbc.co.uk/devon/content/articles/2007/03/20/abolition_navy_feature.shtml |date=3 November 2008 }}. BBC</ref>

Action was also taken against African leaders who refused to agree to British treaties to outlaw the trade, for example against "the usurping King of [[Lagos]]", deposed in 1851. Anti-slavery treaties were signed with over 50 African rulers.<ref>{{cite web|url=http://www.pdavis.nl/Background.htm#WAS|title=The West African Squadron and slave trade|publisher=Pdavis.nl|access-date=18 May 2010|archive-url=https://web.archive.org/web/20100610030306/http://www.pdavis.nl/Background.htm|archive-date=10 June 2010|url-status=live}}</ref> The largest powers of West Africa (the [[Asante Confederacy]], the [[Dahomey|Kingdom of Dahomey]], and the [[Oyo Empire]]) adopted different ways of adapting to the shift. Asante and Dahomey concentrated on the development of "legitimate commerce" in the form of [[palm oil]], [[Cocoa bean|cocoa]], timber and gold, forming the bedrock of West Africa's modern export trade. The Oyo Empire, unable to adapt, collapsed into civil wars.<ref>Simon, Julian L. (1995) ''State of Humanity'', Blackwell Publishing. p. 175. {{ISBN|1-55786-585-X}}</ref>

===Colonialism ===
{{Main|Colonisation of Africa}}
[[File:Scramble-for-Africa-1880-1913.png|thumb|350px|Comparison of Africa in the years 1880 and 1913]]{{Excerpt|Scramble for Africa| only=paragraphs}}

===Independence struggles===
[[File:Map of Africa in 1939.png|thumb|European control in 1939
{{aligned table |cols=2 |fullwidth=true
| {{legend|orange|[[Belgian colonial empire|Belgian]]}} | {{legend|#77DD77|[[Italian Empire|Italian]]}}
| {{legend|#DC143C|[[British Empire|British]]}} | {{legend|#964B00|[[Portuguese Empire|Portuguese]]}}
| {{legend|#2A52BE|[[French colonial empire|French]]}} | {{legend|#7FFFD4|[[Spanish Empire|Spanish]]}}
| {{legend|#f6f6f6|Independent}}
}}
]]
Imperial rule by Europeans would continue until after the conclusion of [[World War II]], when almost all remaining colonial territories gradually obtained formal independence. [[African independence movements|Independence movements in Africa]] gained momentum following World War II, which left the major European powers weakened. In 1951, [[Libya]], a former Italian colony, gained independence. In 1956, [[Tunisia]] and [[Morocco]] won their independence from France.<ref>{{cite book|author=Bély, Lucien|title=The History of France|url=https://books.google.com/books?id=Ltzav890zpIC&pg=PA118|year=2001|publisher=Editions Jean-paul Gisserot|isbn=978-2-87747-563-1|page=118|access-date=5 February 2018|archive-date=11 June 2020|archive-url=https://web.archive.org/web/20200611045035/https://books.google.com/books?id=Ltzav890zpIC&pg=PA118|url-status=live}}</ref> [[Ghana]] followed suit the next year (March 1957),<ref>{{cite book|author1=Aryeetey, Ernest|author2=Harrigan, Jane|author3=Nissanke Machiko|title=Economic Reforms in Ghana: The Miracle and the Mirage|url=https://books.google.com/books?id=87V55ZHppSYC&pg=PA5|year=2000|publisher=Africa World Press|isbn=978-0-86543-844-6|page=5|access-date=5 February 2018|archive-date=11 June 2020|archive-url=https://web.archive.org/web/20200611044656/https://books.google.com/books?id=87V55ZHppSYC&pg=PA5|url-status=live}}</ref> becoming the first of the sub-Saharan colonies to be granted independence. Most of the rest of the continent became independent over the next decade.

Portugal's overseas presence in [[Sub-Saharan Africa]] (most notably in [[Portuguese Angola|Angola]], Cape Verde, [[Portuguese Mozambique|Mozambique]], [[Portuguese Guinea|Guinea-Bissau]] and São Tomé and Príncipe) lasted from the 16th century to 1975, after the [[Estado Novo (Portugal)|Estado Novo]] regime was overthrown in [[Carnation Revolution|a military coup in Lisbon]]. [[Rhodesia]] [[Rhodesia's Unilateral Declaration of Independence|unilaterally declared independence]] from the United Kingdom in 1965, under the [[White minority rule|white minority]] government of [[Ian Smith]], but was not internationally recognized as an independent state (as [[Zimbabwe]]) until 1980, when black nationalists gained power after a [[Rhodesian Bush War|bitter guerrilla war]]. Although South Africa was one of the first African countries to gain independence, the state remained under the control of the country's white minority through a system of racial segregation known as [[South Africa under apartheid|apartheid]] until 1994.

===Post-colonial Africa===
{{see|Decolonisation of Africa}}
[[File:African nations order of independence 1950-1993.gif|upright=1.2|thumb|An animated map shows the order of [[Decolonisation of Africa|independence of African nations]], 1950–2011]]
Today, Africa contains 54 sovereign countries, most of which have borders that were drawn during the era of European colonialism. Since colonialism, African states have frequently been hampered by instability, corruption, violence, and authoritarianism. The vast majority of African states are republics that operate under some form of the [[presidential system]] of rule. However, few of them have been able to sustain democratic governments on a permanent basis, and many have instead cycled through a series of [[Coup d'état|coups]], producing [[military dictatorship]]s.

Great instability was mainly the result of marginalization of ethnic groups, and [[Political corruption|graft under these leaders]]. For [[Divide and rule|political gain]], many leaders fanned ethnic conflicts, some of which had been exacerbated, or even created, by colonial rule. In many countries, the military was perceived as being the only group that could effectively maintain order, and it ruled many nations in Africa during the 1970s and early 1980s. During the period from the early 1960s to the late 1980s, Africa had more than 70 coups and 13 presidential assassinations. Border and territorial disputes were also common, with the European-imposed borders of many nations being widely contested through armed conflicts.
[[File:Africa’s wars and conflicts, 1980–96.jpg|upright=1.2|thumb|Africa's wars and conflicts, 1980–1996]]
[[Cold War]] conflicts between the United States and the [[Soviet Union]], as well as the policies of the [[International Monetary Fund]],<ref>{{Cite book|last=International Monetary Fund.|title=International Monetary Fund.|publisher=IMF|oclc=40951957}}</ref> also played a role in instability. When a country became independent for the first time, it was often expected to align with one of the two superpowers. Many countries in Northern Africa received Soviet military aid, while others in Central and Southern Africa were supported by the United States, France or both. The 1970s saw an escalation of Cold War intrigues, as newly independent [[Angola]] and [[Mozambique]] aligned themselves with the Soviet Union, and the West and South Africa sought to contain Soviet influence by supporting friendly regimes or insurgency movements. In [[Rhodesia]], Soviet and Chinese-backed leftist guerrillas of the [[Patriotic Front (Zimbabwe)|Zimbabwe Patriotic Front]] waged a brutal [[Rhodesian Bush War|guerrilla war]] against the country's white government. There was a [[1983–85 famine in Ethiopia|major famine in Ethiopia]], when hundreds of thousands of people starved. Some claimed that Marxist economic policies made the situation worse.<ref>{{cite news|url=http://news.bbc.co.uk/2/hi/africa/703958.stm|title=BBC: 1984 famine in Ethiopia|date=6 April 2000|access-date=1 January 2010|work=BBC News|archive-date=19 April 2019|archive-url=https://web.archive.org/web/20190419011700/http://news.bbc.co.uk/2/hi/africa/703958.stm|url-status=live}}</ref><ref>Robert G. Patman, ''The Soviet Union in the Horn of Africa'' 1990, {{ISBN|0-521-36022-6}}, pp. 295–96</ref><ref>Steven Varnis, ''Reluctant aid or aiding the reluctant?: U.S. food aid policy and the Ethiopian Famine Relief'' 1990, {{ISBN|0-88738-348-3}}, p. 38</ref> The most devastating military conflict in modern independent Africa has been the [[Second Congo War]]; this conflict and its aftermath has killed an estimated 5.5 million people.<ref>{{cite news|url=https://www.telegraph.co.uk/news/worldnews/africaandindianocean/democraticrepublicofcongo/8792068/Is-your-mobile-phone-helping-fund-war-in-Congo.html|title=Is your mobile phone helping fund war in Congo?|date=27 September 2011|work=The Daily Telegraph|location=London|first=Gordon|last=Rayner|access-date=3 April 2018|archive-date=18 October 2017|archive-url=https://web.archive.org/web/20171018135029/http://www.telegraph.co.uk/news/worldnews/africaandindianocean/democraticrepublicofcongo/8792068/Is-your-mobile-phone-helping-fund-war-in-Congo.html|url-status=live}}</ref> Since 2003 there has been an ongoing [[War in Darfur|conflict in Darfur]] which has become a humanitarian disaster. Another notable tragic event is the 1994 [[Rwandan genocide]] in which an estimated 800,000 people were murdered.

In the 21st century, however, the number of armed conflicts in Africa has steadily declined. For instance, the [[Angolan Civil War|civil war in Angola]] came to an end in 2002 after nearly 30 years. This coincided with many countries abandoning communist-style command economies and opening up for market reforms. The improved stability and economic reforms have led to a great increase in foreign investment into many African nations, mainly from China,<ref name=Africa/> which has spurred quick economic growth in many countries, seemingly ending decades of stagnation and decline. Several African economies are among the world's fastest growing {{as of|2016|lc=y}}. A significant part of this growth, which is sometimes referred to as [[Africa Rising]], can also be attributed to the facilitated diffusion of information technologies and specifically the mobile telephone.<ref>Jenny Aker, Isaac Mbiti, [https://ssrn.com/abstract=1693963 "Mobile Phones and Economic Development in Africa"] {{Webarchive|url=https://web.archive.org/web/20210330032528/https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1693963 |date=30 March 2021 }} SSRN</ref> [[Human migration|Migration]] from African nations has increased dramatically in the last decade.<ref>{{cite news |title=At Least a Million Sub-Saharan Africans Moved to Europe Since 2010 |url=http://www.pewglobal.org/2018/03/22/at-least-a-million-sub-saharan-africans-moved-to-europe-since-2010/ |publisher=Pew Research Center |date=22 March 2018 |access-date=8 June 2018 |archive-url=https://web.archive.org/web/20190301003547/http://www.pewglobal.org/2018/03/22/at-least-a-million-sub-saharan-africans-moved-to-europe-since-2010/ |archive-date=1 March 2019 |url-status=dead }}</ref>

== Geology, geography, ecology and environment ==
{{Main|Geography of Africa}}
[[File:Topography of africa.png|thumb|upright=1.2|Topography of Africa]]
Africa is the largest of the three great southward projections from the largest landmass of the Earth. Separated from Europe by the [[Mediterranean Sea]], it is joined to Asia at its northeast extremity by the [[Suez Canal|Isthmus of Suez]] (transected by the [[Suez Canal]]), {{convert|163|km|mi|abbr=on}} wide.<ref>Drysdale, Alasdair and Gerald H. Blake. (1985) ''The Middle East and North Africa'', Oxford University Press US. {{ISBN|0-19-503538-0}}</ref> ([[Geopolitics|Geopolitically]], [[Egypt]]'s [[Sinai Peninsula]] east of the Suez Canal is often considered part of Africa, as well.)<ref>{{cite web|url=http://www.nationalgeographic.com/xpeditions/atlas/index.html?Parent=africa&Rootmap=&Mode=d|title=Atlas – Xpeditions|publisher=National Geographic Society|date=2003|access-date=1 March 2009|archive-url=https://web.archive.org/web/20090303230811/http://www.nationalgeographic.com/xpeditions/atlas/index.html?Parent=africa&Rootmap=&Mode=d|archive-date=3 March 2009| url-status=live}}</ref>

The coastline is {{convert|26000|km|mi|abbr=on}} long, and the absence of deep indentations of the shore is illustrated by the fact that Europe, which covers only {{convert|10400000|km2|sqmi|abbr=on}} – about a third of the surface of Africa – has a coastline of {{convert|32000|km|mi|abbr=on}}.<ref name=MW/> From the most northerly point, [[Ras ben Sakka]] in Tunisia (37°21' N), to the most southerly point, [[Cape Agulhas]] in South Africa (34°51'15" S), is a distance of approximately {{convert|8,000|km|mi|abbr=on}}.<ref>Lewin, Evans. (1924) ''Africa'', Clarendon press</ref> [[Cap-Vert|Cape Verde]], 17°33'22" W, the westernmost point, is a distance of approximately {{convert|7400|km|mi|abbr=on}} to [[Ras Hafun]], 51°27'52" E, the most easterly projection that neighbours [[Cape Guardafui]], the tip of the Horn of Africa.<ref name=MW>(1998) ''Merriam-Webster's Geographical Dictionary (Index)'', Merriam-Webster, pp. 10–11. {{ISBN|0-87779-546-0}}</ref>

Africa's largest country is [[Algeria]], and its smallest country is [[Seychelles]], an [[archipelago]] off the east coast.<ref name=Hoare>Hoare, Ben. (2002) ''The Kingfisher A–Z Encyclopedia'', Kingfisher Publications. p. 11. {{ISBN|0-7534-5569-2}}</ref> The smallest nation on the continental mainland is [[The Gambia]].

===African plate===
{{main|African Plate}}
[[Image:Motion of Nubia Plate.gif|thumb|upright=1.2|Today, the African Plate is moving over Earth's surface at a speed of 0.292° ± 0.007° per million years, relative to the "average" Earth (NNR-MORVEL56)]]
The African Plate is a major [[tectonic plate]] straddling the [[equator]] as well as the [[prime meridian]]. It includes much of the [[continent]] of Africa, as well as oceanic crust which lies between the continent and various surrounding ocean ridges. Between {{Mya|60}} and {{Mya|10}}, the [[Somali Plate]] began [[rift]]ing from the African Plate along the [[East African Rift]].<ref>{{cite web | title = Somali Plate | publisher = Ashten Sawitsky | url = http://africa-arabia-plate.weebly.com/somali-plate.html | access-date = 30 June 2015 | archive-date = 20 June 2015 | archive-url = https://web.archive.org/web/20150620140517/http://africa-arabia-plate.weebly.com/somali-plate.html | url-status = live }}</ref> Since the continent of Africa consists of crust from both the African and the Somali plates, some literature refers to the African Plate as the ''Nubian Plate'' to distinguish it from the continent as a whole.<ref>{{Cite journal
| last1 = Chu | first1 = D.
| last2 = Gordon | first2 = R.G.
| title = Evidence for motion between Nubia and Somalia along the Southwest Indian ridge
| year = 1999 | journal = Nature | volume = 398 | issue = 6722
| pages = 64–67
| doi = 10.1038/18014| bibcode = 1999Natur.398...64C| s2cid = 4403043
}}</ref>

Geologically, Africa includes the [[Arabian Peninsula]]; the [[Zagros Mountains]] of Iran and the [[Anatolian Plateau]] of Turkey mark where the [[African Plate]] collided with Eurasia. The [[Afrotropical realm]] and the [[Saharo-Arabian Region|Saharo-Arabian desert]] to its north unite the region biogeographically, and the [[Afroasiatic languages|Afro-Asiatic]] [[language family]] unites the north linguistically.

=== Climate ===
{{Main|Climate of Africa}}
The climate of Africa ranges from [[tropical climate|tropical]] to [[Subarctic climate|subarctic]] on its highest peaks. Its northern half is primarily [[desert]], or [[arid]], while its central and southern areas contain both [[savanna]] plains and dense [[jungle]] ([[rainforest]]) regions. In between, there is a convergence, where vegetation patterns such as [[sahel]] and [[steppe]] dominate. Africa is the hottest continent on Earth and 60% of the entire land surface consists of drylands and deserts.<ref name="environmentalatlas">[http://www.africa.upenn.edu/afrfocus/afrfocus061708.html "Africa: Environmental Atlas, 06/17/08."] {{webarchive|url=https://web.archive.org/web/20120105193432/http://www.africa.upenn.edu/afrfocus/afrfocus061708.html |date=5 January 2012 }} [http://www.africa.upenn.edu African Studies Center] {{Webarchive|url=https://web.archive.org/web/20110731143110/http://www.africa.upenn.edu/ |date=31 July 2011 }}, University of Pennsylvania. Accessed June 2011.</ref> The record for the highest-ever recorded temperature, in [[Libya]] in 1922 ({{convert|58|C|F}}), was discredited in 2013.<ref name=newRecord>{{cite journal|last=El Fadli|first=KI|title=World Meteorological Organization Assessment of the Purported World Record 58°C Temperature Extreme at El Azizia, Libya (13 September 1922)|journal=Bulletin of the American Meteorological Society|date=September 2012|doi=10.1175/BAMS-D-12-00093.1|volume=94|issue=2|page=199|display-authors=etal|bibcode=2013BAMS...94..199E|doi-access=free}} (The 136 °F (57.8 °C), claimed by [['Aziziya]], [[Libya]], on 13 September 1922, has been officially deemed invalid by the [[World Meteorological Organization]].)</ref><ref>{{cite web|title=World Meteorological Organization World Weather / Climate Extremes Archive |url=http://wmo.asu.edu/world-highest-temperature |access-date=10 January 2013 |url-status=dead |archive-url=https://web.archive.org/web/20130104143844/http://wmo.asu.edu/world-highest-temperature |archive-date= 4 January 2013}}</ref>

===Ecology and biodiversity===
[[File:Vegetation Africa.png|thumb|upright=1.2|The main biomes in Africa.]]

Africa has over 3,000 [[protected area]]s, with 198 marine protected areas, 50 biosphere reserves, and 80 wetlands reserves. Significant habitat destruction, increases in human population and poaching are reducing Africa's biological diversity and [[arable land]]. Human encroachment, civil unrest and the introduction of non-native species threaten biodiversity in Africa. This has been exacerbated by administrative problems, inadequate personnel and funding problems.<ref name="environmentalatlas"/>

[[Deforestation]] is affecting Africa at twice the world rate, according to the United Nations Environment Programme ([[UNEP]]).<ref>[http://www.africanews.com/site/list_messages/18831 Deforestation reaches worrying level – UN] {{webarchive |url=https://web.archive.org/web/20081206051452/http://www.africanews.com/site/list_messages/18831 |date=6 December 2008 }}. AfricaNews. 11 June 2008</ref> According to the University of Pennsylvania African Studies Center, 31% of Africa's pasture lands and 19% of its forests and woodlands are classified as degraded, and Africa is losing over four million hectares of forest per year, which is twice the average deforestation rate for the rest of the world.<ref name="environmentalatlas"/> Some sources claim that approximately 90% of the original, virgin forests in West Africa have been destroyed.<ref>[http://www.afrol.com/features/10278 Forests and deforestation in Africa – the wasting of an immense resource] {{webarchive |url=https://web.archive.org/web/20090520182556/http://www.afrol.com/features/10278 |date=20 May 2009 }}. afrol News</ref> Over 90% of [[Madagascar]]'s original forests have been destroyed since the arrival of humans 2000 years ago.<ref>{{NatGeo ecoregion|id=at0118|name=Madagascar subhumid forests}}</ref> About 65% of Africa's agricultural land suffers from [[soil degradation]].<ref>[https://www.independent.co.uk/news/world/africa/nature-laid-waste-the-destruction-of-africa-844370.html "Nature laid waste: The destruction of Africa"] {{Webarchive|url=https://web.archive.org/web/20171017221918/https://www.independent.co.uk/news/world/africa/nature-laid-waste-the-destruction-of-africa-844370.html |date=17 October 2017 }}, ''The Independent'', 11 June 2008.</ref>

{{see also|Afrotropical realm|Palearctic realm}}

=== Environmental issues ===
{{Excerpt|Environmental issues in Africa|paragraphs=1-2|file=no}}

=== Water ===
{{Excerpt|Water in Africa|paragraphs=1-2|file=no}}

=== Climate change ===
{{Excerpt|Climate change in Africa|paragraphs=1-2|file=no}}

===Fauna===

{{Main|Fauna of Africa}}
[[File:Zebras, Serengeti savana plains, Tanzania.jpg|thumb|Savanna at [[Ngorongoro Conservation Area]], [[Tanzania]]]]
Africa boasts perhaps the world's largest combination of density and "range of freedom" of [[wild animal]] populations and diversity, with wild populations of large [[carnivore]]s (such as lions, [[hyena]]s, and cheetahs) and [[herbivore]]s (such as [[African buffalo|buffalo]], elephants, camels, and giraffes) ranging freely on primarily open non-private plains. It is also home to a variety of "jungle" animals including snakes and [[primate]]s and [[aquatic ecosystem|aquatic life]] such as crocodiles and [[amphibian]]s. In addition, Africa has the largest number of [[megafauna]] species, as it was least affected by the [[Quaternary extinction event#The Pleistocene or Ice Age extinction event|extinction of the Pleistocene megafauna]].

==Politics==
{{See also|List of political parties in Africa by country}}

=== African Union ===
{{Main|African Union}}
The [[African Union]] (AU) is a [[continental union]] consisting of 55 [[Member states of the African Union|member states]]. The union was formed, with [[Addis Ababa]], [[Ethiopia]], as its headquarters, on 26 June 2001. The union was officially established on 9 July 2002<ref name="African Union 2002">{{cite web|url=http://www.africa-union.org/official_documents/Speeches_&_Statements/HE_Thabo_Mbiki/Launch%20of%20the%20African%20Union,%209%20July%202002.htm |title=Launch of the African Union, 9 July 2002: Address by the chairperson of the AU, President Thabo Mbeki |author=Mbeki, Thabo |date=9 July 2002 |publisher=africa-union.org |location=ABSA Stadium, Durban, South Africa |access-date=8 February 2009 |url-status=dead |archive-url=https://web.archive.org/web/20090503210549/http://www.africa-union.org/official_documents/Speeches_%26_Statements/HE_Thabo_Mbiki/Launch%20of%20the%20African%20Union%2C%209%20July%202002.htm |archive-date= 3 May 2009}}</ref> as a successor to the [[Organisation of African Unity]] (OAU). In July 2004, the African Union's [[Pan-African Parliament]] (PAP) was relocated to [[Midrand]], in South Africa, but the [[African Commission on Human and Peoples' Rights]] remained in Addis Ababa.

The African Union, not to be confused with the [[African Union Commission|AU Commission]], is formed by the [[Constitutive Act of the African Union]], which aims to transform the [[African Economic Community]], a federated commonwealth, into a state under established international conventions. The African Union has a parliamentary government, known as the [[Assembly of the African Union|African Union Government]], consisting of legislative, judicial and executive organs. It is led by the African Union President and Head of State, who is also the President of the [[Pan-African Parliament]]. A person becomes AU President by being elected to the PAP, and subsequently gaining majority support in the PAP. The powers and authority of the President of the African Parliament derive from the Constitutive Act and the [[Pan-African Parliament|Protocol of the Pan-African Parliament]], as well as the inheritance of presidential authority stipulated by African treaties and by international treaties, including those subordinating the Secretary General of the [[Organisation of African Unity|OAU]] Secretariat (AU Commission) to the PAP. The government of the AU consists of all-union, regional, state, and municipal authorities, as well as hundreds of institutions, that together manage the day-to-day affairs of the institution.

Extensive human rights abuses still occur in several parts of Africa, often under the oversight of the state. Most of such violations occur for political reasons, often as a side effect of civil war. Countries where major human rights violations have been reported in recent times include the [[Democratic Republic of the Congo]], [[Sierra Leone]], [[Liberia]], [[Sudan]], [[Zimbabwe]], and [[Ivory Coast]].

===Boundary conflicts ===
{{see|List of conflicts in Africa}}
{{Excerpt|Military history of Africa#Post-colonial|paragraph=1|file=no}}

==Economy==

{{Main|Economy of Africa|List of African countries by GDP (nominal)|List of African countries by GDP (PPP)}}
{{See also|Economy of the African Union}}
[[File:RECs of the AEC.svg|thumb|upright=1.2|Map of the [[African Economic Community]].
{{legend|#691717|[[Community of Sahel-Saharan States|CEN-SAD]]}}
{{legend|#4F4FB1|[[Common Market for Eastern and Southern Africa|COMESA]]}}
{{legend|#E88356|[[East African Community|EAC]]}}
{{legend|#272759|[[Economic Community of Central African States|ECCAS]]}}
{{legend|#C43C7F|[[Economic Community of West African States|ECOWAS]]}}
{{legend|#4DB34D|[[Intergovernmental Authority on Development|IGAD]]}}
{{legend|#D22E2E|[[Southern African Development Community|SADC]]}}
{{legend|#7E8000|[[Arab Maghreb Union|UMA]]}}
]]

Although it has abundant [[natural resource]]s, Africa remains the world's poorest and [[Human Development Index|least-developed]] continent (other than [[Antarctica]]), the result of a variety of causes that may include [[Corruption Perceptions Index|corrupt governments]] that have often committed serious [[human rights violations]], failed [[central planning]], high levels of [[illiteracy]], lack of access to foreign capital, and frequent tribal and military conflict (ranging from [[guerrilla warfare]] to [[genocide]]).<ref>Sandbrook, Richard (1985) ''The Politics of Africa's Economic Stagnation'', Cambridge University Press. passim</ref> Its total nominal GDP remains behind that of the United States, China, Japan, Germany, the United Kingdom, India and France. According to the United Nations' Human Development Report in 2003, the bottom 24 ranked nations (151st to 175th) were all African.<ref>{{cite web|url=http://hdr.undp.org/|title=Human Development Reports – United Nations Development Programme|website=hdr.undp.org|access-date=11 September 2005|archive-date=16 March 2018|archive-url=https://web.archive.org/web/20180316042117/http://hdr.undp.org/|url-status=live}}</ref>

[[Poverty in Africa|Poverty]], illiteracy, [[malnutrition]] and inadequate water supply and sanitation, as well as poor health, affect a large proportion of the people who reside in the African continent. In August 2008, the [[World Bank]]<ref>{{cite web|url=http://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/0,,contentMDK:21882162~pagePK:64165401~piPK:64165026~theSitePK:469382,00.html |title=World Bank Updates Poverty Estimates for the Developing World |publisher=World Bank |date=26 August 2008 |access-date=18 May 2010 |archive-url=https://web.archive.org/web/20100519204804/http://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/0%2C%2CcontentMDK%3A21882162~pagePK%3A64165401~piPK%3A64165026~theSitePK%3A469382%2C00.html |archive-date=19 May 2010 |url-status=dead}}</ref> announced revised global poverty estimates based on a new international poverty line of $1.25 per day (versus the previous measure of $1.00). 81% of the [[Sub-Saharan Africa]] population was living on less than $2.50 (PPP) per day in 2005, compared with 86% for India.<ref>{{cite web|url=http://econ.worldbank.org/external/default/main?pagePK=64165259&piPK=64165421&theSitePK=469372&menuPK=64166093&entityID=000158349_20080826113239|title=The developing world is poorer than we thought, but no less successful in the fight against poverty|publisher=World Bank|access-date=16 April 2009|archive-url=https://web.archive.org/web/20090323214139/http://econ.worldbank.org/external/default/main?pagePK=64165259&piPK=64165421&theSitePK=469372&menuPK=64166093&entityID=000158349_20080826113239|archive-date=23 March 2009|url-status=dead}}</ref>

Sub-Saharan Africa is the least successful region of the world in reducing poverty ($1.25 per day); some 50% of [[Poverty in Africa|the population living in poverty]] in 1981 (200 million people), a figure that rose to 58% in 1996 before dropping to 50% in 2005 (380 million people). The average poor person in sub-Saharan Africa is estimated to live on only 70 cents per day, and was poorer in 2003 than in 1973,<ref>[https://www.un.org/Depts/rcnyo/newsletter/survs/ecasurv2004.doc Economic report on Africa 2004: unlocking Africa's potential in the global economy] {{Webarchive|url=https://web.archive.org/web/20170118033000/http://www.un.org/Depts/rcnyo/newsletter/survs/ecasurv2004.doc |date=18 January 2017 }} (Substantive session 28 June–23 July 2004), United Nations</ref> indicating increasing poverty in some areas. Some of it is attributed to unsuccessful economic liberalization programmes spearheaded by foreign companies and governments, but other studies have cited bad domestic government policies more than external factors.<ref>{{cite web|url=http://www.globalpolitician.com/21498-africa-malawi-poverty |title=Neo-Liberalism and the Economic and Political Future of Africa |publisher=Globalpolitician.com |date=19 December 2005 |access-date=18 May 2010 |url-status=dead |archive-url=https://web.archive.org/web/20100131200200/http://globalpolitician.com/21498-africa-malawi-poverty |archive-date=31 January 2010}}</ref><ref>{{cite web|url=http://science.jrank.org/pages/8526/Capitalism-Africa-Neoliberalism-Structural-Adjustment-African-Reaction.html|title=Capitalism – Africa – Neoliberalism, Structural Adjustment, And The African Reaction|publisher=Science.jrank.org|access-date=18 May 2010|archive-url=https://web.archive.org/web/20100420101742/http://science.jrank.org/pages/8526/Capitalism-Africa-Neoliberalism-Structural-Adjustment-African-Reaction.html|archive-date=20 April 2010|url-status=live}}</ref><ref>{{cite web|url=http://www.turkishweekly.net/news.php?id=58925 |archive-url=https://web.archive.org/web/20080924092909/http://www.turkishweekly.net/news.php?id=58925 |url-status=dead |archive-date=24 September 2008 |title=The Number of the Poor Increasing Worldwide while Sub-Saharan Africa is the Worst of All |publisher=Turkish Weekly |date=29 August 2008 |access-date=7 November 2011 }}</ref>
[[File:Africa and Eurasia at night 2012.jpg|thumb|upright|left|Satellite image of city lights in Africa showing the relatively low modern development on the continent in 2012 as compared to Eurasia.]]
Africa is now at risk of being in debt once again, particularly in Sub-Saharan African countries. The last debt crisis in 2005 was resolved with help from the heavily indebted poor countries scheme (HIPC). The HIPC resulted in some positive and negative effects on the economy in Africa. About ten years after the 2005 debt crisis in Sub-Saharan Africa was resolved, Zambia fell back into debt. A small reason was due to the fall in copper prices in 2011, but the bigger reason was that a large amount of the money Zambia borrowed was wasted or pocketed by the elite.<ref>{{Cite news|url=https://www.economist.com/leaders/2018/09/15/zambias-looming-debt-crisis-is-a-warning-for-the-rest-of-africa|title=Zambia's looming debt crisis is a warning for the rest of Africa|work=The Economist|access-date=19 September 2018|language=en|archive-date=18 September 2018|archive-url=https://web.archive.org/web/20180918163443/https://www.economist.com/leaders/2018/09/15/zambias-looming-debt-crisis-is-a-warning-for-the-rest-of-africa|url-status=live}}</ref>

From 1995 to 2005, Africa's rate of economic growth increased, averaging 5% in 2005. Some countries experienced still higher growth rates, notably [[Angola]], [[Sudan]] and [[Equatorial Guinea]], all of which had recently begun extracting their petroleum reserves or had expanded their [[oil extraction]] capacity.

In a recently published analysis based on [[World Values Survey]] data, the Austrian political scientist [[Arno Tausch]] maintained that several African countries, most notably [[Ghana]], perform quite well on scales of mass support for democracy and the [[market economy]].<ref>{{cite document|doi=10.2139/ssrn.3214715|ssrn=3214715|title=Africa on the Maps of Global Values: Comparative Analyses, Based on Recent World Values Survey Data|date=2018|last1=Tausch|first1=Arno|s2cid=158596579|url=https://mpra.ub.uni-muenchen.de/87966/1/MPRA_paper_87966.pdf|journal=|access-date=26 September 2019|archive-date=11 February 2020|archive-url=https://web.archive.org/web/20200211141227/https://mpra.ub.uni-muenchen.de/87966/1/MPRA_paper_87966.pdf|url-status=live}}</ref>

Tausch's global value comparison based on the [[World Values Survey]] derived the following factor analytical scales: 1. The non-violent and law-abiding society 2. Democracy movement 3. Climate of personal non-violence 4. Trust in institutions 5. Happiness, good health 6. No redistributive religious fundamentalism 7. Accepting the market 8. Feminism 9. Involvement in politics 10. Optimism and engagement 11. No welfare mentality, acceptancy of the Calvinist work ethics. The spread in the performance of African countries with complete data, Tausch concluded "is really amazing". While one should be especially hopeful about the development of future democracy and the market economy in [[Ghana]], the article suggests pessimistic tendencies for [[Egypt]] and [[Algeria]], and especially for Africa's leading economy, South Africa. High [[Human Inequality]], as measured by the [[UNDP]]'s [[Human Development Report]]'s [[Index of Human Inequality]], further impairs the development of [[human security]]. Tausch also maintains that the certain recent optimism, corresponding to economic and human rights data, emerging from Africa, is reflected in the development of a [[civil society]].
[[File:African countries by GDP (PPP) per capita in 2020.png|upright=1.2|thumb|African countries by GDP (PPP) per capita in 2020]]
The continent is believed to hold 90% of the world's [[cobalt]], 90% of its [[platinum]], 50% of its gold, 98% of its [[chromium]], 70% of its [[tantalite]],<ref>"[http://allafrica.com/stories/200802070635.html Africa: Developed Countries' Leverage On the Continent] {{Webarchive|url=https://web.archive.org/web/20121020072131/http://allafrica.com/stories/200802070635.html |date=20 October 2012 }}". AllAfrica.com. 7 February 2008</ref> 64% of its [[manganese]] and one-third of its [[uranium]].<ref>[http://www.timesonline.co.uk/tol/news/world/africa/article3319909.ece Africa, China's new frontier] {{Webarchive|url=https://web.archive.org/web/20110629123044/http://www.timesonline.co.uk/tol/news/world/africa/article3319909.ece |date=29 June 2011 }}. ''Times Online''. 10 February 2008</ref> The [[Democratic Republic of the Congo]] (DRC) has 70% of the world's [[coltan]], a mineral used in the production of [[tantalum capacitor]]s for electronic devices such as cell phones. The DRC also has more than 30% of the world's diamond reserves.<ref>{{Cite news| url=http://news.bbc.co.uk/2/hi/africa/5209428.stm| title=DR Congo poll crucial for Africa| work=BBC| date=16 November 2006| access-date=10 October 2009| archive-date=2 December 2010| archive-url=https://web.archive.org/web/20101202153903/http://news.bbc.co.uk/2/hi/africa/5209428.stm| url-status=live}}</ref> [[Guinea]] is the world's largest exporter of [[bauxite]].<ref>[http://www.thetimes.co.uk/tto/news/world/africa/article2594373.ece China tightens grip on Africa with $4.4bn lifeline for Guinea junta]. The Times. 13 October 2009 {{subscription required}} {{Webarchive|url=https://web.archive.org/web/20150429071020/http://www.thetimes.co.uk/tto/news/world/africa/article2594373.ece |date=29 April 2015 }}</ref> As the growth in Africa has been driven mainly by services and not manufacturing or agriculture, it has been growth without jobs and without reduction in [[Poverty in Africa|poverty]] levels. In fact, the [[2007–08 world food price crisis|food security crisis of 2008]] which took place on the heels of the global financial crisis pushed 100 million people into food insecurity.<ref>[http://www.strategicforesight.com/african_decade.htm The African Decade?] {{Webarchive|url=https://web.archive.org/web/20100613173905/http://www.strategicforesight.com/african_decade.htm |date=13 June 2010 }}. Ilmas Futehally. Strategic Foresight Group.</ref>

In recent years, the People's Republic of China has built increasingly stronger ties with African nations and is Africa's largest trading partner. In 2007, Chinese companies invested a total of US$1 billion in Africa.<ref name=Africa>[http://www.migrationinformation.org/Feature/display.cfm?id=690 Malia Politzer, "China and Africa: Stronger Economic Ties Mean More Migration"] {{Webarchive|url=https://web.archive.org/web/20140129114909/http://www.migrationinformation.org/Feature/display.cfm?id=690 |date=29 January 2014 }}, ''Migration Information Source''. August 2008</ref>

A Harvard University study led by professor [[Calestous Juma]] showed that Africa could feed itself by making the transition from importer to self-sufficiency. "African agriculture is at the crossroads; we have come to the end of a century of policies that favoured Africa's export of raw materials and importation of food. Africa is starting to focus on agricultural innovation as its new engine for regional trade and prosperity."<ref>[https://www.sciencedaily.com/releases/2010/12/101202124337.htm "Africa Can Feed Itself in a Generation, Experts Say"] {{Webarchive|url=https://web.archive.org/web/20171017221141/https://www.sciencedaily.com/releases/2010/12/101202124337.htm |date=17 October 2017 }}, ''[[Science Daily]]'', 3 December 2010</ref>

==Demographics==
{{Main|Demographics of Africa|Child marriage#Africa{{!}}Child marriage in Africa}}
{{Pie chart
| caption= [[List of African countries by population|Proportion of total African population by country]]
| other = yes
| label1 = Nigeria
| value1 = 15.38 | color1=#36A
| label2 = Ethiopia
| value2 = 8.37 | color2=#1A9
| label3 = Egypt
| value3 = 7.65 | color3=#6A5
| label4 = Democratic Republic of the Congo
| value4 = 6.57 | color4=#CC5
| label5 = Tanzania
| value5 = 4.55 | color5=#928
| label6 = South Africa
| value6 = 4.47 | color6=#E33
| label7 = Kenya
| value7 = 3.88 | color7=#E72
| label8 = Uganda
| value8 = 3.38 | color8=#FE3
| label9 = Algeria
| value9 = 3.36 | color9=#A45
}}

Africa's population has rapidly increased over the last 40 years, and is consequently relatively young. In some African states, more than half the population is under 25 years of age.<ref>{{cite web|url=http://www.overpopulation.org/Africa.html|title=Africa Population Dynamics|publisher=overpopulation.org|access-date=26 July 2007|archive-date=17 February 2015|archive-url=https://web.archive.org/web/20150217040305/http://www.overpopulation.org/Africa.html|url-status=live}}</ref> The total number of people in Africa increased from 229 million in 1950 to 630 million in 1990.<ref>[http://www.geohive.com/earth/his_proj_africa.aspx Past and future population of Africa] {{webarchive|url=https://web.archive.org/web/20150924044751/http://www.geohive.com/earth/his_proj_africa.aspx |date=24 September 2015 }}. Source: United Nations, Department of Economic and Social Affairs, Population Division (2013)</ref> As of {{UN_Population|Year}}, the population of Africa is estimated at {{#expr:{{replace|{{UN_Population|Africa}}|,|}} / 1e9 round 1}} billion {{UN_Population|ref}}. Africa's total population surpassing other continents is fairly recent; African population surpassed Europe in the 1990s, while the Americas was overtaken sometime around the year 2000; Africa's rapid population growth is expected to overtake the only two nations currently larger than its population, at roughly the same time – India and China's 1.4 billion people each will swap ranking around the year 2022.<ref>{{Cite news|url=https://www.nytimes.com/2015/07/30/world/asia/india-will-be-most-populous-country-sooner-than-thought-un-says.html|title=India Will Be Most Populous Country Sooner Than Thought, U.N. Says|first=Rick|last=Gladstone|date=29 July 2015|newspaper=The New York Times|access-date=14 February 2017|archive-date=1 December 2020|archive-url=https://web.archive.org/web/20201201022241/https://www.nytimes.com/2015/07/30/world/asia/india-will-be-most-populous-country-sooner-than-thought-un-says.html|url-status=live}}</ref> This increase in number of babies born in Africa compared to the rest of the world is expected to reach approximately 37% in the year 2050, an increase of 21% since 1990 alone.<ref>{{Cite news|url=https://www.economist.com/leaders/2018/09/22/what-to-do-about-africas-dangerous-baby-boom|title=What to do about Africa's dangerous baby boom|work=The Economist|access-date=26 September 2018|language=en|archive-date=25 September 2018|archive-url=https://web.archive.org/web/20180925235351/https://www.economist.com/leaders/2018/09/22/what-to-do-about-africas-dangerous-baby-boom|url-status=live}}</ref>

Speakers of [[Bantu languages]] (part of the [[Niger–Congo languages|Niger–Congo]] family) are the majority in southern, central and southeast Africa. The Bantu-speaking peoples from [[the Sahel]] progressively expanded over most of Sub-Saharan Africa.<ref>Luc-Normand Tellier (2009). ''[https://books.google.com/books?id=cXuCjDbxC1YC Urban world history: an economic and geographical perspective] {{Webarchive|url=https://web.archive.org/web/20150924171325/https://books.google.com/books?id=cXuCjDbxC1YC |date=24 September 2015 }}''. PUQ. p. 204. {{ISBN|2-7605-1588-5}}</ref> But there are also several [[Nilotic]] groups in [[South Sudan]] and East Africa, the mixed [[Swahili people]] on the [[Swahili Coast]], and a few remaining [[Indigenous peoples of Africa|indigenous]] Khoisan ("[[Bushmen|San"]] or "Bushmen") and [[Pygmy peoples]] in southern and central Africa, respectively. Bantu-speaking Africans also predominate in Gabon and Equatorial Guinea, and are found in parts of southern Cameroon. In the [[Kalahari Desert]] of Southern Africa, the distinct people known as the Bushmen (also "San", closely related to, but distinct from "[[Khoikhoi|Hottentots]]") have long been present. The San are physically distinct from other Africans and are the indigenous people of southern Africa. Pygmies are the pre-Bantu indigenous peoples of central Africa.<ref>[http://www.timesonline.co.uk/tol/news/world/article402970.ece Pygmies struggle to survive in war zone where abuse is routine] {{Webarchive|url=https://web.archive.org/web/20100525095020/http://www.timesonline.co.uk/tol/news/world/article402970.ece |date=25 May 2010 }}. ''Times Online''. 16 December 2004</ref>

The peoples of West Africa primarily speak [[Niger–Congo languages]], belonging mostly to its non-Bantu branches, though some [[Nilo-Saharan]] and Afro-Asiatic speaking groups are also found. The Niger–Congo-speaking [[Yoruba language|Yoruba]], [[Igbo language|Igbo]], [[Fulani]], [[Akan language|Akan]] and [[Wolof people|Wolof]] ethnic groups are the largest and most influential. In the central Sahara, [[Mandinka people|Mandinka]] or [[Mande languages|Mande]] groups are most significant. Chadic-speaking groups, including the [[Hausa language|Hausa]], are found in more northerly parts of the region nearest to the Sahara, and Nilo-Saharan communities, such as the [[Songhai people|Songhai]], [[Kanuri people|Kanuri]] and [[Zarma people|Zarma]], are found in the eastern parts of West Africa bordering Central Africa.
[[Image:African_countries_by_HDI_(2019).png|thumb|upright=1.2|left|
{| width="100%" style="background:transparent;"
| Map of Africa indicating [[Human Development Index]] (2018).
|-Africa
|
{{Col-begin}}
{{Col-break}}
{{Legend|#00C400|0.800–0.849}}
{{Legend|#00F900|0.750–0.799}}
{{Legend|#D3FF00|0.700–0.749}}
{{Legend|#FFFF00|0.650–0.699}}
{{Legend|#FFD215|0.600–0.649}}
{{Legend|#FFA83C|0.550–0.599}}
{{Col-break}}
{{Legend|#FF852F|0.500–0.549}}
{{Legend|#FF5B00|0.450–0.499}}
{{Legend|#FF0000|0.400–0.449}}
{{Legend|#A70000|≤ 0.399}}
{{Legend|#D9D9D9|No data}}
{{Col-end}}
|}]]
The peoples of North Africa consist of three main indigenous groups: Berbers in the northwest, Egyptians in the northeast, and Nilo-Saharan-speaking peoples in the east. The [[Arab]]s who arrived in the 7th century AD introduced the Arabic language and Islam to North Africa. The Semitic [[Phoenicia]]ns (who founded [[Carthage]]) and [[Hyksos]], the Indo-Iranian [[Alans]], the Indo- European [[Ancient Greece|Greeks]], [[Ancient Rome|Romans]], and [[Vandals]] settled in North Africa as well. Significant Berber communities remain within [[Morocco]] and [[Algeria]] in the 21st century, while, to a lesser extent, Berber speakers are also present in some regions of Tunisia and Libya.<ref>{{cite news|title=Q&A: The Berbers|url=http://news.bbc.co.uk/2/hi/africa/3509799.stm|access-date=30 December 2013|newspaper=BBC News|date=12 March 2004|archive-date=12 January 2018|archive-url=https://web.archive.org/web/20180112181804/http://news.bbc.co.uk/2/hi/africa/3509799.stm|url-status=live}}</ref> The Berber-speaking [[Tuareg people|Tuareg]] and other often-[[nomad]]ic peoples are the principal inhabitants of the Saharan interior of North Africa. In Mauritania, there is a small but near-extinct Berber community in the north and Niger–Congo-speaking peoples in the south, though in both regions Arabic and Arab culture predominates. In Sudan, although Arabic and Arab culture predominate, it is mostly inhabited by groups that originally spoke Nilo-Saharan, such as the Nubians, Fur, Masalit and Zaghawa, who, over the centuries, have variously intermixed with migrants from the Arabian peninsula. Small communities of Afro-Asiatic-speaking Beja nomads can also be found in Egypt and Sudan.<ref>{{Cite book|chapter=The Linguistic Prehistory of the Sahara|doi=10.1017/9781108634311.014|chapter-url=https://www.cambridge.org/core/books/burials-migration-and-identity-in-the-ancient-sahara-and-beyond/linguistic-prehistory-of-the-sahara/76E34016357D7694FCF23CCF9A7F29D3|access-date=31 May 2020|archive-date=30 August 2020|archive-url=https://web.archive.org/web/20200830165212/https://www.cambridge.org/core/books/burials-migration-and-identity-in-the-ancient-sahara-and-beyond/linguistic-prehistory-of-the-sahara/76E34016357D7694FCF23CCF9A7F29D3|url-status=live|title=Burials, Migration and Identity in the Ancient Sahara and Beyond|year=2019|last1=Blench|first1=Roger|pages=431–463|isbn=9781108634311}}</ref>{{citation needed|date=December 2013}}

In the [[Horn of Africa]], some Ethiopian and Eritrean groups (like the [[Amhara people|Amhara]] and [[Tigrayans]], collectively known as [[Habesha people|Habesha]]) speak languages from the [[Semitic languages|Semitic]] branch of the [[Afroasiatic languages|Afro-Asiatic]] language family, while the [[Oromo people|Oromo]] and [[Somalis|Somali]] speak languages from the [[Cushitic]] branch of Afro-Asiatic.

Prior to the [[decolonization]] movements of the post-[[World War II]] era, [[Ethnic groups in Europe|Europeans]] were represented in every part of Africa.<ref>[http://www.time.com/time/magazine/article/0,9171,901759-3,00.html "We Want Our Country" (3 of 10)] {{Webarchive|url=https://web.archive.org/web/20130723000220/http://www.time.com/time/magazine/article/0,9171,901759-3,00.html |date=23 July 2013 }}. ''Time'', 5 November 1965</ref> Decolonization during the 1960s and 1970s often resulted in the mass emigration of white settlers – especially from Algeria and Morocco (1.6 million ''[[pieds-noir]]s'' in North Africa),<ref>Raimondo Cagiano De Azevedo (1994). ''[https://books.google.com/books?id=N8VHizsqaH0C&pg=PA25 Migration and development co-operation.] {{Webarchive|url=https://web.archive.org/web/20150906025429/https://books.google.com/books?id=N8VHizsqaH0C&pg=PA25 |date=6 September 2015 }}''. Council of Europe, p. 25. {{ISBN|92-871-2611-9}}</ref> Kenya, Congo,<ref>[http://www.time.com/time/magazine/article/0,9171,826488-4,00.html "Jungle Shipwreck"] {{Webarchive|url=https://web.archive.org/web/20130722210703/http://www.time.com/time/magazine/article/0,9171,826488-4,00.html |date=22 July 2013 }}. ''Time'' 25 July 1960</ref> Rhodesia, Mozambique and Angola.<ref>[http://www.economist.com/world/mideast-africa/displayStory.cfm?story_id=12079340 "Flight from Angola"] {{Webarchive|url=https://web.archive.org/web/20130723131954/http://www.economist.com/node/12079340?story_id=12079340 |date=23 July 2013 }}, ''The Economist '', 16 August 1975</ref> Between 1975 and 1977, over a million colonials returned to Portugal alone.<ref>[http://countrystudies.us/portugal/48.htm Portugal – Emigration] {{Webarchive|url=https://web.archive.org/web/20110629081956/http://countrystudies.us/portugal/48.htm |date=29 June 2011 }}, Eric Solsten, ed. Portugal: A Country Study. Washington: GPO for the Library of Congress, 1993</ref> Nevertheless, [[White Africans of European ancestry|white Africans]] remain an important minority in many African states, particularly [[Zimbabwe]], [[Namibia]], Réunion, and the [[Republic of South Africa]].<ref>{{Cite book|first=John A.|last=Holm|title=Pidgins and Creoles: References survey|url=https://books.google.com/books?id=PcD7p9y3EIcC&pg=PA394|publisher=Cambridge University Press|date=1989|page=394|isbn=978-0-521-35940-5|access-date=14 October 2015|archive-date=5 September 2015|archive-url=https://web.archive.org/web/20150905192604/https://books.google.com/books?id=PcD7p9y3EIcC&pg=PA394|url-status=live}}</ref> The country with the largest white African population is South Africa.<ref>[https://www.cia.gov/the-world-factbook/countries/south-africa/ South Africa: People: Ethnic Groups.] {{Webarchive|url=https://web.archive.org/web/20210110042951/https://www.cia.gov/the-world-factbook/countries/south-africa |date=10 January 2021 }} CIA World Factbook</ref> [[Dutch people|Dutch]] and [[British diaspora in Africa|British]] [[diaspora]]s represent the largest communities of European ancestry on the continent today.<ref name=World>{{cite encyclopedia|date=1989|title=Africa|encyclopedia=[[World Book Encyclopedia]]|publisher=World Book, Inc.|location=Chicago|isbn=978-0-7166-1289-6|url-access=registration|url=https://archive.org/details/1989worldbookencyclo22worl}}</ref>

European colonization also brought sizable groups of [[Asian people|Asians]], particularly from the [[Indian subcontinent]], to British colonies. Large [[Non-resident Indian and person of Indian origin|Indian communities]] are found in South Africa, and smaller ones are present in Kenya, Tanzania, and some other southern and southeast African countries. The large [[Indians in Uganda|Indian community in Uganda]] was [[expulsion of Asians from Uganda|expelled]] by the dictator [[Idi Amin]] in 1972, though many have since returned. The islands in the Indian Ocean are also populated primarily by people of Asian origin, often mixed with Africans and Europeans. The [[Malagasy people]] of [[Madagascar]] are an [[Austronesian people]], but those along the coast are generally mixed with Bantu, Arab, Indian and European origins. Malay and Indian ancestries are also important components in the group of people known in South Africa as [[Cape Coloureds]] (people with origins in two or more races and continents). During the 20th century, small but economically important communities of [[Demographics of Lebanon#The Lebanese Diaspora|Lebanese]] and [[Overseas Chinese|Chinese]]<ref name="Africa"/> have also developed in the larger coastal cities of [[West Africa|West]] and East Africa, respectively.<ref>[http://www1.voanews.com/english/news/a-13-2007-07-10-voa46.html Naomi Schwarz, "Lebanese Immigrants Boost West African Commerce"] {{Webarchive|url=https://web.archive.org/web/20111224135631/http://www.voanews.com/english/news/a-13-2007-07-10-voa46.html |date=24 December 2011 }}, VOANews.com, 10 July 2007</ref>

===Religion===
[[File:Religion distribution Africa crop.png|A map showing religious distribution in Africa|thumb|upright=1.1]]
{{Main|Religion in Africa}} {{See also||African divination}}

While Africans profess a wide variety of religious beliefs, the majority of the people respect African religions or parts of them. However, in formal surveys or census, most people will identify with major religions that came from outside the continent, mainly through colonisation. There are several reasons for this, the main one being the colonial idea that African religious beliefs and practices are not good enough. Religious beliefs and statistics on religious affiliation are difficult to come by since they are often a sensitive topic for governments with mixed religious populations.<ref name=stanford>{{cite web|url=http://library.stanford.edu/africa/religion.html|title=African Religion on the Internet |archive-url=https://web.archive.org/web/20060902182749/http://library.stanford.edu/africa/religion.html |archive-date=2 September 2006 |url-status=dead |publisher=[[Stanford University]]}}</ref><ref name=NYT>{{cite news|url=https://www.nytimes.com/2001/11/01/world/rising-muslim-power-in-africa-causes-unrest-in-nigeria-and-elsewhere-963950.html|date=1 November 2001|title=Rising Muslim Power in Africa Causing Unrest in Nigeria and Elsewhere|first=Normitsu|last=Onishi|work=The New York Times|access-date=1 March 2009|archive-date=20 November 2010|archive-url=https://web.archive.org/web/20101120221305/http://query.nytimes.com/gst/fullpage.html?res=9C00EEDC1030F932A35752C1A9679C8B63&sec=&spon=&pagewanted=1|url-status=live}}</ref> According to the ''[[World Book Encyclopedia]]'', [[Islam in Africa|Islam]] and [[Christianity in Africa|Christianity]] are the two largest religions in Africa. According to [[Encyclopædia Britannica]], 45% of the population are Christians, 40% are [[Muslim]]s, and 10% follow [[Traditional African religions|traditional religions]].{{citation needed|date=October 2020}} A small number of Africans are [[Hindu]], [[Buddhist]], [[Confucianist]], [[Baháʼí Faith|Baháʼí]], or [[Judaism in Africa|Jewish]]. There is also a minority of people in Africa who are [[Irreligion in Africa|irreligious]].

===Languages===
{{Main|Languages of Africa}}
{{See also|Writing systems of Africa#Indigenous writing systems}}
By most estimates, well over a thousand [[language]]s ([[UNESCO]] has estimated around two thousand) are spoken in Africa.<ref>{{cite web|url=http://portal.unesco.org/ci/en/ev.php-URL_ID=8048&URL_DO=DO_TOPIC&URL_SECTION=201.html |title=Africa |date=2005 |publisher=UNESCO |access-date=1 March 2009 |archive-url=https://web.archive.org/web/20080602050234/http://portal.unesco.org/ci/en/ev.php-URL_ID%3D8048%26URL_DO%3DDO_TOPIC%26URL_SECTION%3D201.html |archive-date= 2 June 2008 |url-status=dead}}</ref> Most are of African origin, though some are of European or Asian origin. Africa is the most [[Multilingualism|multilingual]] continent in the world, and it is not rare for individuals to fluently speak not only multiple African languages, but one or more European ones as well. There are four major [[language family|language families]] indigenous to Africa:
[[File:Map of African language families.svg|thumb|upright=1.2|A simplistic view of language families spoken in Africa]]
* The [[Afroasiatic languages|''Afroasiatic'']] languages are a language family of about 240 languages and 285 million people widespread throughout the [[Horn of Africa]], North Africa, the [[Sahel]], and Southwest Asia.
* The [[Nilo-Saharan languages|''Nilo-Saharan'']] language family consists of more than a hundred languages spoken by 30 million people. Nilo-Saharan languages are spoken by ethnic groups in [[Chad]], [[Ethiopia]], [[Kenya]], [[Nigeria]], [[Sudan]], [[South Sudan]], [[Uganda]], and northern [[Tanzania]].
* The [[Niger–Congo languages|''Niger-Congo'']] language family covers much of Sub-Saharan Africa. In terms of number of languages, it is the largest language family in Africa and perhaps one of the largest in the world.
* The [[Khoisan languages|''Khoisan'']] languages number about fifty and are spoken in Southern Africa by approximately 400,000 people.<ref>{{cite web|title=Khoisan Languages|url=http://www.languagesgulper.com/eng/Khoisan.html|website=The Language Gulper|access-date=2 January 2017|archive-date=25 January 2017|archive-url=https://web.archive.org/web/20170125082754/http://languagesgulper.com/eng/Khoisan.html|url-status=live}}</ref> Many of the Khoisan languages are [[endangered language|endangered]]. The [[Khoikhoi|Khoi]] and [[Bushmen|San]] peoples are considered the original inhabitants of this part of Africa.

Following the end of [[colonialism]], nearly all African countries adopted [[official language]]s that originated outside the continent, although several countries also granted legal recognition to indigenous languages (such as [[Swahili language|Swahili]], [[Yoruba language|Yoruba]], [[Igbo language|Igbo]] and [[Hausa language|Hausa]]). In numerous countries, English and French (''see [[African French]]'') are used for communication in the public sphere such as government, commerce, education and the media. Arabic, [[Portuguese language|Portuguese]], [[Afrikaans]] and Spanish are examples of languages that trace their origin to outside of Africa, and that are used by millions of Africans today, both in the public and private spheres. Italian is spoken by some in former [[Italian Colonial Empire|Italian colonies]] in Africa. German is spoken in [[Namibia]], as it was a former German protectorate.

===Health===
[[File:HIV in Africa 2011.svg|thumb|upright=1.2|Prevalence of HIV/AIDS in Africa, total (% of population ages 15–49), in 2011 ([[World Bank]])
{| style="width:100%;"
|-
| valign=top |
{{legend|#2b0000|over 15%}}
{{legend|#800000|5–15%}}
{{legend|#d40000|2–5%}}
{{legend|#ff2a2a|1–2%}}
{{legend|#ff9955|0.5-1%}}
{{legend|#ffb380|0.1–0.5%}}
{{legend|#b9b9b9|not available}}
|}]]

More than 85% of individuals in Africa use traditional medicine as an alternative to often expensive allopathic medical health care and costly pharmaceutical products. The [[Organisation of African Unity|Organization of African Unity]] (OAU) Heads of State and Government declared the 2000s decade as the African Decade on [[African traditional medicine]] in an effort to promote The WHO African Region's adopted resolution for institutionalizing traditional medicine in health care systems across the continent.<ref>{{Cite journal|last=Kofi-Tsekpo|first=Mawuli|date=2004|title=Institutionalization of African traditional medicine in health care systems in Africa|journal=African Journal of Health Sciences|volume=11|issue=1–2|pages=i–ii|issn=1022-9272|pmid=17298111|doi=10.4314/ajhs.v11i1.30772}}</ref> Public policy makers in the region are challenged with consideration of the importance of traditional/indigenous health systems and whether their coexistence with the modern medical and health sub-sector would improve the equitability and accessibility of health care distribution, the health status of populations, and the social-economic development of nations within sub-Saharan Africa.<ref>{{Cite journal|last=Dunlop|first=David W.|date=November 1975|title=Alternatives to "modern" health delivery systems in Africa: Public policy issues of traditional health systems|journal=Social Science & Medicine|volume=9|issue=11–12|pages=581–586|doi=10.1016/0037-7856(75)90171-7|pmid=817397|issn=0037-7856}}</ref>

[[HIV/AIDS in Africa|AIDS in post-colonial Africa]] is a prevalent issue. Although the continent is home to about 15.2 percent of the world's population,<ref>{{cite web|url=http://www.nationsonline.org/oneworld/world_population.htm|title=World Population by continents and countries – Nations Online Project|access-date=18 March 2015|archive-date=5 January 2014|archive-url=https://web.archive.org/web/20140105110631/http://www.nationsonline.org/oneworld/world_population.htm|url-status=live}}</ref> more than two-thirds of the total infected worldwide – some 35 million people – were Africans, of whom 15 million have already died.<ref name=":0">{{Cite book|title=Encyclopedia of Africa| first1 = Anthony | last1 = Appiah | first2 = Henry Louis | last2 = Gates | name-list-style = vanc |publisher=Oxford University Press|year=2010|pages=8}}</ref> [[Sub-Saharan Africa]] alone accounted for an estimated 69 percent of all people living with HIV<ref name="2012 Facts">{{Cite web |url=http://www.unaids.org/en/media/unaids/contentassets/documents/epidemiology/2012/gr2012/20121120_FactSheet_Global_en.pdf |title="Global Fact Sheet", Joint United Nations Programme on HIV and AIDS, 20 November 2012 |access-date=18 October 2020 |archive-date=27 March 2014 |archive-url=https://web.archive.org/web/20140327233932/http://www.unaids.org/en/media/unaids/contentassets/documents/epidemiology/2012/gr2012/20121120_factsheet_global_en.pdf |url-status=live }}</ref> and 70 percent of all AIDS deaths in 2011.<ref name="dUNAIDSi ck 2012">{{cite web|title=UNAIDS Report on the Global AIDS Epidemic 2012 | url=http://www.unaids.org/en/media/unaids//documents/epidemiology/2012/gr2012/20121120_UNAIDS_Global_Report_2012_with_annexes_en.pdf | access-date=13 May 2013}}</ref> In the countries of sub-Saharan Africa most affected, AIDS has raised death rates and lowered life expectancy among adults between the ages of 20 and 49 by about twenty years.<ref name=":0" /> Furthermore, the life expectancy in many parts of Africa is declining, largely as a result of the HIV/AIDS epidemic with life-expectancy in some countries reaching as low as thirty-four years.<ref>{{Cite book|title=The Oxford Encyclopedia of The Modern World|last=Stearns|first=Peter N. | name-list-style = vanc |publisher=Oxford University Press|year=2008|pages=556}}</ref>

==Culture==
{{Main|Culture of Africa}}

Some aspects of traditional African cultures have become less practised in recent years as a result of neglect and suppression by colonial and post-colonial regimes. For example, African customs were discouraged, and African languages were prohibited in mission schools.<ref name="pearsonhighered.com">{{Cite web|url=http://www.pearsonhighered.com/assets/hip/us/hip_us_pearsonhighered/samplechapter/0205208606.pdf|archive-url=https://web.archive.org/web/20150501070358/http://www.pearsonhighered.com/assets/hip/us/hip_us_pearsonhighered/samplechapter/0205208606.pdf|url-status=dead|title=Pearsonhighered.com|archive-date=1 May 2015}}</ref> Leopold II of Belgium attempted to "civilize" Africans by discouraging polygamy and witchcraft.<ref name="pearsonhighered.com"/>
[[File:Bet Giyorgis church Lalibela 01.jpg|thumb|The rock-hewn [[Church of Saint George, Lalibela|Church of Saint George]] in [[Lalibela]], [[Ethiopia]] is a [[World Heritage Site|UNESCO World Heritage Site]].]]
Obidoh Freeborn posits that colonialism is one element that has created the character of modern African art.<ref>{{Cite journal|url=http://quod.lib.umich.edu/g/gefame/4761563.0002.103/--crisis-of-appropriating-identity-for-african-art-and-artists?rgn=main;view=fulltext|title=The Crisis of Appropriating Identity for African Art and Artists: The Abayomi Barber School Responsorial Paradigm|journal=Gefame|year=2005|last1=Freeborn|first1=Odiboh|access-date=18 December 2015|archive-date=22 December 2015|archive-url=https://web.archive.org/web/20151222185342/http://quod.lib.umich.edu/g/gefame/4761563.0002.103/--crisis-of-appropriating-identity-for-african-art-and-artists?rgn=main;view=fulltext|url-status=live}}</ref> According to authors Douglas Fraser and Herbert M. Cole, "The precipitous alterations in the power structure wrought by colonialism were quickly followed by drastic iconographic changes in the art."<ref name="books.google.com">{{Cite book|url=https://books.google.com/books?id=sSIxOcgE378C&pg=PA95|title=African Art and Leadership|first1=Douglas|last1=Fraser|first2=Herbert M.|last2=Cole|year=2004|publisher=Univ of Wisconsin Press|isbn=978-0-299-05824-1|page=95|access-date=18 December 2015|archive-date=11 June 2020|archive-url=https://web.archive.org/web/20200611043035/https://books.google.com/books?id=sSIxOcgE378C&pg=PA95|url-status=live}}</ref> Fraser and Cole assert that, in Igboland, some art objects "lack the vigor and careful craftsmanship of the earlier art objects that served traditional functions.<ref name="books.google.com"/> Author Chika Okeke-Agulu states that "the racist infrastructure of British imperial enterprise forced upon the political and cultural guardians of empire a denial and suppression of an emergent sovereign Africa and modernist art."<ref>{{Cite book|url=https://books.google.com/books?id=ojPJBwAAQBAJ&pg=PT63|title=Postcolonial Modernism: Art and Decolonization in Twentieth-Century Nigeria|first=Chika|last=Okeke-Agulu|year=2015|publisher=Duke University Press|isbn=978-0-8223-7630-9|page=63|access-date=18 December 2015|archive-date=11 June 2020|archive-url=https://web.archive.org/web/20200611035844/https://books.google.com/books?id=ojPJBwAAQBAJ&pg=PT63|url-status=live}}</ref> Editors F. Abiola Irele and Simon Gikandi comment that the current identity of African literature had its genesis in the "traumatic encounter between Africa and Europe."<ref>{{cite book |year=2000 |publisher=Cambridge University Press |isbn =9781139054638 |doi=10.1017/CHOL9780521832755 |volume=1 |editor-last=Irele |editor-first=F. Abiola |editor-last2=Gikandi |editor-first2=Simon |title=The Cambridge History of African and Caribbean Literature}}</ref> On the other hand, Mhoze Chikowero believes that Africans deployed music, dance, spirituality, and other performative cultures to (re)asset themselves as active agents and indigenous intellectuals, to unmake their colonial marginalization and reshape their own destinies."<ref>{{Cite book|url=https://books.google.com/books?id=o3y9CgAAQBAJ&pg=PA8|page=8|title=African Music, Power, and Being in Colonial Zimbabwe|first=Mhoze|last=Chikowero|year=2015|publisher=Indiana University Press|isbn=9780253018090|access-date=18 December 2015|archive-date=11 June 2020|archive-url=https://web.archive.org/web/20200611043301/https://books.google.com/books?id=o3y9CgAAQBAJ&pg=PA8|url-status=live}}</ref>

There is now a resurgence in the attempts to rediscover and revalue African traditional cultures, under such movements as the [[African Renaissance]], led by [[Thabo Mbeki]], [[Afrocentrism]], led by a group of scholars, including [[Molefi Asante]], as well as the increasing recognition of traditional spiritualism through decriminalization of [[West African Vodun|Vodou]] and other forms of spirituality.

===Visual art ===
{{Excerpt|African art|paragraph=1,2|file=1}}

===Architecture===
{{Excerpt|Architecture of Africa|paragraph=1,2,3|file=1}}

===Music ===
[[File:Ke-Nako Music-Performance Vienna2008c.jpg|thumb|upright=0.7|A musician from South Africa]]
{{Excerpt|Music of Africa|paragraph=1,2}}

===Dance===
{{Excerpt|African dance|paragraph=1,2|file=no}}

===Sports===
[[File:World cup african countries best results and hosts.png|thumb|Best results of African men's national football teams at the FIFA World Cup]]

Fifty-four African countries have [[Association football|football]] teams in the [[Confederation of African Football]]. Egypt has won the African Cup seven times, and a record-making three times in a row. Cameroon, Nigeria, Senegal, Ghana, and Algeria have advanced to the knockout stage of recent [[FIFA World Cup]]s. South Africa hosted the [[2010 FIFA World Cup|2010 World Cup tournament]], becoming the first African country to do so.

In recent years, the continent has made major progress in terms of state-of-the-art [[basketball]] facilities which have been built in cites as diverse as [[Cairo]], [[Dakar]], [[Johannesburg]], [[Kigali]], [[Luanda]] and [[Rades]].<ref>{{cite news |title=Getting to know Africa's flashy basketball arenas |url=https://www.fiba.basketball/news/getting-to-know-africas-flashy-basketball-arenas |access-date=10 December 2020 |work=[[FIBA]] |date=2 September 2019 |archive-date=7 January 2021 |archive-url=https://web.archive.org/web/20210107193242/https://www.fiba.basketball/news/getting-to-know-africas-flashy-basketball-arenas |url-status=live }}</ref> The number of African basketball players who drafted into the [[National Basketball Association|NBA]] has experienced major growth in the 2010s.<ref>{{cite news |first=Lee |last=Nxumalo |title=Basketball's next frontier is Africa |url=https://www.newframe.com/basketballs-next-frontier-is-africa/ |access-date=11 January 2021 |work=New Frame |date=20 December 2020 |archive-date=16 January 2021 |archive-url=https://web.archive.org/web/20210116062357/https://www.newframe.com/basketballs-next-frontier-is-africa/ |url-status=live }}</ref>

[[Cricket]] is popular in some African nations. [[South Africa national cricket team|South Africa]] and [[Zimbabwe national cricket team|Zimbabwe]] have [[Test cricket|Test]] status, while [[Kenya national cricket team|Kenya]] is the leading non-test team and previously had [[One Day International|One-Day International cricket]] (ODI) status (from [[President's Cup 1997-98|10 October 1997]], until [[2014 Cricket World Cup Qualifier#Super Six|30 January 2014]]). The three countries jointly hosted the [[2003 Cricket World Cup]]. [[Namibia national cricket team|Namibia]] is the other African country to have played in a World Cup. [[Morocco]] in northern Africa has also hosted the [[2002 Morocco Cup]], but the national team has never qualified for a major tournament.

[[Rugby union|Rugby]] is popular in several southern African nations. [[Namibia]] and [[Zimbabwe]] both have appeared on multiple occasions at the [[Rugby World Cup]], while South Africa is the joint-most successful national team (alongside New Zealand) at the Rugby World Cup, having won the tournament on 3 occasions, in 1995, 2007, and 2019.<ref>{{Cite web|url=https://www.rugbyworldcup.com/2023/news/608463/rwc-2023-spotlight-south-africa|title=RWC 2023 Spotlight: South Africa ｜ Rugby World Cup 2023|website=www.rugbyworldcup.com}}</ref>

==Territories and regions==
{{Main|List of regions of Africa|List of sovereign states and dependent territories in Africa}}{{-}}
{| style="width:155px; float:right; margin-right:0.0em;"
|[[File:Africa-regions.png|thumb|upright|[[List of regions of Africa|Regions]] of Africa:
{{legend|#2020FF|[[:en:North Africa|North Africa]]}}
{{legend|#40FF40|[[:en:West Africa|West Africa]]}}
{{legend|#FF00FF|[[:en:Central Africa|Central Africa]]}}
{{legend|#FFD000|[[:en:East Africa|East Africa]]}}
{{legend|#FF0A0A|[[:en:Southern Africa|Southern Africa]]}}]]
|-
| 
|-
| 
|-
|}

{{Africa Labelled Map}}

The countries in this table are categorized according to the [[United Nations geoscheme for Africa|scheme for geographic subregions]] used by the United Nations, and data included are per sources in cross-referenced articles. Where they differ, provisos are clearly indicated.



{| class="wikitable sortable" style="margin-left:auto; margin-right:auto; border:1px solid #aaa;"
|- style="background:#ececec;"
! class="unsortable" style="width:20px" | [[Coat of arms|Arms]]
! class="unsortable" style="width:20px" | [[Flag]]
! Name of region<ref>Continental regions as per [[United Nations geoscheme for Africa|UN categorizations/map]]. </ref> and territory, with [[flag]]
! data-sort-type="number" | [[List of countries and dependencies by area|Area]] (km2)
! data-sort-type="number" | [[List of countries and dependencies by population|Population]]<ref name="uscen">{{cite web|url=https://www.census.gov\/cgi-bin/ipc/idbrank.pl|title=IDB: Countries Ranked by Population|date=28 November 1999|url-status=bot: unknown|archive-url=https://web.archive.org/web/19991128111024/http://www.census.gov/cgi-bin/ipc/idbrank.pl|archive-date=28 November 1999}}</ref>
! Year
! data-sort-type="number" | [[List of countries and dependencies by population density|Density]] (per km2)
! Capital
|- style="background:#eee;"
|colspan="8" style="text-align:center;"|'''North Africa'''
|-
| style="text-align:center" | [[File:Seal of Algeria.svg|25px]]
| style="text-align:center" | {{flagicon|Algeria}}
| [[Algeria]]
|style="text-align:right;"|2,381,740
|style="text-align:right;"|34,178,188
|style="text-align:right;"|2009
|style="text-align:right;"|14
|[[Algiers]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Canary Islands}}
| style="text-align:center" | {{flagicon|Canary Islands}}
|[[Canary Islands]] (Spain)<ref>The Spanish [[Canary Islands]], of which [[Las Palmas de Gran Canaria]] are [[Santa Cruz de Tenerife]] are co-capitals, are often considered part of Northern Africa due to their relative proximity to [[Morocco]] and [[Western Sahara]]; population and area figures are for 2001. </ref>
|style="text-align:right;"|7,492
|style="text-align:right;"|2,154,905
|style="text-align:right;"|2017
|style="text-align:right;"|226
|[[Las Palmas de Gran Canaria]], [[Santa Cruz de Tenerife]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Ceuta}}
| style="text-align:center" | {{flagicon|Ceuta}}
|[[Ceuta]] (Spain)<ref>The Spanish [[exclave]] of [[Ceuta]] is surrounded on land by Morocco in Northern Africa; population and area figures are for 2001. </ref>
|style="text-align:right;"|20
|style="text-align:right;"|85,107
|style="text-align:right;"|2017
|style="text-align:right;"|3,575
|—
|-
| style="text-align:center" | {{Coat of arms|text=none|Egypt}}
| style="text-align:center" | {{flagicon|Egypt}}
|[[Egypt]]<ref>[[Egypt]] is generally considered a [[List of transcontinental countries|transcontinental country]] in Northern Africa (UN region) and Western Asia; population and area figures are for African portion only, west of the [[Suez Canal]]. </ref>
|style="text-align:right;"|1,001,450
|style="text-align:right;"|82,868,000
|style="text-align:right;"|2012
|style="text-align:right;"|83
|[[Cairo]]
|-
| style="text-align:center" | [[File:The emblem on the passport of Libya.svg|25px]]
| style="text-align:center" | {{flagicon|Libya}}
|[[Libya]]
|style="text-align:right;"|1,759,540
|style="text-align:right;"|6,310,434
|style="text-align:right;"|2009
|style="text-align:right;"|4
|[[Tripoli]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Madeira}}
| style="text-align:center" | {{flagicon|Madeira}}
|[[Madeira]] (Portugal)<ref>The Portuguese [[Madeira Islands]] are often considered part of Northern Africa due to their relative proximity to Morocco; population and area figures are for 2001. </ref>
|style="text-align:right;"|797
|style="text-align:right;"|245,000
|style="text-align:right;"|2001
|style="text-align:right;"|307
|[[Funchal]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Melilla}}
| style="text-align:center" | {{flagicon|Melilla}}
|[[Melilla]] (Spain)<ref>The Spanish [[exclave]] of [[Melilla]] is surrounded on land by Morocco in Northern Africa; population and area figures are for 2001. </ref>
|style="text-align:right;"|12
|style="text-align:right;"|85,116
|style="text-align:right;"|2017
|style="text-align:right;"|5,534
|—
|-
| style="text-align:center" | {{Coat of arms|text=none|Morocco}}
| style="text-align:center" | {{flagicon|Morocco}}
|[[Morocco]]
|style="text-align:right;"|446,550
|style="text-align:right;"|35,740,000
|style="text-align:right;"|2017
|style="text-align:right;"|78
|[[Rabat]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Tunisia}}
| style="text-align:center" | {{flagicon|Tunisia}}
|[[Tunisia]]
|style="text-align:right;"|163,610
|style="text-align:right;"|10,486,339
|style="text-align:right;"|2009
|style="text-align:right;"|64
|[[Tunis]]
|-
| style="text-align:center" | [[File:Coat of arms of the Sahrawi Arab Democratic Republic.svg|25px]]
| style="text-align:center" | {{flagicon|Western Sahara}}
| [[Western Sahara]]<ref name="Sahrawi Arab Democratic Republic">The territory of [[Western Sahara]] is claimed by the [[Sahrawi Arab Democratic Republic]] and [[Morocco]]. The [[SADR]] is recognized as a sovereign state by the [[African Union]]. [[Morocco]] claims the entirety of the country as its [[Southern Provinces]]. Morocco administers 4/5 of the territory while the SADR controls 1/5. Morocco's annexation of this territory has not been recognized internationally.</ref>
|style="text-align:right;"|266,000
|style="text-align:right;"|405,210
|style="text-align:right;"|2009
|style="text-align:right;"|2
|[[El Aaiún]]
|- style="background:#eee;"
|colspan="8" style="text-align:center;"|'''East Africa'''
|-
| style="text-align:center" | {{Coat of arms|text=none|Burundi}}
| style="text-align:center" | {{flagicon|Burundi}}
| [[Burundi]]
|style="text-align:right;"|27,830
|style="text-align:right;"|8,988,091
|style="text-align:right;"|2009
|style="text-align:right;"|323
|[[Gitega]]
|-
| style="text-align:center" | [[File:Seal of the Comoros.svg|25px]]
| style="text-align:center" | {{flagicon|Comoros}}
| [[Comoros]]
|style="text-align:right;"|2,170
|style="text-align:right;"|752,438
|style="text-align:right;"|2009
|style="text-align:right;"|347
|[[Moroni, Comoros|Moroni]]
|-
| style="text-align:center" | [[File:Emblem of Djibouti.svg|25px]]
| style="text-align:center" | {{flagicon|Djibouti}}
| [[Djibouti]]
|style="text-align:right;"|23,000
|style="text-align:right;"|828,324
|style="text-align:right;"|2015
|style="text-align:right;"|22
|[[Djibouti (city)|Djibouti]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Eritrea}}
| style="text-align:center" | {{flagicon|Eritrea}}
| [[Eritrea]]
|style="text-align:right;"|121,320
|style="text-align:right;"|5,647,168
|style="text-align:right;"|2009
|style="text-align:right;"|47
|[[Asmara]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Ethiopia}}
| style="text-align:center" | {{flagicon|Ethiopia}}
| [[Ethiopia]]
|style="text-align:right;"|1,127,127
|style="text-align:right;"|84,320,987
|style="text-align:right;"|2012
|style="text-align:right;"|75
|[[Addis Ababa]]
|-
| style="text-align:center" | {{Coat of arms|text=none|French Southern and Antarctic Lands}}
| style="text-align:center" | {{flagicon|French Southern and Antarctic Lands}}
| [[French Southern and Antarctic Lands|French Southern Territories]] (France)
|style="text-align:right;"|439,781
|style="text-align:right;"|100
|style="text-align:right;"|2019
|style="text-align:right;"|—
|[[Saint-Pierre, Réunion|Saint Pierre]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Kenya}}
| style="text-align:center" | {{flagicon|Kenya}}
| [[Kenya]]
|style="text-align:right;"|582,650
|style="text-align:right;"|39,002,772
|style="text-align:right;"|2009
|style="text-align:right;"|66
|[[Nairobi]]
|-
| style="text-align:center" | [[File:Seal of Madagascar.svg|25px]]
| style="text-align:center" | {{flagicon|Madagascar}}
| [[Madagascar]]
|style="text-align:right;"|587,040
|style="text-align:right;"|20,653,556
|style="text-align:right;"|2009
|style="text-align:right;"|35
|[[Antananarivo]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Malawi}}
| style="text-align:center" | {{flagicon|Malawi}}
|[[Malawi]]
|style="text-align:right;"|118,480
|style="text-align:right;"|14,268,711
|style="text-align:right;"|2009
|style="text-align:right;"|120
|[[Lilongwe]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Mauritius}}
| style="text-align:center" | {{flagicon|Mauritius}}
|[[Mauritius]]
|style="text-align:right;"|2,040
|style="text-align:right;"|1,284,264
|style="text-align:right;"|2009
|style="text-align:right;"|630
|[[Port Louis]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Mayotte}}
| style="text-align:center" | {{flagicon|Mayotte|local}}
|[[Mayotte]] (France)
|style="text-align:right;"|374
|style="text-align:right;"|223,765
|style="text-align:right;"|2009
|style="text-align:right;"|490
|[[Mamoudzou]]
|-
| style="text-align:center" | [[File:Emblem of Mozambique.svg|25px]]
| style="text-align:center" | {{flagicon|Mozambique}}
|[[Mozambique]]
|style="text-align:right;"|801,590
|style="text-align:right;"|21,669,278
|style="text-align:right;"|2009
|style="text-align:right;"|27
|[[Maputo]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Réunion}}
| style="text-align:center" | {{flagicon|Réunion}}
| [[Réunion]] (France)
|style="text-align:right;"|2,512
|style="text-align:right;"|743,981
|style="text-align:right;"|2002
|style="text-align:right;"|296
|[[Saint-Denis, Réunion|Saint Denis]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Rwanda}}
| style="text-align:center" | {{flagicon|Rwanda}}
| [[Rwanda]]
|style="text-align:right;"|26,338
|style="text-align:right;"|10,473,282
|style="text-align:right;"|2009
|style="text-align:right;"|398
|[[Kigali]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Seychelles}}
| style="text-align:center" | {{flagicon|Seychelles}}
|[[Seychelles]]
|style="text-align:right;"|455
|style="text-align:right;"|87,476
|style="text-align:right;"|2009
|style="text-align:right;"|192
|[[Victoria, Seychelles|Victoria]]
|-
| style="text-align:center" | [[File:Coat of arms of Somalia.svg|25px]]
| style="text-align:center" | {{flagicon|Somalia}}
|[[Somalia]]
|style="text-align:right;"|637,657
|style="text-align:right;"|9,832,017
|style="text-align:right;"|2009
|style="text-align:right;"|15
|[[Mogadishu]]
|-
| style="text-align:center" | [[File:Emblem of Somaliland.svg|25px]]
| style="text-align:center" | {{flagicon|Somaliland}}
|[[Somaliland]]
|style="text-align:right;"|176,120
|style="text-align:right;"|3,508,180
|style="text-align:right;"|2012
|style="text-align:right;"|25
|[[Hargeisa]]
|-
| style="text-align:center" | {{Coat of arms|text=none|South Sudan}}
| style="text-align:center" | {{flagicon|South Sudan}}
|[[South Sudan]]
|style="text-align:right;"|619,745
|style="text-align:right;"|8,260,490
|style="text-align:right;"|2008
|style="text-align:right;"|13
|[[Juba]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Sudan}}
| style="text-align:center" | {{flagicon|Sudan}}
|[[Sudan]]
|style="text-align:right;"|1,861,484
|style="text-align:right;"|30,894,000
|style="text-align:right;"|2008
|style="text-align:right;"|17
|[[Khartoum]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Tanzania}}
| style="text-align:center" | {{flagicon|Tanzania}}
|[[Tanzania]]
|style="text-align:right;"|945,087
|style="text-align:right;"|44,929,002
|style="text-align:right;"|2009
|style="text-align:right;"|43
|[[Dodoma]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Uganda}}
| style="text-align:center" | {{flagicon|Uganda}}
|[[Uganda]]
|style="text-align:right;"|236,040
|style="text-align:right;"|32,369,558
|style="text-align:right;"|2009
|style="text-align:right;"|137
|[[Kampala]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Zambia}}
| style="text-align:center" | {{flagicon|Zambia}}
|[[Zambia]]
|style="text-align:right;"|752,614
|style="text-align:right;"|11,862,740
|style="text-align:right;"|2009
|style="text-align:right;"|16
|[[Lusaka]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Zimbabwe}}
| style="text-align:center" | {{flagicon|Zimbabwe}}
|[[Zimbabwe]]
|style="text-align:right;"|390,580
|style="text-align:right;"|11,392,629
|style="text-align:right;"|2009
|style="text-align:right;"|29
|[[Harare]]
|-
|colspan="8" style="background:#eee; text-align:center;"|'''Central Africa'''
|-
| style="text-align:center" | [[File:Emblem of Angola.svg|25px]]
| style="text-align:center" | {{flagicon|Angola}}
|[[Angola]]
|style="text-align:right;"|1,246,700
|style="text-align:right;"|12,799,293
|style="text-align:right;"|2009
|style="text-align:right;"|10
|[[Luanda]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Cameroon}}
| style="text-align:center" | {{flagicon|Cameroon}}
|[[Cameroon]]
|style="text-align:right;"|475,440
|style="text-align:right;"|18,879,301
|style="text-align:right;"|2009
|style="text-align:right;"|40
|[[Yaoundé]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Central African Republic}}
| style="text-align:center" | {{flagicon|Central African Republic}}
|[[Central African Republic]]
|style="text-align:right;"|622,984
|style="text-align:right;"|4,511,488
|style="text-align:right;"|2009
|style="text-align:right;"|7
|[[Bangui]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Chad}}
| style="text-align:center" | {{flagicon|Chad}}
|[[Chad]]
|style="text-align:right;"|1,284,000
|style="text-align:right;"|10,329,208
|style="text-align:right;"|2009
|style="text-align:right;"|8
|[[N'Djamena]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Republic of the Congo}}
| style="text-align:center" | {{flagicon|Republic of the Congo}}
|[[Republic of the Congo]]
|style="text-align:right;"|342,000
|style="text-align:right;"|4,012,809
|style="text-align:right;"|2009
|style="text-align:right;"|12
|[[Brazzaville]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Democratic Republic of the Congo}}
| style="text-align:center" | {{flagicon|Democratic Republic of the Congo}}
|[[Democratic Republic of the Congo]]
|style="text-align:right;"|2,345,410
|style="text-align:right;"|69,575,000
|style="text-align:right;"|2012
|style="text-align:right;"|30
|[[Kinshasa]]
|-
| style="text-align:center" | [[File:Coat of arms of Equatorial Guinea.svg|25px]]
| style="text-align:center" | {{flagicon|Equatorial Guinea}}
|[[Equatorial Guinea]]
|style="text-align:right;"|28,051
|style="text-align:right;"|633,441
|style="text-align:right;"|2009
|style="text-align:right;"|23
|[[Malabo]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Gabon}}
| style="text-align:center" | {{flagicon|Gabon}}
|[[Gabon]]
|style="text-align:right;"|267,667
|style="text-align:right;"|1,514,993
|style="text-align:right;"|2009
|style="text-align:right;"|6
|[[Libreville]]
|-
| style="text-align:center" | [[File:Coat of arms of São Tomé and Príncipe.svg|25px]]
| style="text-align:center" | {{flagicon|São Tomé and Príncipe}}
| [[São Tomé and Príncipe]]
|style="text-align:right;"|1,001
|style="text-align:right;"|212,679
|style="text-align:right;"|2009
|style="text-align:right;"|212
|[[São Tomé]]
|- style="background:#eee;"
|colspan="8" style="text-align:center;"|'''Southern Africa'''
|-
| style="text-align:center" | {{Coat of arms|text=none|Botswana}}
| style="text-align:center" | {{flagicon|Botswana}}
|[[Botswana]]
|style="text-align:right;"|600,370
|style="text-align:right;"|1,990,876
|style="text-align:right;"|2009
|style="text-align:right;"|3
|[[Gaborone]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Eswatini}}
| style="text-align:center" | {{flagicon|Eswatini}}
|[[Eswatini]]
|style="text-align:right;"|17,363
|style="text-align:right;"|1,123,913
|style="text-align:right;"|2009
|style="text-align:right;"|65
|[[Mbabane]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Lesotho}}
| style="text-align:center" | {{flagicon|Lesotho}}
|[[Lesotho]]
|style="text-align:right;"|30,355
|style="text-align:right;"|2,130,819
|style="text-align:right;"|2009
|style="text-align:right;"|70
|[[Maseru]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Namibia}}
| style="text-align:center" | {{flagicon|Namibia}}
| [[Namibia]]
|style="text-align:right;"|825,418
|style="text-align:right;"|2,108,665
|style="text-align:right;"|2009
|style="text-align:right;"|3
|[[Windhoek]]
|-
| style="text-align:center" | {{Coat of arms|text=none|South Africa}}
| style="text-align:center" | {{flagicon|South Africa}}
|South Africa
|style="text-align:right;"|1,219,912
|style="text-align:right;"|51,770,560
|style="text-align:right;"|2011
|style="text-align:right;"|42
|[[Bloemfontein]], [[Cape Town]], [[Pretoria]]<ref>[[Bloemfontein]] is the judicial capital of South Africa, while [[Cape Town]] is its legislative seat, and [[Pretoria]] is the country's administrative seat. </ref>
|- style="background:#eee;"
|colspan="8" style="text-align:center;"|'''West Africa'''
|-
| style="text-align:center" | {{Coat of arms|text=none|Benin}}
| style="text-align:center" | {{flagicon|Benin}}
| [[Benin]]
|style="text-align:right;"|112,620
|style="text-align:right;"|8,791,832
|style="text-align:right;"|2009
|style="text-align:right;"|78
|[[Porto-Novo]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Burkina Faso}}
| style="text-align:center" | {{flagicon|Burkina Faso}}
| [[Burkina Faso]]
|style="text-align:right;"|274,200
|style="text-align:right;"|15,746,232
|style="text-align:right;"|2009
|style="text-align:right;"|57
|[[Ouagadougou]]
|-
| style="text-align:center" | [[File:Coat of arms of Cape Verde.svg|25px]]
| style="text-align:center" | {{flagicon|Cape Verde}}
| [[Cape Verde]]
|style="text-align:right;"|4,033
|style="text-align:right;"|429,474
|style="text-align:right;"|2009
|style="text-align:right;"|107
|[[Praia]]
|-
| style="text-align:center" | {{Coat of arms|text=none|The Gambia}}
| style="text-align:center" | {{flagicon|The Gambia}}
| [[The Gambia]]
|style="text-align:right;"|11,300
|style="text-align:right;"|1,782,893
|style="text-align:right;"|2009
|style="text-align:right;"|158
|[[Banjul]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Ghana}}
| style="text-align:center" | {{flagicon|Ghana}}
| [[Ghana]]
|style="text-align:right;"|239,460
|style="text-align:right;"|23,832,495
|style="text-align:right;"|2009
|style="text-align:right;"|100
|[[Accra]]
|-
| style="text-align:center" | [[File:Coat of arms of Guinea-new.svg|25px]]
| style="text-align:center" | {{flagicon|Guinea}}
| [[Guinea]]
|style="text-align:right;"|245,857
|style="text-align:right;"|10,057,975
|style="text-align:right;"|2009
|style="text-align:right;"|41
|[[Conakry]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Guinea-Bissau}}
| style="text-align:center" | {{flagicon|Guinea-Bissau}}
| [[Guinea-Bissau]]
|style="text-align:right;"|36,120
|style="text-align:right;"|1,533,964
|style="text-align:right;"|2009
|style="text-align:right;"|43
|[[Bissau]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Ivory Coast}}
| style="text-align:center" | {{flagicon|Ivory Coast}}
|[[Ivory Coast]]
|style="text-align:right;"|322,460
|style="text-align:right;"|20,617,068
|style="text-align:right;"|2009
|style="text-align:right;"|64
|[[Abidjan]],<ref>[[Yamoussoukro]] is the official capital of [[Ivory Coast]], while [[Abidjan]] is the ''de facto'' seat.</ref> [[Yamoussoukro]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Liberia}}
| style="text-align:center" | {{flagicon|Liberia}}
|[[Liberia]]
|style="text-align:right;"|111,370
|style="text-align:right;"|3,441,790
|style="text-align:right;"|2009
|style="text-align:right;"|31
|[[Monrovia]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Mali}}
| style="text-align:center" | {{flagicon|Mali}}
|[[Mali]]
|style="text-align:right;"|1,240,000
|style="text-align:right;"|12,666,987
|style="text-align:right;"|2009
|style="text-align:right;"|10
|[[Bamako]]
|-
| style="text-align:center" | [[File:Seal of Mauritania (2018).svg|25px]]
| style="text-align:center" | {{flagicon|Mauritania}}
|[[Mauritania]]
|style="text-align:right;"|1,030,700
|style="text-align:right;"|3,129,486
|style="text-align:right;"|2009
|style="text-align:right;"|3
|[[Nouakchott]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Niger}}
| style="text-align:center" | {{flagicon|Niger}}
|[[Niger]]
|style="text-align:right;"|1,267,000
|style="text-align:right;"|15,306,252
|style="text-align:right;"|2009
|style="text-align:right;"|12
|[[Niamey]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Nigeria}}
| style="text-align:center" | {{flagicon|Nigeria}}
|[[Nigeria]]
|style="text-align:right;"|923,768
|style="text-align:right;"|166,629,000
|style="text-align:right;"|2012
|style="text-align:right;"|180
|[[Abuja]]
|-
| style="text-align:center" | {{Coat of arms|text=none|United Kingdom}}
| style="text-align:center" | {{flagicon|Saint Helena, Ascension and Tristan da Cunha}}
|[[Saint Helena, Ascension and Tristan da Cunha]] (United Kingdom)
|style="text-align:right;"|420
|style="text-align:right;"|7,728
|style="text-align:right;"|2012
|style="text-align:right;"|13
|[[Jamestown, Saint Helena|Jamestown]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Senegal}}
| style="text-align:center" | {{flagicon|Senegal}}
|[[Senegal]]
|style="text-align:right;"|196,190
|style="text-align:right;"|13,711,597
|style="text-align:right;"|2009
|style="text-align:right;"|70
|[[Dakar]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Sierra Leone}}
| style="text-align:center" | {{flagicon|Sierra Leone}}
|[[Sierra Leone]]
|style="text-align:right;"|71,740
|style="text-align:right;"|6,440,053
|style="text-align:right;"|2009
|style="text-align:right;"|90
|[[Freetown]]
|-
| style="text-align:center" | {{Coat of arms|text=none|Togo}}
| style="text-align:center" | {{flagicon|Togo}}
|[[Togo]]
|style="text-align:right;"|56,785
|style="text-align:right;"|6,019,877
|style="text-align:right;"|2009
|style="text-align:right;"|106
|[[Lomé]]
|- style="font-weight:bold; background:#eee;"
| colspan="3" | Africa Total
|style="text-align:right;"|30,368,609
|style="text-align:right;"|1,001,320,281
|style="text-align:right;"|2009
|style="text-align:right;"|33
|style="background:#eee;"|
|}


==See also==
{{Portal|Africa}}

* [[Index of Africa-related articles]]
* [[African historiography]]
* [[Outline of Africa]]

==References==
{{reflist}}

==Sources==
* {{Cite book|last=Malone|first=Jacqui|url= |title=Steppin' on the Blues: the Visible Rhythms of African American Dance|date=1996|publisher=University of Illinois Press|oclc=891842452}}
* {{Cite book|last=Welsh-Asante|first=Kariamu|url=https://books.google.com/books?id=WrbrTfSO3fwC|title=African Dance|date=2009|publisher=Infobase Publishing|isbn=978-1-4381-2427-8|language=en}}

==Further reading==
{{see also|Africa Bibliography}}
{{refbegin}}
* {{cite book|last=Asante|first=Molefi|author-link=Molefi Asante|title=The History of Africa|publisher=Routledge|location=US|date=2007|isbn=978-0-415-77139-9}}
* {{cite book|last=Clark|first=J. Desmond|author-link=J. Desmond Clark|title=The Prehistory of Africa|publisher=Thames and Hudson|location=London|date=1970|isbn=978-0-500-02069-2}}
* {{cite book|last=Crowder|first=Michael|title=The Story of Nigeria|publisher=Faber|location=London|date=1978|isbn=978-0-571-04947-9}}
* {{cite book|last=Davidson|first=Basil|author-link=Basil Davidson|title=The African Past: Chronicles from Antiquity to Modern Times|publisher=Penguin|location=Harmondsworth|date=1966|oclc=2016817}}
* {{cite book|last=Gordon|first=April A.|author2=Donald L. Gordon|title=Understanding Contemporary Africa|publisher=Lynne Rienner Publishers|location=Boulder|date=1996|isbn=978-1-55587-547-3}}
* {{cite book|last=Khapoya|first=Vincent B.|title=The African experience: an introduction|publisher=Prentice Hall|location=Upper Saddle River, NJ|date=1998|isbn=978-0-13-745852-3|url=https://archive.org/details/africanexperienc00khap}}
* Moore, Clark D., and Ann Dunbar (1968). ''Africa Yesterday and Today'', in series, ''The George School Readings on Developing Lands''. New York: Praeger Publishers.
* [[V. S. Naipaul|Naipaul, V.S.]] ''The Masque of Africa: Glimpses of African Belief''. Picador, 2010. {{ISBN|978-0-330-47205-0}}
* {{cite journal|last1=Wade|first1=Lizzie|title=Drones and satellites spot lost civilizations in unlikely places|journal=Science|doi=10.1126/science.aaa7864|year=2015|doi-access=free}}
{{refend}}

==External links==
{{Sister project links|n=Category:Africa|voy=Africa}}
;General information
* {{curlie|Regional/Africa}}
* [https://www.loc.gov/rr/amed/ African & Middle Eastern Reading Room] from the United States [[Library of Congress]]
* [http://www-sul.stanford.edu/depts/ssrg/africa/ Africa South of the Sahara] from [[Stanford University]]
* [http://www.afrika.no/index/ The Index on Africa] from ''The Norwegian Council for Africa''
* [http://www.aluka.org/ Aluka] Digital library of scholarly resources from and about Africa
* [https://web.archive.org/web/20100117005910/http://www.usaraf.army.mil/MAP_INTERACTIVE/INTERACTIVE_MAP.swf Africa Interactive Map] from the [[United States Army Africa]]

;History
* [https://web.archive.org/web/20190519102901/http://www.africankingdoms.com/ African Kingdoms]
* [https://www.bbc.co.uk/worldservice/africa/features/storyofafrica/index_section10.shtml The Story of Africa] from [[BBC World Service]]
* [http://www.africa.upenn.edu/Urgent_Action/menu_APIC.html Africa Policy Information Center (APIC)]
* [https://www.scribd.com/doc/161017303/Hungarian-military-forces-in-Africa-%E2%80%93-past-and-future-Recovering-lost-knowledge-exploiting-cultural-anthropology-resources-creating-a-comprehensive Hungarian military forces in Africa] {{Webarchive|url=https://web.archive.org/web/20131103230525/http://www.scribd.com/doc/161017303/Hungarian-military-forces-in-Africa-%E2%80%93-past-and-future-Recovering-lost-knowledge-exploiting-cultural-anthropology-resources-creating-a-comprehensive |date=3 November 2013 }}

;News media
* [http://allafrica.com/ allAfrica.com] current news, events and statistics
* [https://www.bbc.co.uk/worldservice/africa/features/focus_magazine/index.shtml ''Focus on Africa''] magazine from [[BBC World Service]]

{{Africa topics}}
{{Africa}}
{{Navboxes
|title = Articles Related to Africa
|list =
{{African Trade Agreements}}
{{Continents of the world}}
{{Regions of the world}}
}}

{{Authority control}}

[[Category:Africa| ]]
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[[Category:Regions]]

Absement

2021-04-28T12:47:59Z

IntegralPython: /* Higher integrals */ rm self-redirect

{{short description|Measure of sustained displacement of an object from its initial position}}
{{Infobox physical quantity
| name = Absement
| image = MotionParticleIntegralSign.svg
| caption = {{longitem|When an object moves, its motion can be described by the integrals of displacement, including absement, absity, abseleration, etc., as well as the derivatives of displacement, including velocity, acceleration, jerk, jounce, etc.}}
| symbols = '''A'''
| unit = Meter second
| baseunits = m·s
| dimension = '''L''' '''T'''
}}
{{Classical mechanics |Fundamentals |width=20.5em}}

[[File:MotionIntegralsDerivativesAbsementActergy.svg|400px|thumb|Integrals and derivatives of displacement, including absement, as well as integrals and derivatives of energy, including actergy. (Janzen et al. 2014)]]

In [[kinematics]], '''absement''' (or '''absition''') is a measure of sustained [[displacement (vector)|displacement]] of an object from its initial [[position (vector)|position]], i.e. a measure of how far away and for how long. The word ''absement'' is a [[portmanteau]] of the words ''absence'' and ''displacement''. Similarly, ''absition'' is a portmanteau of the words ''absence'' and ''position''.<ref name=mann2006>{{cite conference|title=Hydraulophone design considerations: absement, displacement, and velocity-sensitive music keyboard in which each key is a water jet|last1=Mann|first1=Steve|last2=Janzen|first2=Ryan|last3=Post|first3=Mark |date=2006|conference=MM06: The 14th ACM International Conference on Multimedia|location=Santa Barbara, CA|pages=519–528|doi=10.1145/1180639.1180751|publisher=Association for Computing Machinery}}</ref><ref>{{Cite web|url=http://www.thespectrumofriemannium.com/2012/11/10/log053-derivatives-of-position/|title=LOG#053. Derivatives of position|date=2012-11-10|website=The Spectrum Of Riemannium|access-date=2016-03-08|author=Amarashiki}}</ref>

Absement changes as an object remains displaced and stays constant as the object resides at the initial position. It is the first time-[[integral]] of the displacement<ref name=PWTMG>{{cite journal |last1=Pei |first1=Jin-Song |last2=Wright |first2=Joseph P. |last3=Todd |first3=Michael D. |last4=Masri |first4=Sami F. |last5=Gay-Balmaz |first5=François |title=Understanding memristors and memcapacitors in engineering mechanics applications |journal=Nonlinear Dynamics |date=2015 |volume=80 |issue=1–2 |pages=457–489 |doi=10.1007/s11071-014-1882-3|quote=for example, a new concept and state variable called “absement,” the time integral of deformation, emerge}}</ref><ref name=Jeltsema>{{cite journal |last1=Jeltsema |first1=Dimitri |title=Memory Elements: A Paradigm Shift in Lagrangian Modeling of Electrical Circuits |journal=IFAC Proceedings Volumes |date=2012 |volume=45 |issue=2 |pages=445–450 |doi=10.3182/20120215-3-AT-3016.00078|arxiv=1201.1032|quote=Although time-integrated charge is a somewhat unusual quantity in circuit theory, it may be considered as the electrical analogue of a mechanical quantity called absement.}}</ref> (i.e. absement is the area under a displacement vs. time graph), so the displacement is the rate of change (first time-[[derivative]]) of the absement. The [[dimensional analysis|dimension]] of absement is [[length]] multiplied by [[time in physics|time]]. Its [[SI unit]] is '''meter second''' (m·s), which corresponds to an object having been displaced by 1 [[meter]] for 1 [[second]]. This is not to be confused with a [[meter per second]] (m/s), a unit of [[velocity]], the time-derivative of position.

For example, opening the gate of a gate valve (of rectangular cross section) by 1 mm for 10 seconds yields the same absement of 10 mm·s as opening it by 5 mm for 2 seconds. The amount of water having flowed through it is linearly proportional to the absement of the gate, so it is also the same in both cases.<ref>[[Maya Burhanpurkar]]. ''Absement: Direct Evidence of the Time-Integral of Distance''. Canada-Wide Science Fair 2014.</ref>

==Occurrence in nature==
Whenever the rate of change {{var|f}}′ of a quantity {{var|f}} is proportional to the displacement of an object, the quantity {{var|f}} is a linear function of the object's absement. For example, when the fuel [[volumetric flow rate|flow rate]] is proportional to the position of the throttle lever, then the total amount of fuel consumed is proportional to the lever's absement.

The first published paper on the topic of absement introduced and motivated it as a way to study
flow-based musical instruments, such as the [[hydraulophone]], to model empirical observations of some hydraulophones in which obstruction of a water jet for a longer period of time resulted in a buildup in [[Loudness|sound level]], as [[water]] accumulates in a [[sounding mechanism]] (reservoir), up to a certain maximum filling point beyond which the sound level reached a maximum, or fell off (along with a slow decay when a water jet was unblocked).<ref name=mann2006/> Absement has also been used to model artificial muscles,<ref>ROBUST CONTROL LAW FOR PNEUMATIC ARTIFICIAL MUSCLES, Jonathon E. Slightam and Mark L. Nagurka, Proceedings of the ASME/Bath 2017 Symposium on Fluid Power and Motion Control, FPMC 2017, October 16-19, 2017, Sarasota, USA</ref> as well as for real muscle interaction in a physical fitness context.<ref>Effectiveness of Integral Kinesiology Feedback
for Fitness-based Games, Steve Mann, Max Lv Hao, Ming-Chang Tsai, Maziar Hafezi, Amin Azad, and Farhad Keramatimoezabad, 2018 IEEE Games, Entertainment, Media Conference (GEM), pages 43-50</ref> Absement has also been used to model human posture.<ref>Postural strategy for mediolateral weight shifting in healthy adult, J Tousignant, C Cherriere, A Pouliot-Laforte, É Auvinet, Gait & Posture, 2018 - Elsevier</ref>

As the displacement can be seen as a mechanical analogue of [[electric charge]], the absement can be seen as a mechanical analogue of the time-integrated charge, a quantity useful for modelling some types of memory elements.<ref name=Jeltsema/>

==Applications==
In addition to modeling fluid flow and for lagrangian modeling of electric circuits (Jeltsema 2012), absement is used in physical fitness and kinesiology to model muscle bandwidth, and as a new form of physical fitness training.<ref>"Actergy as a Reflex Performance Metric: Integral-Kinematics Applications", Janzen etal., in Proceedings of the IEEE GEM 2014, pp. 311-2. {{doi|10.1109/GEM.2014.7048123}}</ref><ref>"Integral Kinematics (Time-Integrals of Distance, Energy, etc.) and Integral Kinesiology", by Mann etal, in Proceedings of the IEEE GEM 2014, pp. 270-2.</ref> In this context, it gives rise to a new quantity called ''actergy'', which is to energy as energy is to power. Actergy has the same units as action (joule-seconds) but is the time-integral of total energy (time-integral of the Hamiltonian rather than time-integral of the Lagrangian).

Fluid flow in a throttle:
{{quote|A vehicle's distance travelled results from its throttle's absement. The further the throttle has been opened, and the longer it's been open, the more the vehicle's travelled.|}}

==Relation to PID controllers==
[[PID controller]]s are controllers that work on a signal that is proportional to a physical quantity (e.g. displacement, proportional to position) and its integral(s) and derivative(s), thusly defining PID in the context of Integrals and Derivatives of a position of a control element in the Bratland sense<ref>{{cite journal |last1=Bratland |first1=Magne |last2=Haugen |first2=Bjørn |last3=Rølvåg |first3=Terje |title=Modal analysis of active flexible multibody systems containing PID controllers with non-collocated sensors and actuators |journal=Finite Elements in Analysis and Design |date=2014 |volume=91 |pages=16–29 |doi=10.1016/j.finel.2014.06.011}}</ref>

{{quote|depending on the type of sensor inputs, PID controllers can contain gains proportional to position, velocity, acceleration or the time integral of position (absement)…||Bratland et al.}}

Example of PID controller (Bratland 2014):
* P, Position;
* I, Absement;
* D, Velocity.

==Higher integrals==
Just as displacement and its derivatives form kinematics, so do displacement and its integrals form "integral kinematics" (Janzen ''et al.'' 2014), giving rise to the ordered list of ''n''-th derivatives of displacement:
{{ordered list
| start=-7
|3=Absop|4=Absackle|5=Absnap|6=Abserk|7=Abseleration|8=Absity|9='''Absement'''|10=[[Displacement (vector)|Displacement]]|11=[[Velocity]]|12=[[Acceleration]]|13=[[Jerk (physics)|Jerk/Jolt]]|14=[[Snap (physics)|Snap/Jounce]]|15=[[Crackle_(physics)|Crackle/Flounce]]|16=[[Pop (physics)|Pop/Pounce]]}}

==Strain absement==
Strain absement is the time-integral of [[strain (mechanics)|strain]], and is used extensively in mechanical systems and memsprings:
: a quantity called absement which allows mem-spring models to display hysteretic response in great abundance.<ref name=PWTMG/>

==Anglement==
Absement originally arose in situations involving valves and fluid flow, for which the opening of a valve was by a long "T"-shaped handle, which actually varied in angle rather than position. The time-integral of angle is called "anglement" and it is approximately equal or proportional to absement for small angles, i.e. the sine of an angle is approximately equal to the angle for small angles.<ref>Integral Kinesiology Feedbackfor Weight and Resistance Training, 2019 15th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS), http://wearcam.org/sitis2019.pdf</ref>

==References==
{{reflist}}

{{Kinematics}}
{{Classical mechanics derived SI units}}

[[Category:Motion (physics)]]
[[Category:Vector physical quantities]]

Talk:Infinite monkey theorem

2021-03-25T21:45:08Z

IntegralPython: /* Archiving request */ consensus?

{{Talk header}}
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{{dashboard.wikiedu.org assignment | course = Wikipedia:Wiki_Ed/Rowan_College_at_Burlington_County/Society,_Ethics,_and_Technology_(Summer) | assignments = [[User:Transuejames|Transuejames]] | start_date = 2019-05-16 | end_date = 2019-08-24 }}
{{Annual readership}}

== Real Monkeys ==
The section Real Monkeys is one of the funniest things I ever had the joy to read. Thank you very much. [[Special:Contributions/78.22.179.27|78.22.179.27]] ([[User talk:78.22.179.27|talk]]) 13:42, 18 November 2016 (UTC)

:Much as I appreciate Shakespeare, sometimes I think the "performance art" of the real monkeys has more relevance to our contemporary world.--[[User:Jack Upland|Jack Upland]] ([[User talk:Jack Upland|talk]]) 09:17, 21 November 2016 (UTC)

I agree wholeheartedly; this entire article made me laugh, but this section was the best [[User:IntegralPython|IntegralPython]] ([[User talk:IntegralPython|talk]]) 19:20, 7 December 2018 (UTC)

=== Date Correction Needed? ===
Is the date for the real monkeys incorrect? Wikipedia says the experiment occurred in 2003, but the copyright date in the linked PDF says 2002. Where did the 2003 date come from? [[Special:Contributions/99.251.11.183|99.251.11.183]] ([[User talk:99.251.11.183|talk]]) 04:45, 30 August 2020 (UTC)

== Swift and Pascal bits are missing ==

The history section mentions Swift and Pascal in passing, but unlike the Aristotle and Cicero bits, the Swift and Pascal references are never explained or quoted. I found the Swift bit in the popular culture article (which I am working on at the moment), but don't know where to find the Pascal bit. The Swift bit should be added:
*1782 - [[Jonathan Swift]]'s ''[[Gulliver's Travels]]'' (1782) anticipates the central idea of the theorem, depicting a [[professor]] of the Grand Academy of Lagado who attempts to create a complete list of all knowledge of science by having his students constantly create random strings of letters by turning cranks on a mechanism (Part three, Chapter five): although his intention was more likely to parody [[Ramon Llull]].
And someone should find and add the Pascal bit. Thanks. [[User:Carcharoth|Carcharoth]] 14:27, 12 August 2007 (UTC)

::Hi. I have found a book by Swift published 1774 (there are many editions - I found this particular edition on [http://www.jischistoricbooks.ac.uk/ JISC Historic Books]). ''The works of Jonathn. Swift, D.D.: D.S.P.D. with notes historical and critical.'' pg 176-182 (By J. Hawkesworth, L.L.D. and others. Printed for J. Williams, Dublin Library : Bodleian Library (Oxford) - Accessed via www.jischistoricbooks.ac.uk) In this is the Part III Chapter 5. The machine created by the professor in the academy has words, pronouns, punctuation and all the parts of speech written on individual tablets. The tablets are all put into the machine and the handle is turned. The students of the professor then read off the sentence that is formed and it is written down. "..whereby by his contrivance the most ignorant person at a reasonable charge and with a little bodily labour might write books in philosophy, poetry, politics, law, mathematics, and theology, without the least assistance from genius or study.".
:: There is a note to this chapter in this edition by Lord Charles Boyle, Earl of Orrery:
:::"The project for a more easy and expeditious method of writing a treatise in any science, by a wooden engine, is entertainingly satirical; and is aimed at those authors who, instead of receiving materials from their own thoughts and observations, collect from dictionaries and common-place books, an irregular variety, without order, use or design: "ut nec pes nec caput uni, reddatur formae". Orrery." (The latin is a reference to [[Horace]] "where the feet and the head have no relation to the other parts").
::The addition of the quote from Orrery in this edition of Swift's Works is interesting. It puts Swift and Bentley in the same group. Bentley's ''Confutation of Atheism'' (1692) was part of the Boyle Lectures. (See my section below ''History - references preceding Borel''). And according to the page, [[Gullivers Travels]], the book was read by almost everyone when it was published in 1726. Thus almost everyone would have this idea of a machine pumping out sentences from random assortments of words and letters.
::In the same edition is Johnathan Swift's essay, ''A Tritical Essay upon the Faculties of the Mind'', about which Swift's footnote reads, "in a farcical, satiric light, designed purely to expose the folly and temerity of those brainless, illiterate scriblers (sic), who are eternally plaguing their contemporaries with a parcel of wild, incoherent, nonsensical trash. Swift." It is in this essay that Swift talks about the jumbling up of letters - but in a similar vein to Bentley without the monkey:
:::"And if this be so, how can the Epicurean's opinion be true, that the universe was formed by a fortuitous concourse of atoms; which I will no more believe, than that the accidental jumbling of letters of the alphabet could fall by chance into a most ingenious and learned treatise of philosophy. ''Risum teneatis amici?'' [[Horace]] (latin: "Could you refrain from laughing, my friend?" from the same passage of Horace as Orrery quotes.)
::In fact another book by [[William Wotton]] ''Reflections upon ancient and Modern Learning'', Wotton complains about what he considers to be Swift's satirical attack on himself and Richard Bentley in Swift's ''Tale of the Tub''. [[User:Zorgster|Zorgster]] ([[User talk:Zorgster|talk]]) 16:11, 23 January 2012 (UTC)

== Later history section? ==

Is there a possibility for a rigorously sourced selection of examples from the 'in popular culture' article being integrated to this article under the title 'Later history' or 'Recent history'? At the moment, I'm rigorously sourcing the examples and re-ordering them in date order. I'm also turning up papers that mention not the mathematics or the early history and development of the idea, but rather of the current history and usage of the idea in literature and elsewhere. In other words, rather than being a "here are some examples", it would become "here is a history of the later use of the idea". [[User:Carcharoth|Carcharoth]] 14:34, 12 August 2007 (UTC)

== New paper on this topic - Monkeying Around with Text ==

Please see [http://www.chass.utoronto.ca/epc/chwp/CHC2005/Butler/Butler.htm Monkeying Around with Text], Terry Butler, University of Alberta, ''Computing in the Humanities Working Papers'', January 2007. I've used this as a reference over at [[Infinite monkey theorem in popular culture]], and I think it will be useful here as well. [[User:Carcharoth|Carcharoth]] 18:26, 12 August 2007 (UTC)

== Expanded summary for popular culture section ==

I've now added a summary for the popular culture section, based on my rewrite of the daughter article [[Infinite monkey theorem in popular culture]]. See my edit [http://en.wikipedia.org/w/index.php?title=Infinite_monkey_theorem&diff=151588967&oldid=150735379 here]. [[User:Carcharoth|Carcharoth]] 12:11, 16 August 2007 (UTC)

== Stephen Ballentyne ==

Editors of this page may wish to be wary of including material by the philosopher of mathematics [[Stephen Ballentyne]] until there is evidence that such an individual exits ''and'' has been published. --[[User:Digby Tantrum|Mark H Wilkinson]] ([[User talk:Digby Tantrum|t]], [[Special:Contributions/Digby Tantrum|c]]) 19:12, 16 September 2007 (UTC)

:Yes, even though Ballentyne seems to exist, publishing is key. The edits to this article are confused, and if they are based on a published source, I would be very interested in learning which publisher endorsed it and what exactly it said. [[User:Melchoir|Melchoir]] 04:41, 17 September 2007 (UTC)

::Apparently this person is now teaching Religious Studies at [[Uppingham School]].[http://www.uppingham.co.uk/rpmserver/Uppingham/rpmHtml_content/autoGen15-552.htm] Unless he publishes in a reputable journal, and his point of view is discussed by multiple independent published sources, his opinions are just as non-notable and unencyclopedic as those expressed at the local pub.  --[[User talk:Lambiam|Lambiam]] 07:04, 17 September 2007 (UTC)

:::Or that could be another individual entirely. The Ballentyne edits have been introduced by two new user accounts, apparently set up for the sole purpose of pushing this issue ([http://en.wikipedia.org/wiki/Special:Contributions/Merisalis], [http://en.wikipedia.org/wiki/Special:Contributions/Yoshikawa]).

:::At least we're getting a better class of vandal. --[[User:Digby Tantrum|Mark H Wilkinson]] ([[User talk:Digby Tantrum|t]], [[Special:Contributions/Digby Tantrum|c]]) 09:03, 17 September 2007 (UTC)

:::It's a shame that the Internet appears to be the sole source of verification for published source materials and that no evidence of Ballentyne's published material exists (as yet) on the Internet. My fellow students and I will strive to correct this omission by writing to the editors of the journals and books he has contributed to. It is sad that material about monkeys urinating on typewriters and repetitive computerised random number experiments are present in this article, instead of the dynamic range of mathematical philosophies that exist on the subject. --[[User talk:Merisalis|Merisalis]] 04:33, 18 September 2007 (UTC)

::::Funny thing about the internet is that the work of noted academics tends to turn up in Google searches. For example: [http://www.google.co.uk/search?hl=en&q=%22j+f+toland%22&meta= Prof JF Toland] is exceedingly easy to find. --[[User:Digby Tantrum|Mark H Wilkinson]] ([[User talk:Digby Tantrum|t]], [[Special:Contributions/Digby Tantrum|c]]) 10:41, 18 September 2007 (UTC)

== Evolution? ==

Excuse me, what is the relevance of this to the article?

"Various Christian apologists on the one hand, and Richard Dawkins on the other, have argued about the appropriateness of the monkeys as a metaphor for evolution." —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/128.122.20.71|128.122.20.71]] ([[User talk:128.122.20.71|talk]]) 14:23, 18 September 2007 (UTC) 

:Evolution is probably the most common and most important context for the infinite monkey theorem in modern popular culture, and there is a section of the article describing how. The [[Wikipedia:Lead section]] provides a one-sentence summary of that section.
:I've reverted a bunch of recent edits to the lead that had little basis and removed information. I've also reverted the "However" paragraph from the "Real monkeys" section; the section already states that monkeys are not random number generators, and we don't need arguments that experiments are "unnecessary". [[User:Melchoir|Melchoir]] 01:00, 3 November 2007 (UTC)

== I don't get it... :P ==

Infinity is an unending period of time. Why does the term "almost surely" apply? Shouldnt it just be "certainly"? After all, there are no limits. -- [[User:Anonymous Dissident|Anonymous Dissident]][[User_talk:Anonymous Dissident|Talk]] 09:42, 21 December 2007 (UTC)

:Imagine that a fair coin is tossed infinitely often. Denoting the two sides as 0 and 1, this gives an infinite sequence of bits, something like
::0111001110001111111111010011110100101001110111100010010111001011011100010011101110101110110010001010...
:(space limitations do not allow to show the full sequence). What is the chance of getting ''exactly'' this sequence? In fact, all possible sequences are just as likely, such as the sequence
::1011010001100111000011101111110110011111011011110011000111011011101111110111001011011110010100011011...,
:or, for that matter, the sequence
::0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000....
:So if it is ''certain'' that the last sequence will not occur, it is ''just as certain'' that any other sequence, such as the two above it, will not occur, including the one that actually ''is'' the result of tossing the coin infinitely often. Under the normal meaning of the word ''certain'', that is a contradiction.  --[[User talk:Lambiam|Lambiam]] 18:56, 21 December 2007 (UTC)

I had the exact identical question back in the section entitled (appropriately enough) "Question". Refer to Dcoetzee's answer there. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/75.163.233.26|75.163.233.26]] ([[User talk:75.163.233.26|talk]]) 17:35, 29 March 2008 (UTC) 

This looks like a case of approaches infinity vs. infinity. Similarly p(x)->1 not p(x) identical to 1. [[Special:Contributions/68.144.80.168|68.144.80.168]] ([[User talk:68.144.80.168|talk]]) 12:30, 19 June 2008 (UTC)

It is definitely not surely. If a monkey presses a button at random, that means it is possible that it can repeatedly press the same button each time. The probability of this continually happening is very small, but it can happen. For this reason it is impossible to 'guarantee' that the works of Shakespeare would ever be produced. It only becomes more likely as more time is given. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/79.68.196.135|79.68.196.135]] ([[User talk:79.68.196.135|talk]]) 14:31, 27 June 2008 (UTC) 

== "Huxley" &c. ==

Does it not seem a little silly to put a picture of Huxley up when the article does not actually owe much to the man? He simply was attributed wrongly, and had a nice little quip about the bishop. The only other illustration contained in this article illustrates the article beautifully. Huxley, however, has little to do with this. I'm just being picky, but it threw me for a second. It seems like unnecessary emphasis on an only mildly meaningful character (in the context of this article).[[Special:Contributions/130.101.14.214|130.101.14.214]] ([[User talk:130.101.14.214|talk]]) 19:35, 2 February 2010 (UTC)

The "Evolution" section currently begins:
:In his 1931 book ''The Mysterious Universe'', Eddington's rival [[James Hopwood Jeans|James Jeans]] attributed the monkey parable to a "Huxley", presumably meaning [[Thomas Henry Huxley]]. This attribution is incorrect.
The footnote for this bold statement includes the following citation: '''{{cite journal |first=Thanu |last=Padmanabhan |title=The dark side of astronomy |journal=Nature |volume=435 |pages=20–21 |year=2005 |doi=10.1038/435020a}}'''. However, I cannot figure out how that article relates to the Infinite Monkeys, let alone to whether or not "Huxley" invented that formulation. Am I missing something, or has there been an error?

Also:
:Borges follows the history of this argument through [[Blaise Pascal]] and [[Jonathan Swift]], then observes that in his own time, the vocabulary had changed. By 1939, the idiom was "that a half-dozen monkeys provided with typewriters would, in a few eternities, produce all the books in the British Museum."

Interestingly, when Borges says this, he too attributes it to a "Huxley":
:Siglo y medio más tarde, tres hombres justifican a Demócrito y refutan a Cicerón. En tan desaforado espacio de tiempo, el vocabulario y las metáforas de la polémica son distintos. Huxley (que es uno de esos hombres) no dice que los "caracteres de oro" acabarán por componer un verso latino, si los arrojan un número suficiente de veces; dice que media docena de monos, provistos de máquinas de escribir, producirán en unas cuantas eternidades todos los libros que contiene el British Museum.
I would be interested in knowing the history of this attribution. Is there a kernel of truth in it? I guess I'll have to check ''Respectfully quoted: a dictionary of quotations'' (the other authority cited in the aforementioned footnote) when I get home, but in the meantime can anyone elucidate this? It seems like we should have some firm sources if we're going to categorically state that the "attribution is incorrect," right? --[[User:Iustinus|Iustinus]] ([[User talk:Iustinus|talk]]) 23:44, 28 December 2007 (UTC)
:Should have specified: since Borges wrote in 39, he's presumably repeating the claim from Jeans. But it's still very interesting that hementions it. --[[User:Iustinus|Iustinus]] ([[User talk:Iustinus|talk]]) 00:50, 29 December 2007 (UTC)

::I think the attribution to Huxley is correct. To Julian Huxley, not Thomas Huxley. Six monkeys, infinite years... (one source: [http://books.google.co.uk/books?id=0YiXM-x--4wC&pg=PA210#v=onepage&q=Huxley%20six%20monkeys%20Julian&f=false Universal Book of Mathematics]). Eddington, Jeans, Huxley were all contemporaries (see [http://books.google.co.uk/books?id=KE46AAAAMAAJ&q=Huxley God and the Universe]) [[User:Zorgster|Zorgster]] ([[User talk:Zorgster|talk]]) 20:05, 16 January 2012 (UTC)

:: I have found an article from ''Irishman's Diary'' (The Irish Times, May 18, 1939. Accessed via ProQuest Historical Newspapers: The Irish Times (1859-2007)). There is a short piece titled "Monkeys by the Million", the author reports listening to a radio program in which the host mentioned a friend from Aberdeen told him "if you had a sufficiently large number of monkeys thumping the keys of an equally large number of typewriters for an indefinite period of time, you eventually would produce all Shakespeare's plays.". A Scottish friend of the author disputes this, "..because if you covered the earth with typewriters and monkeys, with a patch three-foot square in the middle for the Editor of the ''Irish Times'' to sit on - if it would hold him - and to take observations; and if you set these said monkeys knocking the keys of each typewriter about a line a minute, say 40 characters, the chances that at end of the year one of them had produced 'To be or not to be, that is the question' is about one in a million million million million million million million". This therefore differs from the comment above that "by 1939, the idiom was" - it is highly likely that by 1939 the idiom had taken a multitude of forms. [[User:Zorgster|Zorgster]] ([[User talk:Zorgster|talk]]) 17:28, 21 January 2012 (UTC)

:: From The Manchester [[The_Guardian|Guardian]] Sept 12th 1928, pg 5, ''The British Association'':
:: The article discusses the talk given by Professor [[Frederick George Donnan]] at the Annual Meeting. He is discussing a discovery made by physiologist [[Archibald Hill|A V. Hill]] regarding cell respiration and the constant need for oxygen. It is not clear whether the Guardian reporter is making this comment, or whether Prof. Donnan said it:
:::"... but according to the statistical theory of probability if we waited long enough anything that was possible, no matter how improbable, would happen. If six monkeys were set before six typewriters it would be a long time before they produced by mere chance all the written books in the British Museum, but it would not be an infinitely long time." (Guardian, Sep 12 1928, pg 5) [[User:Zorgster|Zorgster]] ([[User talk:Zorgster|talk]])

:: From [[The Guardian]] - Feb 5th 1927 - ''A Definition of Extreme Improbability'':
:: This article discusses a talk on '''Feb 4th 1927''' by [[Arthur Eddington]] at the 2nd [[Gifford Lectures]] titled ''The Nature of the Physical World'' (of which the book is referenced in the main article), in which he discusses entropy in the universe. Eddington's metaphor for entropy is of air spreading out in a box and the article adds (or reports) that the chance of all the air ending up in one half is very low... "The chances against this are greater than that an army of monkeys drumming on an array of typewriters should by accident compose all the books in the British Museum.". This attribution is earlier than that cited in the main article, but I cannot confirm it is the words of the reporter or Eddington. [[User:Zorgster|Zorgster]] ([[User talk:Zorgster|talk]]) 19:11, 21 January 2012 (UTC)

== assumptions ==

It must be made very clear here what we take on faith in our definition of "infinite". Even if "infinity" as a logical abstraction is comfortable and acceptable in the exact branches of mathematics, bringing it into statistics raises innumerable difficulties, not all of which are mathematical. --[[User:VKokielov|VKokielov]] ([[User talk:VKokielov|talk]]) 05:05, 4 February 2008 (UTC)

== One of the worst Wikipedia articles ? ==

And it also has an error - in the Direct Proof section, please note the while the individual keystrokes are independent (by assumption), the blocks of 6 letters are NOT, since they overlap. and if they not overlap, there is the possibility of |1Qp'''BAN'''| |'''ANA'''hlp|, which is not accounted for. Either way, the proof is incorrect.

[[User:Zermalo|zermalo]] ([[User talk:Zermalo|talk]]) 20:02, 17 March 2008 (UTC)

:Your claim is not constructive. Please point out which particular sentence you believe is incorrect by directly quoting it here. [[User:Melchoir|Melchoir]] ([[User talk:Melchoir|talk]]) 01:32, 18 March 2008 (UTC)
:I think this is covered pretty well by footnote 1. Does it need to be in the body of the article? [[User talk:Algebraist|Algebraist]] 14:29, 5 May 2008 (UTC)

Whether or not the text is organized into blocks is irrelevant to the proof. Organizing the text into blocks of, say, 6 implies that we will be making 1/6th as many samples which in turn implies that the time needed will be 6 times longer. The samples are still independant. Let us examine the alternative. No blocks: There are two variations to this problem. Case 1: Will "poem" be replicated. In this case, overlapping samples can be considered independant. 'Nuff said. Case 2: searching for the first instance of "poem". This case is harder, (as samples demonstrate dependance), but we can still fall back on the blocks (or even case 1). [[Special:Contributions/68.144.80.168|68.144.80.168]] ([[User talk:68.144.80.168|talk]]) 12:48, 19 June 2008 (UTC)

:The proof is correct, because if the probability of BANANA occurring on a block boundary is 1, then any ''more'' likely event has the same probability. There's no need for a precise analysis here. [[User:Dcoetzee|Dcoetzee]] 01:45, 28 June 2008 (UTC)

::Oh, dear! I hadn't noticed the claim. It is wrong, but the errors also really do not matter. See the "Definitely wrong but morally right" section ''infra''. [[User:JoergenB|JoergenB]] ([[User talk:JoergenB|talk]]) 20:20, 7 October 2008 (UTC)

==The Brainiac Experiment==

When I came to this page I immediately thought of an experiment conducted on Brainiac testing this idea. They sat several monkeys and several "Brainiacs" down at computers, and the closest they could get to Shakespeare's works was when one monkey typed "alas" in the whole six hours. I would put this on the page, but I don't think I'd do a very good job of it and I'm relatively new to Wikipedia, so please could someone else who knows more about what they're doing add this? Thanks:) [[User:Welsh-girl-Lowri|Lowri]] ([[User talk:Welsh-girl-Lowri|talk]]) 17:27, 22 July 2008 (UTC)

:Do you have a reliable source for this story? Surely anyone of the "Brainiacs" could have typed ''To be or not to be''; in fact almost any English-speaking "Joe Shmoe" would know that much Shakespeare. What have the computers to do with it?  --[[User talk:Lambiam|Lambiam]] 00:04, 27 July 2008 (UTC)
::Is the story itself not reliable enough? Well, it doesn't matter anyway as their experiment really didn't prove anything, they probably just did it for the "lulz". [[Special:Contributions/193.44.6.146|193.44.6.146]] ([[User talk:193.44.6.146|talk]]) 21:24, 31 July 2008 (UTC)

== How tiny is very tiny? ==

In a recent edit, the sentence
:"''The [[probability]] of a monkey typing a given string of text as long as, say, ''[[Hamlet]]'', is so tiny that, were the experiment conducted, the chance of it actually occurring during a span of time of the order of the [[age of the universe]] is minuscule but not zero.''"
was changed to:
:"''The [[probability]] of a monkey typing a given string of text as long as, say, ''[[Hamlet]]'', is very tiny, but not zero.''"
The edit summary stated:
:"''This is redundant and wrong'' ".
I don't see what is wrong with this, assuming the [[age of the universe]] is not significantly more than 10100000 years and that the monkey does not type significantly faster than 10100000 characters per second. Please enlighten me.  --[[User talk:Lambiam|Lambiam]] 23:41, 12 August 2008 (UTC)
:<s>You might want to knock a zero off the second exponent (I'm not sure of the average word length of Hamlet, so I'm being conservative), but otherwise</s> You're right. [[User talk:Algebraist|Algebraist]] 23:52, 12 August 2008 (UTC)
::I see the full edit summary was ''This is redundant and wrong, especially interesting to be wrong only 2 lines after explaining the perils of reasoning in this exact way...:)'', referring to 'the perils of reasoning about infinity by imagining a vast but finite number, and vice versa'. Perhaps [[User:Diza|Diza]] was imagining an infinite age of the universe? [[User talk:Algebraist|Algebraist]] 23:55, 12 August 2008 (UTC)

Yes, the shortened version, while true, is a bit of an empty statement - it's a bit obvious. But the original is also vague - we've no idea what "the chance of" is actually referring to, or even what "the experiment" is. The theorem is only meaningful in an idealised, thought-experiment sense. I suspect any re-write eliminating this vagueness would render it too wordy for an introductory paragraph. I can't imagine calculations about the universe or the laws of physics having any place here - there's plenty of scope for them in the main body. I'd say the sentence is both redundant and vague, and the section works better without it.[[User:Bobathon71|Bob D]] ([[User talk:Bobathon71|talk]]) 00:14, 13 August 2008 (UTC)

:The antecedent of "''it''" in the original is obviously "''a monkey typing a given string of text as long as, say, ''[[Hamlet]]". In the preceding sentences of the lede it has just been explained that the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a [[random sequence]] of letters [[ad infinitum]], which in an [[infinite]] amount of time will almost surely produce a given text, such as the complete works of [[William Shakespeare]]. At least to me, it appears so obvious that the experiment is to let an abstract device produce a random sequence of letters for an indefinite amount of time that I don't think this needs to be explicated.  --[[User talk:Lambiam|Lambiam]] 13:05, 17 August 2008 (UTC)

::Since 'it' has no reference to a time scale or rate, so there is no meaning to the 'chance of it occurring' within a given time.[[User:Bobathon71|Bob D]] ([[User talk:Bobathon71|talk]]) 06:28, 18 August 2008 (UTC)

:::The "chance of it occurring" in question is, of course, a non-constant function of the typing rate. This does not make it meaningless, nor does it prevent us from observing that it is miniscule in any experiment. [[User:Melchoir|Melchoir]] ([[User talk:Melchoir|talk]]) 07:01, 18 August 2008 (UTC)

== 130.000 (or actually more) ? ==

Take Hamlet from http://www.gutenberg.org/dirs/etext98/2ws2610.txt and pipe it through

perl -pe 's/\[.*?\]//; s/[\s\,\.\-\;]//g'|wc

and you'll see it's just a little less than 130.000 characters.

Not that it matters much, but still :) [[User:Rkarlsba|Rkarlsba]] ([[User talk:Rkarlsba|talk]]) 03:55, 3 September 2008 (UTC)

== Definitely wrong but morally right ==

The section '''Direct proof''' contains a fallacious statement (whence of course its proof also is not quite correct). At the same time, the error is of no importance for the thesis of the article; instead of the proposed exactly [[exponent]]ial expression, you get an approximately exponential expression, not with the same but with a rather similar [[base]], and thus the conclusion is not at all involved. Thus, the argument ought to be rephrased. I'm afraid there is no doubt about the results in themselves; they are the most simple cases in studies of growths of dimension sequences related to finitely presented algebras, which were studied in the '70's e.g. by Victor Ufnarovski, Warren Dicks, and myself. The simple, purely combinatoric situations, independently later have been rediscovered by combinatorians (coming to the same conclusion, of course); and I suspect that the simple "one forbidden word" enumeration problem also has been covered by independent discoveries several other times, both before and after the first publication of it that I know about (by Govorov in 1972).

Here is the result, and an outline of the proof. A much more general result, covering any finite number of "forbiodden words", was proved and puublished by V. E. Govorov in the ''Mat. Zametki'' '''12''' (1972), pp. 197-204. I concentrate on "BANANA" and 50 letters, however.

Assume given an alphabet of 50 letters, and among these the letters A, B, and N. For any ("European") natural number ''n'', let ''an'' be the number of strings of length ''n'' in the 50 letters, that '''do not''' contain BANANA as a subword; let us call "BANANA" ''forbidden'', and the strings without an occurrence thereof ''allowed''. Then, for <math>0 \leq n \leq 5</math>, clearly <math>a_n = 50^n</math>. For <math>n \geq 6</math>, <math>a_n</math> fulfils the recursion formula
:<math>a_n = 50 a_{n-1} - a_{n-6}</math>.
The reason is simply this: We may prepend any letter to any one of the <math>a_{n-1}</math> allowed strings of length <math>n-1</math>, and in that manner we get <math>50 a_{n-1}</math> strings. Obviously, these strings include all the allowed strings of length ''n''. However, some of them are forbidden; namely those beginning with "BANANA". For any one of these forbidden strings, the last <math>n-6</math> letters will form an allowed string, say '''''S''''' (since otherwise already the string ANANA'''''S''''' would be forbidden, before prepending the initial "B"). Thus, out of the <math>50 a_{n-1}</math> candidates for allowed strings, exactly <math>a_{n-6}</math> fail.

Now, the most neat way to sum up these recursive properties are by the generating formal power series (as it ought to be called), alias the [[generating function]] (as it usually '''is''' called). In fact, it is not hard to see that
:<math>\sum_{n=0}^\infty a_n x^n = {1 \over 1-50x+x^6}\,</math>,
by means of the usual methods taught in an elementary coursis including combinatoric enumeration by means of [[formal power series]]. However, our purpose here is slightly different; we want an estimate of the ''probability'' for finding or not finding "BANANA" as a subword of an arbitrary length ''n'' string. Assuming that the letters are chosen [[i.i.d.]] and with equal probabilities, all the <math>50^n</math> strings are equally probable; whence the probability of '''not''' having "BANANA" as a subword is exactly <math>a_n/50^n</math>. Now, as also taught in these elementary enumeratoric combinatorics courses, a rational expression for a formal power series, such as the one ''supra'', may be converted to a polynomial expression for the coefficients, in terms of the roots of denominator polynomial of the rational expression; or, more precisely, of the associated "auxiliary equation"
:<math>y^6-50y^5+1 = 0\,</math>.
A sixth degree equation is a bit hard to solve by elementary means (as [[Niels Henrik Abel|Abel]] proved); but it is easy to see that this one does not have double roots (by taking the g.c.d of it and its derivative). This yields, that there are constants <math>c_1,\ldots,c_6</math>, such that for any <math>n \geq 0</math> we have
:<math>a_n = \sum_{j=1}^6 c_j r_j^n\,</math>,
where the <math>r_j</math> are the roots of the auxiliary polynomial. The absolutely largest of these roots, say <math>r_1</math>, is a positive real number; and thus already for moderately large ''n'' the quotient <math>a_n/50^n</math> will be rather close to <math>(r_1/50)^{-n}</math>. However, as you may see by simple means, the auxiliary equation does not have any rational root, whence ''a fortiori''
:<math>r_1 \neq 50 - 50^{-5}\,</math>.

Now, I know that arguments involving probabilities often invoke hot sentiments; and I also know that nobody likes being told that someone else knows much more about these subjects. I'm really not trying to bully anyone; but I did write part of my ph.d. thesis about these things, and have published a few articles about this later, whence I would be lying if I told you that this just is a guess. I've really tried to explain ''why'' you get the results, perhaps too lengthily; I'm absolutely prepared to discuss them further in detail, here or on my user talk page; but, as I said, the differences are actually not that important. The base of the exponential expression is a number somewhat smaller than 1, that's all that matters here, actually. I'd like to rewrite the text slightly, weakening the claims a bit (and '''not''' including the Govorov et al. precise results with proofs, if you don't mind), but providing a correct and sufficient estimate. However, I won't try this, if there are too many "unresolved issues" about the mathematics left. [[User:JoergenB|JoergenB]] ([[User talk:JoergenB|talk]]) 21:50, 7 October 2008 (UTC)
:What's wrong with the direct proof as it stands? [[User talk:Algebraist|Algebraist]] 11:54, 8 October 2008 (UTC)
:The entire point of the simple proof is just to demonstrate a simple case of the theorem and why it's true in that case. It doesn't need to provide an accurate estimate of the probability, only to bound it from below, and then show that that bound tends to 1. A more accurate estimate may be relevant, but for this initial example it's overkill and defeats the point of having a simple example. [[User:Dcoetzee|Dcoetzee]] 21:45, 8 October 2008 (UTC)
::I believe JoergenB meant we should use the phrase "Sketch of proof" if we don't meant to have the exact details to be correct. [[User:K61824|K61824]] ([[User talk:K61824|talk]]) 05:10, 10 May 2009 (UTC)

==Link 24: the monkey Shakespeare simulator==
This site bombarded me with Java errors and made Firefox crash. Is this a common issue with the site? Should the link be removed if the site is unusable because of this? - [[Special:Contributions/84.27.9.117|84.27.9.117]] ([[User talk:84.27.9.117|talk]]) 22:21, 25 October 2008 (UTC)

might be just you. worked fine on ie[[User:Firl21|Firl21]] ([[User talk:Firl21|talk]]) 15:13, 3 September 2009 (UTC)

== Kolmogorov Complexity and monkeys typing on a computer ==

One of the most interesting sidenotes to the infinite monkey theorem is the fact that the monkeys would be significantly faster if they used a computer instead of a typewriter. I first heard about this in Cover and Thomas, "Elements of Information Theory".

Given that the [[Information entropy]] of the English language is about 1.5 bits per character, Shakespeare's works could probably be compressed by a factor of at least three. Thus, the probability that the monkeys come up with a compressed version of Hamlet (or a computer program which prints Hamlet) is much higher than the probability that they produce the full text.

I just find that interesting... what do others think about it? If there are enough people that motivate me (write to j dot b dot w at gmx dot ch) I'll create a Wikipedia account and write it down properly. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/128.178.83.79|128.178.83.79]] ([[User talk:128.178.83.79|talk]]) 11:00, 19 December 2008 (UTC) 
:Yes, that's interesting - I've added your paragraph (with minor changes) to the Probabilities section as a footnote to the text.[[User:Bobathon71|Bobathon]] ([[User talk:Bobathon71|talk]]) 13:48, 11 January 2009 (UTC)
:It may be appropriate to create a section for this somewhere within the article - it has a significant bearing on all calculations relevant to this subject (though clearly not on the final "finite but non-zero" conclusion)[[User:Bobathon71|Bobathon]] ([[User talk:Bobathon71|talk]]) 13:54, 11 January 2009 (UTC)

== Note ==

There are some broken equations in the Solution section. Could somebody more knowledgeable than I fix that? Thanks! –[[User:Juliancolton|Juliancolton]] [[User talk:Juliancolton|'''T'''ropical]] [[Special:contributions/Juliancolton|'''C'''yclone]] 02:53, 25 February 2009 (UTC)

==Weasel misconception==
I have corrected a misconception in the article which stated that "Dawkins has his Weasel program produce the Hamlet phrase ''METHINKS IT IS LIKE A WEASEL'' by typing random phrases but '''constantly freezing''' those parts of the output which already match the goal". The bold text is incorrect; in the context of this article, it's not a big deal, but it is part of the toolkit used by those who oppose evolution, and is plainly wrong.

The current [[Weasel program]] article does not clearly address the issue, but it is well covered in the [[Talk:Weasel_program#The_Algorithm|discussion]]. The correct letters ''appear'' to lock because the program chooses the best match from mutated progeny (so mutations which make a good letter bad will usually not be the next parent). However, a [[Talk:Weasel_program#Recent_events|video]] shows that the program does not lock correct letters, and the words used by Dawkins in ''[[The Blind Watchmaker]]'' make it obvious that his simulations apply random mutations to ''all'' locations. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 08:55, 2 May 2009 (UTC)

== How was the specific probability calculated? ==

Just wondering, how was the number 3.4 × 10183,946 obtained? This needs to be explained somewhere, or else the number should be removed for being unverifiable original research. [[User:Gracefool|··gracefool]][[User talk:gracefool|☺]] 18:54, 8 May 2009 (UTC)
:It's just 26^130000. [[User talk:Algebraist|Algebraist]] 20:00, 8 May 2009 (UTC)
::How do you work that out? [[User:Gracefool|··gracefool]][[User talk:gracefool|☺]] 10:38, 9 May 2009 (UTC)
:::It was clear from the immediately preceding text that 26^130000 was the number that ''should'' have appeared there, so I just checked (using Google calculator) that it was. [[User talk:Algebraist|Algebraist]] 18:37, 9 May 2009 (UTC)

:::To calculate n = 26^130000 we take log of both sides: log(n) = 1300000*log(26) = 183946.5352
:::Therefore n = 10^0.5352 * 10^183946 = 3.429 * 10^183946 [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 00:29, 10 May 2009 (UTC)
Thanks. So including punctuation the figure is about 10^360783 (26 letters x2 for capitalisation, + 12 for punctuation characters = 64, log(64)*199749 characters). That makes a big difference to the number! [[User:Gracefool|··gracefool]][[User talk:gracefool|☺]] 01:18, 10 May 2009 (UTC)

I've added this stuff to the article. [[User:Gracefool|··gracefool]][[User talk:gracefool|☺]] 05:36, 25 June 2009 (UTC)

::::I once slapped the keyboard at random and got "iloveyouall". [[User:Professor Fiendish|Professor]] [[User talk:Professor Fiendish|M.]] [[Special:Contributions/Professor Fiendish|Fiendish]], [[User:Professor Fiendish/Page of Doom!|Esq.]] 04:51, 13 September 2009 (UTC)

== "Almost surely" ==

The phrase "almost surely" has ABSOLUTELY NOTHING to do with the monkey metaphor! It's a precise mathematical term with a precise meaning that has NOTHING to do with metaphors. Please fix the article so that it makes sense! —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/174.20.91.244|174.20.91.244]] ([[User talk:174.20.91.244|talk]]) 18:04, 15 August 2009 (UTC) 

== Broken external link ==
The first external link (to the Baltimore Examiner) is broken, and just redirects to the homepage of the Washington Examiner.
[[Special:Contributions/170.148.198.156|170.148.198.156]] ([[User talk:170.148.198.156|talk]]) 09:52, 9 November 2009 (UTC)
:You meant the first link in the External Links, correct? I couldn't find it either. Looks like it got lost when baltimoreexaminer merged into washingtonexaminer. Lots of googling no go. Worse news, see here: http://web.archive.org/web/*/http://www.baltimoreexaminer.com/opinion/The_million_monkey_room.html Looks like baltimoreexaminer never let IA archive their stuff, so it'll be real hard to find again. :-( —[[User:Aladdin Sane|Aladdin Sane]] ([[User talk:Aladdin Sane|talk]]) 10:43, 9 November 2009 (UTC)

== Probability section fails ==

That whole section is [[wp: or]], not to mention wrong. It assumes that the monkey is only striking keys that produce letters the monkey could strike any of the function keys or numbers etc. It cites no sources except the bottom where it takes a quote. The main trouble is that this section falsely presents a low probabilty by restricting the origional terms of the theory to the life of the known universe rather then infinity which is how it is supposed to be. Any probability repeated with a time frame of infinity will be come 1, the life of the universe isnt even a warm up phase compared to infinity. This section needs to be removed entirely it is not helpful [[User:Smitty1337|Smitty1337]] ([[User talk:Smitty1337|talk]]) 23:04, 21 April 2010 (UTC)
:I agree that OR is an issue but I support the section ([[WP:IAR]] or whatever) since it is verifiable and useful to readers. The restriction to 26 letters is just a commonly-made simplification – if the monkey had more than 26 choices the probability would be lower than the effectively zero value shown in the article. No one is denying that in some metaphorical sense randomly striking keys would eventually produce a sentence, but it is valuable to learn that in practical terms the outcome is impossible. The concept has sometimes been used to assert that freak things will eventually occur by blind chance (i.e. physical things in a practical universe). Whereas that is true for many very rare phenomena, it is not true in this universe for an event such as randomly typing a particular book. Also, note that there ''is'' a very reliable reference that supports the conclusion of the section. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:02, 22 April 2010 (UTC)
:: this section shows the odds at 3.4 × 10183,946 . if we assume 1 letter per second and 130,000 total seconds thats 2166.66 hours or 90 days roughly 1 quarter of a year, per full book attempted (and this is generous because it doesn't assume failure on letter 3 throws the book out and starts over right there). if the 3.4 × 10183,946 attempts required to get 1 right is all that's statistically probable, then the monkey should write one copy every 76.5 x 10183,946 years which divided by infinity will happen infinite times so in 76.5 x 101839460 years we'd have the 10th copy (collectors edition i presume). I'm being absurd to get my point across, this theorem is meant to prove a point, not be taken literally, the point is that anything that has a probability that is not zero no matter how excessively large the denominator is on that fraction, the concept of infinity makes that number minisucle to the point of if it "can" happen it will happen repeatedly in 345235324532452345234523453245234523452345234532 years (not even .00001% of infinity of course). this is all OR of course which is why i'd never push to say such on the article, but neither should some silly probability section give a false notion of unlikelyhood when infinity makes the chance 100% (eventually) [[User:Smitty1337|Smitty1337]] ([[User talk:Smitty1337|talk]]) 10:33, 22 April 2010 (UTC)
:::From the reliable source: {{xt|As Kittel and Kroemer put it, "The probability of Hamlet is therefore zero in any operational sense of an event…"}}. The numbers used for input to the calculation are shown in the "Direct proof" section. Don't you find it interesting that the chance of randomly typing ''Hamlet'' is essentially zero, even if you have as many monkeys as there are particles in the universe, and each types 1,000 keystrokes per second for 100 times the life of the universe? [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 11:20, 22 April 2010 (UTC)
:::: The trouble with making that statement is that it is without context. The probability may be operationally zero, as the sourcer says, and that is infact true, but the problem is that any probability given infinite repetition will become 1, and if not stopped upon successful completiion, then technically the book will be made and remade infinite times. the statement is true but by itself its misleading because its just the probability without moving to the next logical step of infinite repeatition, and everything above it is [[wp: or]] [[User:Smitty1337|Smitty1337]] ([[User talk:Smitty1337|talk]]) 00:43, 23 April 2010 (UTC)

From a Darwinian approach to the question "How long would monkeys type the works of Shakespeare"?, the solution is the period intelligent man has evolved from primates. A monkey , trapped on earth a million years ago with nothing but to mutate itself into an intelligent being, discover words, concoct writing, invent the typewriter and finally produce among its descendants a literary genius would take a few million years. This period has been historically tested and is much much less than the predicted statistical outcome with the assumption that the monkey will genetically remain stagnant without hope for increased brain capacity but just random act in his eternal life. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/149.136.17.253|149.136.17.253]] ([[User talk:149.136.17.253|talk]]) 20:16, 1 September 2010 (UTC) 
:You have missed the point entirely, as the article says right in the lead, its a metaphorical moneky, as in one monkey, not a series of generations, there is no evolution involved in this article and its only meant to imply a concept of probability and infinity. [[User:Smitty1337|Smitty1337]] ([[User talk:Smitty1337|talk]]) 18:29, 2 September 2010 (UTC)
::Sorry to engage in forum talk, but while of course you are correct, I think 149.136.17.253 probably knows that, and the point they made is interesting because it highlights how human commonsense can fail when extrapolated too far. Our brains provide a model of the world whereby we are happy to talk about an ideal monkey typing for millions of years (which is fine), but in the time frames under discussion lots of things will change in ways that we struggle to appreciate (the Atlantic ocean is expected to disappear in 200 million years or so, due to plate tectonics; can't find it on en.wiki). [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 02:35, 3 September 2010 (UTC)

None of this is right at all. A correct neuron configuration is required to type out a William's Shakesphere Play. Given infinite time, the neuron configuration of the brain will never reach precisely the right locations and synapses as that so the monkey will be typing out an entirely play by chance because it is deterministically a zero probability. In order for an action to occur, a neuron must be fired, and certain neurons specifically. —Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/209.159.183.114|209.159.183.114]] ([[User talk:209.159.183.114|talk]]) 13:33, 20 November 2010 (UTC) 

== "Experiment" ==

"''Popular interest in the typing monkeys is sustained by numerous appearances in literature, television, radio, music, and the Internet. In 2003, an experiment was performed with six [[Celebes Crested Macaque]]s. Their literary contribution was five pages consisting largely of the letter 'S'.''"
#This wrongly suggests that it was a scientific experiment, rather than a student art project.
#I don't see that this is significant enough to mention in the lead. [[User:Feezo|Feezo]] [[User_talk:Feezo|(Talk)]] 03:05, 8 February 2011 (UTC)
#:I agree that the sentence should be removed from the lead. I suppose the mention in the "Real monkeys" section is warranted, although it adds little of value to the article other than to show that the topic is of general interest. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 05:46, 8 February 2011 (UTC)

== Weird sentence in the lede ==

''"The theorem illustrates the perils of reasoning about infinity by imagining a vast but finite number, and vice versa. "'' - what's "perilous" about it? What, I'm gonna get shot if I think about infinity by imaging a vast but finite number? What the hell is this sentence even supposed to mean? Nonsense.[[User:Volunteer Marek|Volunteer Marek]] ([[User talk:Volunteer Marek|talk]]) 21:24, 11 April 2011 (UTC)

And what for monkey's sake does the "vice versa" refer to? Reasoning about "finity" by imagining a minuscule but infinite number? Seriously.[[User:Volunteer Marek|Volunteer Marek]] ([[User talk:Volunteer Marek|talk]]) 21:26, 11 April 2011 (UTC)

== almost surely ==

"almost surely" implies the change the money wont type the play is considerable, but the chance is actually infinitesimal or zero. [[Special:Contributions/173.183.79.81|173.183.79.81]] ([[User talk:173.183.79.81|talk]]) 22:35, 14 April 2011 (UTC)
:It's a technical term; it's not up to us to change it. It's explained in the text, which is really all we can do. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 22:45, 14 April 2011 (UTC)

::"Almost surely" is the wrong term. It is not a technical term at all, but colloquial English. The concept is that an infinite amount of time, or monkeys, is available to write the complete works of Shakespeare. Given the unlimited extent of infinity the works of Shakespeare WILL be written. Not "almost surely".[[Special:Contributions/125.237.105.102|125.237.105.102]] ([[User talk:125.237.105.102|talk]]) 04:51, 20 September 2014 (UTC)
:::No, "[[almost surely]]" has a rather strict mathematical definition. In my calculus-oriented mind, it means that the probability of not typing a given work has [[limit (mathematics)|limit]] 0 as the amount of typing n tends to infinity (i.e. for any possible positive probability there always exists an n where the probability of not typing the work is less than that given positive probability); the linked article has a better definition. But there exists ''no'' n for which the probability of ''not'' typing a given work is actually 0. To assume otherwise implies the [[gambler's fallacy]] (if the typing is truly at random).--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 04:58, 20 September 2014 (UTC)

== typing for infinity ==

If the apes could type for infinity, and there was a small amount of probability that they could type the works of some author, is it also possible there is a small chance they write something that hasn't been written yet (assuming nobody saw the writings and somehow copied them)? Could the typing somehow predict a future? If they could, they could also predict another piece of writing that will never happen, or never has happened, creating something new, couldn't they? I may be asking something that has been gone over already, but I don't think so. — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/66.165.17.192|66.165.17.192]] ([[User talk:66.165.17.192|talk]]) 03:47, 3 July 2011 (UTC) 
:anything about the article ? '''[[User:Arjun024|Arjun]]'''[[User talk:Arjun024|codename'''024''']] 05:21, 3 July 2011 (UTC)
: For the record, there is no "predictive" value to monkeys typing for billions of years, because any process which ''found'' some coherent text in the randomness (say, a collection of poetry about probability) would in effect be "writing" that text by searching for it. Overall, readable random texts are way way way less frequent than unreadable ones, unless you have an evolution-like process that keeps the readable stuff and ditches the unreadable, in which case it's a somewhat different puzzle.
: However, if we're talking about an infinite length of time, then yes, even one monkey will indeed produce ''every possible finite sequence of text'' with probability 1, including readable books that no human had ever written — an infinite number of such books, in fact. There's nothing special about Shakespeare here, it's just used to illustrate this surprising idea. ± [[User:Lenoxus|Lenoxus]] ([[User talk:Lenoxus|" *** "]]) 00:32, 13 October 2011 (UTC)
::Nothing surprising about it when you consider how long infinite time is. [[User:Gracefool|··gracefool]][[User talk:gracefool|☺]] 03:22, 3 November 2011 (UTC)
:[[User:Lenoxus|Lenoxus]] has it right, but in other words: You can't predict the future with randomness, because to recognize it as something apart from randomness, you have to already know the information in question. If there was no Shakespeare, you couldn't recognize Hamlet as being something special, any more than any of the other millions of readable books likely to be typed before that. If there were a specific prediction, eg. "The World Trade Center will be destroyed on September 11, 2011", how would you tell its accuracy over a million other predictions of the same thing happening on a different date? [[User:Gracefool|··gracefool]][[User talk:gracefool|☺]] 03:22, 3 November 2011 (UTC)

== The picture in the article is wrong also ==

That's a chimpanzee, not a monkey. And that's a camera, not a typewriter (I think). — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/71.98.215.115|71.98.215.115]] ([[User talk:71.98.215.115|talk]]) 05:45, 10 September 2011 (UTC) 
: It's a very old typewriter: zoom in on it to see the detail (the base, the keys, the paper holding structure at the top). As for the ape historically 'monkey' was used for both apes and monkeys. Certainly when this phrase arose it would be common usage. Even now it is still used informally like that.--[[User:JohnBlackburne|JohnBlackburne]][[User_talk:JohnBlackburne|words]][[Special:Contributions/JohnBlackburne|deeds]] 12:15, 26 September 2011 (UTC)

::Typical. When the monkeys produce a work of literature, it's credited to an ape!--[[User:Jack Upland|Jack Upland]] ([[User talk:Jack Upland|talk]]) 09:19, 21 November 2016 (UTC)

== Title of first (non-lead) section ==

Why is the first section under the TOC headed ''Solution''? Solution of what? ''Problems'' have solutions, but the theorem is not stated as a problem.
I've looked through the history and it appears to have been that way for quite a while so I don't want to change it rashly, but surely we can do better than that. Suggestions? Maybe the inelegance of this word indicates a more structural difficulty with the article, and suggests moving the proof sketch further down? --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:13, 12 October 2011 (UTC)
:Yes, it is odd, although I don't think moving it would be particularly helpful. Perhaps "Analysis"? Also, in "proof of this theorem", the "this" should be replaced with "the infinite monkey". [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 00:31, 13 October 2011 (UTC)

== Similar concepts ==
Why aren't there links to similar concepts? Someone could include a link to [[Bogosort]] or something, which I believe is a great example.[[Special:Contributions/98.119.209.61|98.119.209.61]] ([[User talk:98.119.209.61|talk]]) 09:09, 15 November 2011 (UTC)
:Good question! I'll add some later, if I can remember to. In the meantime, feel free to [[WP:BEBOLD|be bold]]. Evanh2008, Super Genius [[User:Evanh2008|Who am I?]] [[User talk:Evanh2008|You can talk to me...]] 09:43, 15 November 2011 (UTC)

== Why the link at the top about "not to be confused with..."? ==

Why exactly is there a link at the top of the article staying this should not be confused with the hundreth monkey effect? They are totally unrelated things and in my opinion, are not easily confused. If this statement stays, then I think we should also add the following statement: "Not to be confused [[12 Monkeys]]." I mean, there is a number at the front of the statement and it says monkeys... a reader might be confused. [[User:Krohn211|Krohn211]] ([[User talk:Krohn211|talk]]) 19:09, 1 December 2011 (UTC)
: I agree: if this title were ambiguous it should link to a disambiguation page but it clearly isn't. Further I can't see anyone confusing this with the [[hundredth monkey effect]] which seems not to be widely known: from the article it's a discredited crank theory, not part of mainstream science. I've moved it to 'See also' but I have no objection to it being removed altogether.--[[User:JohnBlackburne|JohnBlackburne]][[User_talk:JohnBlackburne|words]][[Special:Contributions/JohnBlackburne|deeds]] 19:35, 1 December 2011 (UTC)
:: I am not a heavy wikipedia user so I don't know the goal of the "See Also" section. From my understanding, this section is meant for related topics and I don't believe this is a related topic. If it were up to me, I'd remove all references to it from this article. Looking at the history I can't tell who put this there in the first place. Since I didn't put it there, I don't want to be the one to remove it. But if I had a vote, I'd say remove it altogether. [[User:Krohn211|Krohn211]] ([[User talk:Krohn211|talk]]) 03:53, 2 December 2011 (UTC)
The above issue was fixed in {{diff|Infinite monkey theorem|prev|463544027|this edit}} by JohnBlackburne. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 06:45, 2 December 2011 (UTC)

== Unnecessary paragraph ==

"Primate behaviorists Cheney and Seyfarth remark that real monkeys would indeed have to rely on chance to have any hope of producing Romeo and Juliet. Monkeys lack a theory of mind and are unable to differentiate between their own and others' knowledge, emotions, and beliefs. Even if a monkey could learn to write a play and describe the characters' behavior, it could not reveal the characters' minds and so build an ironic tragedy"

Summary: monkeys are too dumb to write Shakespeare.

This is taking stating the obvious to a new level; it reads like a tabloid article or something from a waiting room magazine. Perhaps this somehow meets a guideline I don't know about, but surely a fact that no reader is ever likely to not already be aware of expressed in 73 words with "sciency" language and name-dropping of behaviourists does not belong on Wikipedia. [[User:Kombucha|Kombucha]] ([[User talk:Kombucha|talk]]) 00:05, 23 December 2011 (UTC)

:We should lose the name-drop at the least. [[User:Kombucha|Kombucha]] ([[User talk:Kombucha|talk]]) 00:11, 23 December 2011 (UTC)
::Feel free to remove the whole "Real monkeys" section because it is just unrelated commentary (i.e. it is nothing to do with the actual "theorem", and is essentially nonsense, no doubt expressed in impressive language in the original). [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 01:49, 23 December 2011 (UTC)

== History - references preceding Borel ==

I am fairly new to editing Wikipedia articles, not new to Wikipedia. I have been researching this topic to find an earlier reference to "the likeliness of monkeys writing great works". I found a reference by Richard Bentley in a sermon (originally 1692/3) regarding the probability that a monkey scribbling away could ever write Hobbes' ''Leviathan'' as a comparison to the probability of creation. I dispute that Borel is the first person to use the idea of monkeys typing works by chance in the context of probability. (As an aside, in the French, I have also found 'singe' with the meaning of 'Mime Artist' or 'people who mimic' ("les singes de Balsac (sic)" appeared in a dictionary to denote the plethora of authors all trying to mimic the style of Balzac.)) Bentley's prose is in the context of probability (albeit in terms of creation and not mathematics). True there were no typewriters in 1692, but this merely means that Borel had re-framed the concept of 'monkey scribbles' to 'les singes a frapper' on a typewriter (if it had not already been re-framed previously to him). Also in the re-framing one monkey becomes one million monkeys.

My edit was made in haste... and was removed as 'misplaced and original research'... I would like to discuss this. (Source: [http://books.google.co.uk/books?id=VB43AAAAMAAJ&dq=Monky&pg=RA1-PA63#v=onepage&q=Monky&f=false Richard Bentley's The folly and unreasonableness of atheism]) [[User:Zorgster|Zorgster]] ([[User talk:Zorgster|talk]]) 04:41, 15 January 2012 (UTC)
:It was {{diff|Infinite monkey theorem|prev|471391778|this edit}} that was reverted. I haven't wanted to take the time to investigate this issue (I saw your edit and decided to leave it). However, my guess is that the editor who reverted your edit thinks that we would need a [[WP:SECONDARY|secondary source]] that makes an association between the information you found and the topic of this article. If writing an article at some other place, it would appear under the writer's name and any views in the article would clearly be the views of the author. However, there is no author here, and all statements need to be verifiably related to the topic, rather than likely associations that an editor has located. I have not formed any firm views on the issue, and mention this for background. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 06:06, 15 January 2012 (UTC)

::Hi.. thank you for taking the time to explain that. I see what you are saying. I could only speculate myself. ...that the sermon, which was part of the [[Boyle Lectures]] and so re-published in 1737 (A Defence of Revealed and Natural Religion), in 1809 (Eight Sermons, Oxford), in 1838 (The Works of Richard Bentley, Vol 3), was possibly used in arguments against atheism - or to strengthen religion. It would have been well circulated amongst scientist and clergy alike. Bentley's 'Confutation of Atheism' also discusses the ideas of Isaac Newton. I would suggest that any scientist, including Huxley, would be familiar with this text. And the argument about monkeys scribbling Hobbes would have been read by many an academic. Huxley may have used it in his crossings with Owen and Wilberforce (in Oxford, 1860) - in debates of Darwinism and religion. The paragraphs around Bentley's mention of the monkey in the 1838 edition, talk a great deal about probability and chance... and as is said in the philosophy of history... we tend, when reading historical prose, to frame the usage of words in the past using our understanding of the present... The human body was seen then (1692) as the creation of the divine... A body was often compared to a book... and the comparison here is that you can deny the 'hand of god' in the creation of man, as much as you could conceive of a monkey ever scribbling ''Leviathan'' out of pure chance ([http://books.google.co.uk/books?id=B5s1QqExVtQC&lpg=PA104&dq=bentley%20hobbes%20monkey&pg=PA104#v=onepage&q=Leviathan&f=false Kristine Haugen's Richard Bentley: Poetry and Enlightenment]). Again, one can only speculate... and one needs to spend time looking into it... I've used my quota of spare time, too :-) [[User:Zorgster|Zorgster]] ([[User talk:Zorgster|talk]]) 06:31, 16 January 2012 (UTC)

== Possible FAR ==

Referencing on this article is still sub-par and has been tagged as such for close to 5 months (Criteria 1c); the prose in some parts, such as "in popular culture", is rough (Criteria 1a). This should be fixed, if possible. — [[User:Crisco 1492|Crisco 1492]] ([[User talk:Crisco 1492|talk]]) 23:58, 31 May 2012 (UTC)

== numbers of bibles etc ==

I'm sorry if someone has already mentioned this but has anyone considered how many Bibles that contain one or more errors
would be produced in order to produce one without errors.
One way of looking at this would be if the monkeys started at the same time and typed at the same rate how many monkeys would we need before all the material in the universe had been turned into faulty bibles. And many correct letters would there be in the one copy that was correct.

Secondly I thought one of the Bernoulli's said that where a probability was exceedingly low, as in this case, one could
completely ignore it because the probability of almost anything else, for example, the existence of a god with a sense of humour, explaining the result would be astronomically higher.

john f
[[Special:Contributions/212.183.128.84|212.183.128.84]] ([[User talk:212.183.128.84|talk]]) 11:48, 2 May 2013 (UTC)

== The Simpsons' Did it! ==
The Simpsons TV show parodied this idea in some episode, with what I believe to be one of the cleverest lines in the show. Mr. Burns goes into a special room of his which houses a number of monkeys dutifully (and fearfully I believe, thanks to Mr. Burns' reputation) typing away at their assigned computers. Mr. Burns walks over to see the result of a random monkey's work and reads out "It was the best of times, it was the BLURST OF TIMES!", and smacks the monkey or something. 'Worst' was spelled incorrectly, and the line is not Shakespeare's, but still pretty good for monkeys. This could be added under a 'cultural references' section, since it's so damn funny. Also, a picture of Mr. Burns reading the transcript should be included, for completeness. [[User:Jakepapp|Jake Papp]] ([[User talk:Jakepapp|talk]]) 14:14, 7 August 2013 (UTC)
:That is already on [[Infinite monkey theorem in popular culture]]; it does not need to also be in the main article. [[User:Meshach|meshach]] ([[User talk:Meshach|talk]]) 16:41, 7 August 2013 (UTC)

Why is there a whole article when there is already a section for popular culture on the original article's page?

== Citations ==

The beginning paragraphs (before contents) need more citations to back up the claims they make. In fact, the whole article could use more citations backing up their claims, especially the Direct Proof section.
[[User:Bibliophile scribe|Bibliophile scribe]] ([[User talk:Bibliophile scribe|talk]]) 08:23, 3 December 2013 (UTC)

== Twitch Plays Pokemon ==
More than 50 thousands of players are playing ''the same'' videogame. Can this be a practical approach to the theorem?
http://www.twitch.tv/twitchplayspokemon — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:190.55.94.168|190.55.94.168]] ([[User talk:190.55.94.168|talk]] • [[Special:Contributions/190.55.94.168|contribs]]) 19:00, February 19, 2014 (UTC)
:No is is just a coincidence [[User:Meshach|meshach]] ([[User talk:Meshach|talk]]) 06:21, 20 February 2014 (UTC)

== Proof sources? Original research? ==

Are there [[WP:RS|sources]] for the mathematical proofs, and other information, in the {{section link||Solution}} section? Because it doesn’t really seem to cite any. Someone not learned in probability theory would have to take Wikipedia’s word for it on basically that whole section, which seems to go against the whole idea of WP. —[[User:Frungi|Frungi]] ([[User talk:Frungi|talk]]) 07:23, 13 March 2014 (UTC)

:On the other hand, the mathematics presented is pretty straightforward. The combined probability of ''n'' independent events is the product of their individual probabilities, just as it is stated in the section, and the conclusions are just the result of simple algebra. Personally, I would prefer an [http://david.tribble.com/text/monkeys.html example] using only 27 possible keys (26 letters plus a space), but it does not change the idea behind the math shown. We have to assume ''some'' level of reader competence, and the section does have links to [[statistically independent|more detailed]] math articles for readers who want more information about [[probability]]. — [[User:Loadmaster|Loadmaster]] ([[User talk:Loadmaster|talk]]) 23:14, 13 March 2014 (UTC)

== Relevance of picture? ==

I do not think the top page image should be there. I think that the infinite monkey theorem is an extremely important scientific principle and that a picture of this novelty drags the article down. Discuss. [[User:Mackatackastewart|Mackatackastewart]] ([[User talk:Mackatackastewart|talk]]) 12:17, 23 March 2014 (UTC)
:The picture is fine. It is entirely in keeping with an article titled "infinite monkey theorem". [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 00:43, 24 March 2014 (UTC)
::It should at least be of a monkey, not a chimp.--[[Special:Contributions/99.253.58.249|99.253.58.249]] ([[User talk:99.253.58.249|talk]]) 11:32, 30 August 2018 (UTC)

== Wilberforce debate summary bias ==

Is the conclusion to the wilberforce debate bias? It cites an uninternettable paper by [[Nicholas Rescher]]. Obviously the person is worth citing but I don't know enough about them to say whether they may have a slant worth mentioning. It seems to question the historicity.
Other mentions are here [[Thomas_Henry_Huxley#Debate_with_Wilberforce]] and the main article is [[1860_Oxford_evolution_debate]] — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/216.15.26.233|216.15.26.233]] ([[User talk:216.15.26.233|talk]]) 00:27, 27 March 2014 (UTC) 

== Wrong definition. ==

It *should* read more like this:-

The '''infinite monkey theorem''' states that the complete works of [[William Shakespeare]] can be produced by getting an infinite number of monkeys to type [[randomness|randomly]] on a [[typewriter keyboard]].

It's an "'''infinite monkey'''" theorem, not a '''single''' monkey for an infinite amount of '''time'''. — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/120.151.160.158|120.151.160.158]] ([[User talk:120.151.160.158|talk]]) 15:21, 19 June 2014 (UTC) 
:See the third paragraph:
::Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters.
:In any case, whether it is one monkey or infinite monkeys, if they are typing for an infinite amount of time, the effect is the same. Although of course if you have infinite monkeys you can do without infinite time, you only need an amount of time equal to the shortest possible time it could take a monkey to type the works of Shakespeare. ··[[User:Gracefool|gracefool]][[User talk:gracefool|☺]] 10:43, 22 June 2014 (UTC)

::With an ''infinite'' number of monkeys, all it takes is a ''single'' random keystroke from all of them at once to produce every written work, including the entire Shakespeare corpus. This is simply because a single random keystroke from an infinite number of monkeys instantly produces an infinitely long set of random characters. Somewhere within the set of all infinite typed characters is a subset for any given work (as well as any given unpublished work, or any given sequence of random gibberish). This is true even if you order the monkeys in a sequence, i.e., assign each monkey a specific place ([[natural number|number]] or index) within the sequence of all monkeys; somewhere among that sequence is a subsequence of characters exactly matching any chosen text (assuming completely random typing). In fact, any given (finite) subsequence will occur an infinite number of places within the complete sequence. — [[User:Loadmaster|Loadmaster]] ([[User talk:Loadmaster|talk]]) 22:57, 21 September 2014 (UTC)

Off-topic peeve alert: Please don't say "infinite monkeys" when what you mean is infinitely ''many'' monkeys. Infinite monkeys would be more than one monkey, each of which is infinite. What it means for a monkey to be infinite, I'm not sure, but that's another discussion. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 23:39, 21 September 2014 (UTC) 

== Formula? ==

How about a simple general formula calculating the odds that at least one of ''m'' monkeys (effectively random-character generators) on typewriters with ''k'' keys each (the size of the alphabet) will eventually, after typing ''i'' keystrokes at random, turn out producing a string of the length ''n'' with a probability of ''p''? That would make it easier for the lay reader to follow the examples. --[[User:Florian Blaschke|Florian Blaschke]] ([[User talk:Florian Blaschke|talk]]) 16:33, 14 December 2014 (UTC)

== Recent edits ==

Some recent edits by myself and others were block-reverted on the the grounds that they hadn't been discussed[https://en.wikipedia.org/w/index.php?title=Infinite_monkey_theorem&diff=656188241&oldid=656146854]. I think my changes had resulted in an overall improvement, but I'm not greatly attached to the revisions I made to the lead. I have restored the edit I made to the reference to Eddington in the Statistical Mechanics section as I have no doubt that this clarifies his point. Any suggestions or comments? [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 12:45, 17 April 2015 (UTC)
:I'm OK with the clarification of Eddington's meaning. I'm not OK with characterizing the entire subject of the article as a "misunderstanding". The theorem as stated, and with the elaborations on "monkey" and so on in the following paragraph, really is true; it's not a misunderstanding. Physical realizability is not really the point; we're talking about Platonistic mathematical truth, not physics. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 18:24, 17 April 2015 (UTC)

== Confused ==

The more I look at this article, the more I find it totally confused. Is there actually a reliable source that refers to the assertion as a ''theorem'', rather than for example a postulate or an illustration? The reference to infinity is poorly framed and sloppily presented. In so far as mathematics treats the notion of infinity, there are various orders of infinity, and the enumeration of the set of natural numbers is the lowest order. It does not follow that an infinite sequence of symbols contains every conceivable sub-sequence. It could not contain the complete sequence of the decimal representation of pi for example, or even the representation of the square root of 2.

The mathematically rigorous way to express the so-called "proof" would be to state that the limiting value of the probability of any given sequence appearing approaches unity as the sample size increases without limit. This does not imply that the probability ever reaches unity. Even if it did, it is a misconception to treat 'Probability of 1' as being equivalent to 'certain to occur'.

Finally since there are not even an infinite number of atoms in the universe, much less an infinite number of monkeys, the premise is counterfactual. In formal logic, a false premise entails any conclusion, true or false. Therefore this "theorem" is trivially true only in the sense that "If 1 = 0 then I am the Pope" is a true (but vacuous) proposition. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 12:23, 14 May 2015 (UTC)
#Hmm. On the naming issue you may have a point — I don't really know whether "infinite monkey theorem" can be said to be a standard name for the result. ("Postulate" certainly does not seem to be an improvement, though.)
#No one claims that an infinite sequence necessarily contains every possible subsequence. If there is such a claim in the article, please point out where it occurs, so that it can be corrected.
#The distinction between "probability 1" and "certainty" is explicitly mentioned in the article.
#A mathematically rigorous presentation does not in fact necessarily need to mention limits. A [[probability measure]] can be defined for the entire space of possible outcomes, each considered as a completed infinite totality. However, limits may be a more accessible way of describing the result for most readers, and I don't exclude that it might be an improvement to mention it.
#The situation is obviously intended as a [[counterfactual]], and from the point of view of [[conceptual analysis]], that does not in fact make the result vacuous. For a false assertion ''p'', the claim "if ''p'' then ''q''" is automatically true, but the counterfactual "if ''p'' were true then ''q'' would be true" is not (though the interpretation of the latter is obviously not truth-functional; it's something more complicated).
#However, in point of fact, no one knows whether the universe contains a finite or infinite number of atoms, or even monkeys. See [[shape of the universe#Infinite or finite]]. You can find lots of claims in print about the "number of atoms in the universe", but almost always, these are (rather sloppily) using the term "universe" to mean [[observable universe]]. This point is somewhat of a digression and probably not all that relevant to the article, so we shouldn't belabor it. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 14:54, 14 May 2015 (UTC)

Regarding point 2: surely this is the heart of the matter - if the sequence representing the works of Shakespeare does not of necessity occur in the infinitely long sequence produced by the monkeys, the surely the 'theorem' is false? More to the point, what do you see as being the insight which is being given by the assertion of this theorem? [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 15:05, 14 May 2015 (UTC)
:It does not ''necessarily'' occur. But it does occur [[almost surely]]. As to what insight I see, that's a bit off-topic, because the article is not about my insights, but it may be of interest that, given an infinite (or even merely unlimited) amount of time or number of trials, extremely unlikely occurrences become almost guaranteed to happen. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 15:20, 14 May 2015 (UTC)

Apologies for not having expressed myself clearly; I wasn't asking about your personal insights, I was querying what point is being illustrated by this so-called theorem. It is not controversial that the probability of any very unlikely event can be increased to any given value by a sufficient number of repetitions of the trial. However, it appears to me merely to promote misunderstanding to refer to events so unlikely that they could not be elevated to high levels of probability even in timescales orders of magnitude greater than the age of the universe. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 15:50, 14 May 2015 (UTC)
:Well, I think the whole point is that your sentence starting "[i]t is not controversial" really does apply even to events that are just that unlikely. That's not a "misunderstanding". That's just true. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 16:53, 14 May 2015 (UTC)

It might be more illuminating to illustrate the probability of some specific character sequence (eg "Tomorow and tomorrow, and tomorrow, creeps in this petty pace from day to day.") being generated by a random-character generator operating at say one character per microsecond for the timespan of this universe. This would still not equal unity, so speaking of infinite time spans (or infinite numbers of generators) is merely speculation about counterfactual hypotheticals. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 10:36, 15 May 2015 (UTC)
:No speculation involved. It's a theorem. You can prove it. It's true. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 14:51, 15 May 2015 (UTC)
The "proof" given is entirely invalid since it relies on the assertion that one divided by infinity equals zero, a proposition that no mathematician would incorporate into any rigorous proof. The entire article contains a great deal of original research, editorialising, undisciplined speculation and muddled thinking. Is there even any reliable source that even describes this statement as a "theorem"? The overall thrust is misleading, as it seems to be implying that it is possible for random processes to produce a structured output, whereas the context in which this image is used - for example by Eddington - is precisely to illustrate the absurdity of such a suggestion. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 09:49, 29 June 2015 (UTC)
:Uff. No, it does not "rely on the assertion that one divided by infinity equals zero". The notion of what constitutes a zero probability is all perfectly standard and rigorous. It's part of [[measure theory]], and you need to learn something about it.
:As to whether this result in particular is called a "theorem" in reliable sources, that may actually be a valid criticism. But that goes more to the article name than to the content. I don't know what is the best name for the article; it's a genuine issue.
:As for your last sentence, the point is that it ''is'' possible for a random process to produce a structured output, and that is in fact true. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:26, 29 June 2015 (UTC)
::I do not agree with [[User:Trovatore|Trovatore]] that the "infinite monkey theorem" really is a specific theorem; I think it allows a range of different interpretations that produce different mathematical statements. I don't think all who cite this "theorem" have the same exact interpretation in mind, but I do believe that, when calling it a theorem:
::* they do have assumptions in mind on how the "monkeys" behave that cause the resulting statement to be a theorem (even if those assumptions are not the same for everybody), and
::* those assumptions are not a description of how they would expect real monkeys to behave.
::A reasonable further specification of "typing randomly" would be to assume [[memorylessness]]: that is, to assume that the choice of the next key and the time until that key is hit are both completely independent of what has happened until then. This gives you a precise mathematical interpretation of the problem; under that interpretation, the problem is much simplified and it is easy to prove that for every string its probability of being produced converges to 1.
::Another reasonable interpretation is to assume that the monkeys type at a fixed rate, and (as before) that each key has the same probability of being chosen each time.
::You may argue that these interpretations are too strict, that we cannot assume that the monkeys hit each key with the same frequency or that their typing speed remains constant. But these assumptions can be relaxed quite a bit before the theorem no longer holds; and if you relax them beyond that, you need a really convincing argument that what your monkeys are doing is still random typing. For instance, if the monkeys' typing speed halves every minute, or the frequency of the letter Q being chosen halves every minute, I don't think we'd call what they're doing random typing anymore. [[User:Rp|Rp]] ([[User talk:Rp|talk]]) 20:05, 29 June 2015 (UTC)
:::Rp, you write ''I do not agree with [[User:Trovatore|Trovatore]] that the "infinite monkey theorem" really is a specific theorem.'' But I did not in fact say that it was. I actually think that name for the article is at least potentially problematic; that's one small point of common ground between me and Dave Apter. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:02, 29 June 2015 (UTC)
I will briefly note that it is extremely important for readers to realize that this is in the limit of infinite keystrokes and that no ''finite'' number of keystrokes can achieve that probability. This already confused some readers before (see some of the previous sections on this talk page).--[[User:Jasper Deng|Jasper Deng]] [[User talk:Jasper Deng|(talk)]] 00:36, 30 June 2015 (UTC)
:That's certainly true, but not obviously related to Dave Apter's points, unless I've completely misunderstood them. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 00:48, 30 June 2015 (UTC)
::Oh, just a quibble — it's true that the limit of the probability of getting whatever text, as the number of keystrokes approaches infinity, is one, but that's not what it ''means'' that the probability is one when you have a completed infinity of keystrokes. The latter statement is to be understood as a statement about [[probability measure]]s on the space of infinite sequences of keystrokes, not about limits. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 00:56, 30 June 2015 (UTC) 
Thank you {{U|Rp}} and {{U|Jasper Deng}} for your contributions to the discussion. For the record, your remark definitely ''is'' related to my points. [[Almost surely]] is defined as having a probability of 1 (which is not the same as 'certain to occur'). All we can say is that the probability approaches a limit of 1 as the time increases without limit. This is uncontroversial but not particularly illuminating. Talking about what happens "in an infinite amount of time" is mathematically sloppy and would fail any maths exam. Furthermore it is counterfactual speculation since there is no such thing as an infinite amount of time. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 10:48, 2 July 2015 (UTC)
:So first of all, '''''there is nothing wrong with counterfactuals'''''.
:But supposing there were — I'm gonna have to call "citation needed" on "there is no such thing as an infinite amount of time". How do you know that, exactly? --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 01:48, 8 July 2015 (UTC)
::It is irrelevant whether there is physically an infinite amount of time or not, since this is about an abstract [[Gedankenexperiment]] about probability, and not a discussion about real monkeys or typewriters. — [[User:Loadmaster|Loadmaster]] ([[User talk:Loadmaster|talk]]) 16:34, 18 July 2015 (UTC)
(a) I never said there was anything "wrong" with counterfactuals. (b) Of course I don't "know" whether there there is an infinite amount of time, and neither does anyone else, but it's my understanding that this is the current consensus among astrophysicists. But enough of this sophistry. I've amended the lead section to give a clearer and more accurate summary. Please feel free to discuss further improvements. There's still a great deal of cruft needing to be trimmed out of the rest of the article. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 10:41, 15 July 2015 (UTC)
::Dave, I give you credit, that version is better than your earlier attempts. But I don't really think it's better than what was there. For one thing, it's too focused on Borel's initial entry, whereas the formulations have evolved. Also it doesn't cover the notion of "almost surely" up front, which is one of the most important ideas to get across. I have reverted to the version of 03:03, 15 July 05. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:02, 16 July 2015 (UTC)
I think we are going to need to get some other opinions on this. The lead as it stood conveyed precisely the '''opposite''' impression of what Borel and Eddington were driving at. Rather than suggesting that such an outcome from such a process should bear consideration, they were attempting to illustrate the absurdity of the idea. I've kept your elision of the words 'popular' and 'misnomer', although I think that's an accurate description. Borel did not describe the idea as a "theorem", and so far as I can see it isn't one. I don't think the purported "proof" is actually a rigorous demonstration that the proposition is [[almost surely]] true. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 08:47, 18 July 2015 (UTC)
:On the naming issue you may have a point. I think we could profitably discuss that separately.
:Whatever ''Borel and Eddington'' were driving at, specifically, is not really the point. The idea is not specific to Borel and Eddington, not by any means. It's just a fact that, with infinitely many random independent identically distributed trials, for any event that has positive probability in a given trial, you get probability one for the infinitely many trials. (By the way, this is can be expressed with complete mathematical rigor in terms of completed infinite collections — you seem to think that it needs to be expressed in terms of limits, but that is not so. However that is a side angle.)
:You seem to have a dislike of this fact, for some reason, but it doesn't change it, it is still a fact. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 09:03, 18 July 2015 (UTC)

I've started a [[Talk:Infinite monkey theorem#Renaming this article|new section]] about renaming this article. — [[User:Loadmaster|Loadmaster]] ([[User talk:Loadmaster|talk]]) 16:27, 18 July 2015 (UTC)

== Applications and criticism: every possible text ==

An obvious criticism that isn't mentioned: The theorem tells us that the monkey typing for an infinite time will not only type ''Hamlet'', but '''every''' other finite text. Given that, how can there be anything notable about it typing ''Hamlet''? What conclusion can be drawn from the theorem that has any possible application? Isn't it entirely meaningless?

It's amusing to think about, for instance it also types this Wikipedia article, and every previous version of this Wikipedia article... and the entirety of Wikipedia, and the entirety of the Internet and everything ever written and spoken, both in chronological order and reverse chronological order and alphabetical order and every other possible order... as well as literally ''every'' other thing you can possibly imagine, and every possible variation or misspelling of it, so long as it's not infinitely long.
··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 11:25, 18 May 2015 (UTC)
:I think that's dealt with to some extent in the quote by Borges. I would sort of hope it's obvious, anyway. But I'm not in principle against saying it somewhere more prominent.
:Suggestions? What should we say, where do we put it, how do we source it? --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 18:39, 18 May 2015 (UTC)
::That only deals with the question of authorship, not relevance to probability or the anthropic principle (evolution). So it's a new point worth saying, we just need to find a source. I'm sure there are some good ones out there but I don't know how to find them, it's not an easy thing to search for. ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 11:34, 21 May 2015 (UTC)
:::I'm not sure what you mean by "authorship". The Borges quote covers your point about "every other finite text" pretty well.
:::It's possible I hadn't read your first paragraph in your first comment carefully. I don't see how it makes it "entirely meaningless".
:::Consider: Suppose that the universe is infinite and basically homogeneous. That may well be the case, so this consideration is not vacuous. Then, on the shores of some unimaginably distant ocean, light and dark sand grains have arranged themselves into a sharp readable copy of the King James Bible, except with the book of Judges replaced by a subtly distorted Quechua translation of the third chapter of ''Atlas Shrugged''. Not for sure, but almost for sure. How is that meaningless, just because there are other worlds with the same text, except that the name Adam is replaced by Alex? --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 14:30, 21 May 2015 (UTC)
::::I meant it's only dealing with the question of accidentally writing something, rather than the broader picture like evolution.
::::It would be meaningless because everything would be meaningless. Everything would exist almost infinite times, and there would be an almost infinite multiple of those in slight variations, and an almost infinite multiple of those in slightly larger variations, etc... So for starters, by definition no single thing in any particular world could be significant. Everything happening is equivalent in meaning to nothing happening, there's no basis for saying that any variation is more meaningful than any other.
::::But it's worse than that. Extremely improbable things become probable. For every world, there would be a near-infinite number of alternative, almost-identical worlds, but where a thought you had based on evidence is replaced with a thought based on hallucination, psychosis or other illusion. Although this is improbable, it still happens in a huge number of worlds. Thus it is unreasonable for anyone to trust their own thoughts, or anyone else's, and all rationality is unfounded. Thus the argument defeats itself.
::::There are huge numbers of worlds where *every* thought you have is unconnected to reality. If you take this all the way you end up with [[Boltzmann brain]] worlds (a thought experiment created for the purpose of arguments like this) - worst of all, [[Boltzmann brain]] worlds should be ''vastly more common'' than worlds like ours. ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 07:27, 23 May 2015 (UTC)
::::: Oh, I misunderstood you. You weren't saying that the ''statement'' was meaningless (in the sense of not having an interpretation); you were saying that it makes ''existence'' or ''experience'' meaningless (in the sense of having no ultimate importance). I don't really see it that way, but the way that I do see it is probably not relevant to the article so I won't get into it here.
::::: What might be relevant to the article is if there are notable thinkers that have interpreted the result that way and if we can find RSs for those thoughts. I don't know of any, but if there were such, it strikes me as at least plausible that we'd consider mentioning it here, even though it's maybe a bit of a tangent. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 17:29, 23 May 2015 (UTC)
::::::Yeah you get me. Though the statement is also meaningless in a fashion: if a corollary of a statement is that all statements are unreliable or meaningless, then it's self-refuting. ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 00:38, 24 May 2015 (UTC)

== RfC: Which of these versions of the lead is the more accurate and informative? ==

{{Archive top|result= 1) Trovatore's version was preferred; 2) Reliable sources classify this as a theorem; 3) Reliable sources state accepted proofs of this theorem. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 16:31, 28 July 2015 (UTC)}}
#Which of these two versions of the lead is the more accurate and informative: [https://en.wikipedia.org/w/index.php?title=Infinite_monkey_theorem&diff=next&oldid=671965663]?
#Is there any reference anywhere in the literature that classes this proposition as a ''theorem''?
#Is the purported "proof" given in this article a mathematically valid demonstration of the proposition stated in the lead (in the more recent version from the above diff) (With a [[wp:rs|reliable source]])? [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 10:37, 18 July 2015 (UTC)

*'''Comment''' — A general phrasing of the idea is appropriate for the lede section. Details about its origin and the subsequent modifications to it belong in a separate "History" or "Background" section. — [[User:Loadmaster|Loadmaster]] ([[User talk:Loadmaster|talk]]) 16:36, 18 July 2015 (UTC)

*'''Comment'''. Regarding (1), the first sentence of the original revision reads:
::{{quote|The '''infinite monkey theorem''' states that a monkey hitting keys at [[randomness|random]] on a [[typewriter keyboard]] for an infinite amount of time will [[almost surely]] type a given text, such as the complete works of [[William Shakespeare]].}}
:The first sentence of the new revision, that was reverted, reads:
::{{quote|The '''infinite monkey theorem''' is the name often used to refer to an idea from [[Émile Borel | Emile Borel's]] book on [[probability]], published in [[1909]].}}
:The first is a clear statement of the proposition. The second is not, and I do not think many sources place special emphasis on Borel's role, even if he was the first to formulate the theorem. (Also [[WP:YEARLINK|why is the year linked]]?) The next sentence is:
::{{quote| The book introduced the concept of "dactylographic[[#Footnote|1]] [[monkey]]s" seated in front of [[typewriter]] keyboards and hitting keys at random.}}
:This still lacks a clear statement, and brings in a word "dactylographic" which is not explained and does not appear in my dictionary. A reader lacking a knowledge of Greek might legitimately wonder if dactylographic was a synonym for "infinite", the subject of the article presumably being about "infinite monkeys". Whether there are infinitely many monkeys, a single monkey with infinite time (or neither) is never made clear, regardless of the intended meaning of the neologism. This is a grave omission, since it fails to articulate the conditions under which the "theorem" holds.

:The third sentence is:
::{{quote|Borel exemplified a proposition in the [[probability theory|theory of probability]] called [[Kolmogorov's zero-one law]] by saying that the probability is 1 that such a monkey will eventually type every book in [[France | France's]] [[Bibliothèque nationale de France|National Library]].}}
:Even now, we lack a clear statement of the result. This version still does not mention ''infinite'' monkeys, just a single monkey, so the relation to the article remains obscure. Furthermore, the article does not discuss the relation to Kolmogorov's zero-one law, so I think discussing this in the lead (without a source) is ipso facto problematic. Moreover, given the statement presented in the new lead, it is not clear that Kolmogorov's law is even applicable because the event "The Bibliotheque nationale eventually appears" is not a tail event, so any invocation of the 0-1 law needs explanation (with a source). The actual tail event that might be intended here is "The monkey types the text infinitely often", but this is not discussed in the article, and the 0-1 law would only give probability 0 or 1 for this event (it has probability 1, but not by Kolmogorov). So, that the IMT "exemplifies" the Kolmogorov zero-one law is dubious at best.

:The next sentence is
::{{quote|There need not be infinitely many monkeys; a single monkey who executes infinitely many keystrokes suffices.}}
:Which is perfectly true, but not very helpful in the context where it is used, since this is the first time the reader is actually told that there were infinitely many monkeys at all: the previous sentence already appears to have been about the actions of ''a single monkey''. The final sentence is a further remark on the red herring of Kolmogorov's 0-1 law, which borders on original research.

:So, given that the proposed revision is not accurate and does not contain a clear statement of the subject of the article, I conclude in regards to (1) that: ''the original revision is more accurate and informative''.

:Now, regarding point (2), I have found the following sources that use the exact term "infinite monkey theorem":
::* Marc Paolella (2007) ''Intermediate Probability: A Computational Approach'', Wiley.
::* Simon N. Chandler-Wilde, Marko Lindner (2011) ''Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices'', Memoirs of the American Mathematical Society.
::* Ian Stewart (2010) ''Professor Stewart's hoard of mathematical treasures'', Profile Books.
::* Christopher R S Banerji, Toufik Mansour, and Simone Severini (2014) "A notion of graph likelihood and an infinite monkey theorem", Journal of Physics A, ''47'' 035101 doi:10.1088/1751-8113/47/3/035101
::* Eric S. Raymond (1996) ''The New Hacker's dictionary'', MIT Press.
::* Prakash Gorroochurn (2012) ''Classic problems of probability'', Wiley.
::* Edward B. Burger, Michael P. Starbird (2005) ''The Heart of Mathematics: An Invitation to Effective Thinking'', Springer.
::* G. Spencer-Brown (1957) ''Probability and scientific inference'', Longmans-Green. (Refers to it as the "monkey theorem".)
:So, clearly yes, this is regarded as a theorem by many references.

:I feel that point (3) must be a trick question. I count two proofs. The first is an intuitive argument, which I don't think is intended to be completely rigorous, but it can be made so without too much effort. I would say that constitutes a "mathematically valid demonstration", modulo quibbles about standards of rigor. Here is a more rigorous version of the same argument. Let <math>E_n</math> denote the event that the word "banana" fails to appear after ''n'' blocks of 6 letters have been typed. Then, assuming that each character is independent and uniformly distributed, we compute <math>p(E_n) = (1-1/50^6)^n</math>. Observe that the <math>E_n</math> are a nested sequence of subsets of the σ-algebra: <math>E_1\supset E_2\supset \cdots</math>. Let <math>E=\bigcap_{n=1}^\infty E_n</math>. Then the event ''E'' is that the word banana never appears. Then ''E'' is a measurable set, and <math>p(E) = \lim_{n\to\infty} p(E_n) = 0</math>. Thus, the complement of the event ''E'' has probability 1. That is, the word "banana" must appear almost surely. These details are sufficiently routine that anyone with a passing familiarity with the mathematical foundations of probability can supply them, and I do not think the article would benefit from such added details.
:One thing that could be made slightly clearer is that the same argument applies with the word "banana" replaced by any particular (finite) string of length k (e.g., the collected works of Shakespeare), but with <math>p(E_n)= (1-1/50^k)^n</math> instead. The simple word "banana" is just being used for illustration purposes.
:The second proof shows that the theorem is a straightforward application of the second Borel-Cantelli lemma. I would also call this proof "mathematically valid", although I think it should be clarified that the infinite string is broken into non-overlapping blocks of length ''k'' (otherwise the events referred to there are not independent). [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Slawekb|talk]]) 12:55, 20 July 2015 (UTC)

*'''Comment''' on 1) Trovatores version is a better lede than DaveApter's in the diff posted in the RFC. On 2)A quick google for "infinite monkey" produces about 350 results of which about 290 contain "infinite monkey theorem" so clearly that is the common term. I have no comment on 3)[[User:SPACKlick|SPACKlick]] ([[User talk:SPACKlick|talk]]) 11:34, 21 July 2015 (UTC)
*'''Comment'''. (1) Trovatore's version is better, because one can read it without reading anything from the rest of the article and still learn something useful about the subject. That is not true for the other version. (2) Yes, it can be classified as a theorem, but that's the wrong question: the right question is whether "infinite monkey theorem" is a [[WP:COMMONNAME|common name]] of this subject, ignoring whether it's actually an accurate name. As SPACKlick's search results suggest, the answer to the right question is also yes. (3) There is more than one proof in the article, and yes, they are valid mathematical demonstrations of a mathematical abstraction of the claim. The scare quotes in your question are inappropriate editorialization. You might argue that because they prove an abstraction rather than the actual real-world claim, the answer should be no, but I think that would be a mistake: the claim itself is already an abstraction . The part in the claim about an "infinite amount of time" should have been a clue to that. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:05, 22 July 2015 (UTC)
*'''Comment'''. 1) Trovatores's version is better, because it's less technical and more readily understandable. 2) I've always encountered it as "infinite monkey theorem", and a Web of Science search finds some results as well, so yes. 3) The theorem is intuitively obvious to me, and the "direct proof" more than suffices to 'prove' the theorem to my satisfaction. I'm not a mathematician, and they may have different and far more rigourous standards, but for the purposes of an encyclopedia I think it's sufficient. [[User:Banedon|Banedon]] ([[User talk:Banedon|talk]]) 08:54, 22 July 2015 (UTC)
**OK, thanks for the support all, but just for the record, it isn't ''my'' version. I don't think I've even contributed much to it. It's just the longstanding version that I think is preferable to Dave's changes (which is certainly not to say that it can't be improved, or even that some full rewrite might not be indicated — just not one along Dave's current line of thinking). --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 16:26, 22 July 2015 (UTC) 
*'''Clarification'''. Thanks to everyone who has commented so far. It's clear that the consensus on point (1) is that my attempt at a boldly revised wording is not an overall improvement, and I accept that conclusion. However, I get the impression that there seems to be some misunderstanding of what I am driving at with the other two questions:
**Regarding my second question, I'm not disputing that "infinite monkey theorem" is the common name for this proposition, or suggesting that the title of the article should be changed. My point is this (and it is interrelated with the third question): is there an authoritative source that states explicitly states that this is a ''theorem'', rather than just acknowledging that this is the name colloquially used to refer to it? If merely the latter, I should have thought that it would be important to establish this clearly at the outset. An earlier draft of my lead said that it was "...a common misnomer for an idea...", but {{U|Trovatore}} objected to that and instantly reverted it, hence the weaker wording "...is the name often used to refer to an idea...".
**My concern regarding the proof is as follows: the concluding line is that 'As n approaches infinity, the probability Xn approaches zero.' Entirely uncontroversial, and actually obvious; but '''this was not the proposition to be demonstrated'''. To prove that "''a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text,''", the conclusion of the proof would have to be 'At infinity the probability Xn is equal to 0.' Talking about limits as a value approaches infinity is precise and rigorous; talking about results "after an infinite amount of time" is sloppy and can lead to paradoxical conclusions (such as this one!). If the proof doesn't establish the proposition, then it isn't a theorem. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 10:20, 22 July 2015 (UTC)
*'''Comment''' on the "clarification". Limits are not required to make the statement of the theorem rigorous. See the more rigorous version of the first proof that I articulated. If <math>E_n</math> is the event "The word ''banana'' does not appear after ''n'' blocks of 6 characters have been typed", then the event <math>E=\bigcap_{n=1}^\infty E_n</math> is the event "The word "banana" never appears". The concept of a limit is not required; one simply can prove the equality of these two sets in the usual way, by establishing two inclusions. A fully expanded version of the first proof would not need the use of limits either, just the Archimedean property of the real numbers, and the monotonicity of the probability measure. One can show that <math>p(E) = 0</math> in this way, although it is probably simplest (as the article does) just to show that <math>p(E)=\lim_{n\to\infty}p(E_n)=0</math>. This is true, not because the statement of the theorem requires limits to be invoked, but because of the [[probability axioms]] (see also [[probability measure]]). In any case, fully rigorous formulations of two versions of the theorem appear in the section "Infinite strings". Neither of these statements requires limits. One simply has a probability measure on the space of all infinite strings.
:I think the objection has more to do with a reluctance to consider the idea that an infinite string is meaningful. While there are schools of mathematics and philosophy (for example [[ultrafinitism]]) in which the concept of an infinite string would be rejected as meaningless, in the [[ZFC|conventional axiomatization of mathematics]], such infinite objects are indeed allowed and meaningful statements can be made about them independently of any notion of limit. In fact, the [[axiom of infinity]] is logically prior to the concept of a limit, so one can have infinite sets without a concept of limit, but not the other way around. So arguing that infinite sets or infinite sequences are meaningless in favor of infinite limits is just begging the question. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Slawekb|talk]]) 13:21, 22 July 2015 (UTC)
::So is that your personal assessment of the validity, or are there reliable sources that could be cited? [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 15:16, 22 July 2015 (UTC)
:::I've already given sources. The cited source by Gut shows in fact that the probability of the event "The monkey types 'banana'" infinitely often is equal to one. Limits do not come into it, although they are of course used in the proof. Clear formulations appear in several of the abovementioned works as well. The cited source by Isaacs, for example, contains a lengthy discussion (although it is written in a somewhat chatty style which someone lacking background in the mathematical formulations of probabilty might find fault with). The proof given there is almost identical with the expanded version I gave at length.
:::Not one of these sources points out that the correct formulation should be, as you hold, that "'As n approaches infinity, the probability Xn [sic] approaches zero.'" So, if you're going to challenge the validity of such statements of the theorem, it might be better if you showed some indication of having read and understood thso e cited in the article and RfC. Insisting that educated readings of probability sources by individuals with some familiarity with the fundamentals of probability theory is just a "personal assessment" is an unconstructive ad hominem dismissal. ''You'' brought up limits, because you reject the idea of infinite time as mathematically meaningful in this context. I was pointing out why this is a misapprehension. But you don't show any indication if having understood the reply, the sources, or the comments above. So I don't think further discussion is likely to be constructive. Consensus is clear. [[WP:STICK]]. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Slawekb|talk]]) 16:04, 22 July 2015 (UTC)
::::I don't see any reference to a source from 'Gut' on this page - where is that? Regarding my quote above, this is at the end of the third paragraph in the section 'Direct proof' in this article. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 16:24, 22 July 2015 (UTC)
::::: It's referenced in the article: "The first theorem is proven by a similar if more indirect route in Gut, Allan (2005). ''Probability: A Graduate Course. Springer''. pp. 97–100. {{ISBN|0-387-22833-0}}."
::::: I don't see why you think that a statement in the article regarding limits means that the statement of the theorem should be changed. Indeed, one of the defining properties of a [[probability measure]] is that <math>\lim_{n\to\infty}p(E_n)=p(E)</math> if <math>E_n</math> is a nested family of measurable sets ("events") whose intersection of ''E''. Notice that a limit appears on one side of this equation, but not the other. So if we had <math>\lim_{n\to\infty}p(E_n)=0</math>, we would be justified in saying that <math>p(E)=0</math> as well. The statement "<math>p(E)=0</math>" is then just as true as the equivalent statement "<math>\lim_{n\to\infty}p(E_n)=0</math>", but does not involve the use of the limit concept. That's generally what [[identity (mathematics)|identities]] are good for. You will often see this property used without comment in sources on probability (see any textbook on the foundations of probability or measure theory, e.g., Gut, Theorem 1.3.1, first chapter of Rudin's "Read and complex analysis", etc).
::::: I have already indicated that, just because ''a proof'' uses limits, does not mean that ''the theorem'' is about limits. Also, I have explained how the limits that appear in the proof are merely expedient; they are not actually essential to the proof anyway. An example is the theorem, that the area of the plane region bounded by the curves <math>y=x,y=0,x=1</math> is equal to one half. This is a true theorem of plane geometry, and there are several ways to prove it. One is using elementary methods, and other is to calculate the integral <math>\int_0^1x\,dx</math>. This is a kind of limit. But the statement of the theorem is not really about limits, it's about plane areas. It's true that the limit <math>\lim_{n\to\infty}\frac{1}{n^2}\sum_{k=1}^nk</math> is also equal to one half, but one still has to prove that this limit gives the area of the plane region. The theorem is not really about that limit, even if one proof can be reduced to computing this limit at some level. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Slawekb|talk]]) 17:08, 22 July 2015 (UTC)
::::::I was merely expressing a preference for a convention well established since the time of Euclid whereby the final line of the proof states the proposition to be proved - indicated by 'Quod erat Demonstrandum'. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 10:07, 27 July 2015 (UTC)
::::::: The thirteen books of Euclid were originally written in Greek, and would not have included such Latin text (he wrote ὅπερ ἔδει δεῖξαι at the end of propositions). Anyway, this practice is now deprecated in most modern mathematics writing. For example, Goursat's 1905 treatise ''Cours d'analyse mathématique'' does not use such a phrase or abbreviation to denote the end of a proof. The end of a proof environment in the standard AMSTeX maintained by [[American Mathematical Society]] is a small square box at the right margin of the page. Since Wikipedia is not a textbook, we don't usually include "proofs" as such. In this case, we are giving a mathematically valid argument that can be made into a rigorous proof without much effort. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Slawekb|talk]]) 12:23, 27 July 2015 (UTC)
::::::I can't be sure whether you misunderstood my point, or are just deliberately being obtuse. Strange to say, I did realise that Euclid wrote in Greek and used a Greek phrase rather than a Latin one. Nor is my point whether the end of the proof is indicated by the annotation 'QED', by a square, or any other convention. My point was that the purported 'direct proof' given in the article stops short of deriving the proposition. Regardless of whether Wikipedia is a textbook or not, it would serve readers if sections identified as proofs are rigorous expositions. The purpose of this page is for discussion of how to improve the article (which is sorely needed in this case), not to indulge in displays of erudition. The questions I asked could readily be answered by giving a '''precise''' quotation and exact reference for (a) a reputable mathematician stating that the proposition ''is'' a theorem (strictly speaking); and (b) a mathematician endorsing proof of same. It would be preferable (though not of course essential) if the ref can be checked out online without having to track down a hard copy of the text. For instance, the online source I found for Gut returned 'not found' when I searched for Monkey; and I didn't see anything in the Contents listing that gave a clue where there might be some discussion of this. Once these two points have been settled, maybe we could move on to address some of the other numerous shortcomings in this article. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 14:18, 27 July 2015 (UTC)
::::::: @DaveApter: I'm puzzled by your claim that this does not appear in the cited (2005) work of Allan Gut, ''Probability: a graduate course''. The article refers to pages 97-100 of that work, on which pages there is a discussion of "the monkey and the typewriter". The moniker "infinite monkey theorem" is sometimes used to refer to Theorem 18.4 of Gut, which is the second theorem stated in the "infinite strings" section, modulo some trivial details (see, for instance, Chandler-Wilde and Lindner, p. 94, where they refer to this principle in this way). Prakash Gorroochurn states on p. 209, "The result we have just proved is the so-called ''infinite monkey theorem''", referring to Gut and Isaac for a fuller discussion. Isaac has the "direct proof" on pages 48-50 in his (1991) ''The pleasures of probability'', where he says on page 50: "a theorem can be proved asserting that the monkey will type out the works of Shakespeare not only once but actually ''infinitely often'' with certainty." ''Bio-inspired computation in telecommunications'' (2015) by Xin-She Yang, Su Fong Chien, T.O. Ting has, on page 4, a statement of the infinite monkey theorem: "the probability of producing any given text will almost surely be 1 if an infinite number of monkeys randomly type for an infinitely long time". [[Ian Stewart]] (2012), in ''Professor Stewart's Hoard of Mathematical Treasures'', has a heading "The infinite monkey theorem" wherein it is said on p. 225, "if a monkey sat at a typewriter and kept hitting keys at random, then eventually it would type the complete works of Shakespeare". So, yes, this is regarded as a "theorem" in the literature. Proofs aimed at various levels of rigour can be found there. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Slawekb|talk]]) 15:30, 27 July 2015 (UTC)
::::::Thanks very much - that is helpful. Incidentally, I didn't 'claim that it doesn't appear' in Gut - just said that the search (on wherever I was looking, Google books or Amazon look inside etc) didn't return any search results; obviously some quirk of the indexing, or of the selection of the pages available. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 15:53, 27 July 2015 (UTC)
{{Archive bottom}}

== Renaming this article ==

It has been mentioned that using the word "theorem" in the title is not entirely accurate. If we are serious about changing the title of this article, how about ''Infinite Monkey Conjecture'' or ''Infinite Monkey Principle''? — [[User:Loadmaster|Loadmaster]] ([[User talk:Loadmaster|talk]]) 16:26, 18 July 2015 (UTC)

:As to whether the subject is a real [[mathematical theorem]] or not, it is actually fairly easy to cast it as one: given a monkey (or the more general case of ''N'' monkeys) typing a truly random infinite sequence of characters, it is [[almost surely]] <del>(some would argue a [[probability|certainty]])</del> that within that sequence lies the complete corpus of Shakespeare, and indeed every other chosen finite sequence. — [[User:Loadmaster|Loadmaster]] ([[User talk:Loadmaster|talk]]) 16:29, 18 July 2015 (UTC)

The name "Infinite Monkey Theprem" is widely used (e.g. see [https://www.google.com/search?q=%22Infinite+Monkey+Theorem%22+-wikipedia&tbm=bks&ei=O4CqVcTXNcS3oQTT472wBQ&start=10&sa=N]), and I would guess it is by far the most common way of referring to this topic. Whether it is actually a valid mathematical theorem or not (though I'm convinced it is) is, as regards the aricles name, irrelevant, it only matters that it is most commonly called a theorem. [[User:Paul August|Paul August]] [[User_talk:Paul August|☎]] 16:49, 18 July 2015 (UTC)
:I guess what I'm wondering is whether the article itself propagated the name. That's the bad case that we don't want to get into (it's a little like "citogenesis"). Even if it's so, though, I don't know whether it can be fixed now, and I also have no alternative naming suggestion. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 17:29, 18 July 2015 (UTC)
:: I'm sure there is a certain amount of propagation. But not all of the sources I cited above are from the past 10 years or so, and one is from 1996. It's clear to me that we didn't invent the term. Moreover, "infinite monkey principle" only [https://www.google.com/search?client=ubuntu&channel=fs&q=infinite+monkey+probability&ie=utf-8&oe=utf-8&gfe_rd=cr&ei=xd6sVci9EuHE8gfZuaXgDg#channel=fs&tbm=bks&q=%22infinite+monkey+principle%22 gets one hit] on Google books, compared to [https://www.google.com/search?client=ubuntu&channel=fs&q=infinite+monkey+probability&ie=utf-8&oe=utf-8&gfe_rd=cr&ei=xd6sVci9EuHE8gfZuaXgDg#channel=fs&tbm=bks&q=%22infinite+monkey+theorem%22+-wikipedia over 500] for "infinite monkey theorem". Whatever the reason, "infinite monkey theorem" seems to be the standard term for this now and it's too late to try to legislate an alternative term. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Slawekb|talk]]) 13:01, 20 July 2015 (UTC)

== Correspondence between strings and numbers... ==

This section is a mixture of original research and complete and utter garbage. I'm not sure which has the highest proportion. — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/86.182.137.225|86.182.137.225]] ([[User talk:86.182.137.225|talk]]) 19:37, 5 September 2015 (UTC) 
:I don't actually see anything in the section that isn't ''true'', and I'm sort of curious what you think is "garbage" and why. But my curiosity is not really on topic here, and I do have to say that I'm not sure what the point of the section is in this particular article. If kept, it would need to be sourced, but before going to that effort, someone should probably explain the rationale for having it at all. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:10, 5 September 2015 (UTC)
::I don't see the point of it either. Let's [[wp:be bold|be bold]] and remove it. Otherwise we could wait a few weeks for possible objections. ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 10:04, 8 September 2015 (UTC)
::I think the main point is to link the subject of the article explicitly with the concept of a normal number. It seems to me that this is worth doing. We can cite Borel's 1909 paper, if necessary, where he proves both the Borel-Cantelli lemma, and uses it to show that almost all numbers are normal. [[User:Slawekb|<big>S</big>ławomir]] [[User talk:Slawekb|Biały]] 10:47, 8 September 2015 (UTC)
:::Why do we need to link it to normal numbers? ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 22:32, 8 September 2015 (UTC)
::::Because (a version of) the infinite monkey theorem is that almost every real number is normal. (See the cited work by Alan Gut, which we summarize with "Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.") Normality is a slight strengthening of this result that, with probability one, the frequency of any given word in a random infinite string is equal to the natural frequency of that word relative to all words of that size. [[User:Slawekb|<big>S</big>ławomir]] [[User talk:Slawekb|Biały]] 22:56, 8 September 2015 (UTC)

== Paradox ==

I just added "It is a [[veridical paradox]] (a [[Validity|valid]] [[Syllogism|argument]] with a seemingly [[Contradiction|absurd]] conclusion) that demonstrates [[Intuition (psychology)|counterintuitive]] properties of infinity." to the opening and was immediately reverted. I took this from [[Hilbert's paradox of the Grand Hotel]]. This theorem is clearly similarly counterintuitive. Why am I wrong? ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 22:38, 8 September 2015 (UTC)

:The article does not discuss the paradoxical nature of the theorem, and neither do any of the sources I have read. There is nothing especially paradoxical about the notion that certain enormously improbable events can happen with a small positive probability. In the context of the original examples by Borel and Eddington, our intuition is actually ''correct'', that a monkey sitting at a typewriter typing out the play Hamlet is an enormously improbable event. So it isn't really a "paradox" at all construed in this way. The point of their metaphor is that, while we have a pretty good idea how unlikely it is to get quality output from monkeys, we don't have much intuition for what 10^23's of molecules are doing. [[User:Slawekb|<big>S</big>ławomir]] [[User talk:Slawekb|Biały]] 23:13, 8 September 2015 (UTC)
::The paradox is that anything, no matter how ridiculously improbable, becomes almost certain. There's nothing intuitive about a process creating every piece of literature ever written, as well as a myriad of slight misspellings of each, and a variation of each where people are replaced by carrots, etc for everything imaginable... ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 01:49, 9 September 2015 (UTC)
::: Fine. You're welcome to think of it as a paradox or not. I, for one, see nothing paradoxical in small probabilities getting larger with many repetitions. (And I do not think I am unique in that regard.) But anyway, to write an encyclopedia article claiming that it's a paradox requires sources. The references in the article do ''not'' use this metaphor as a paradox. ''On the contrary'', they employ it as an ''intuitive'' result to illustrate some ''counterintuitive'' consequences of statistical mechanics. That's something like the opposite of a paradox. [[User:Slawekb|<big>S</big>ławomir]] [[User talk:Slawekb|Biały]] 02:12, 9 September 2015 (UTC)
:::: Fair enough. ··[[User:Gracefool|gracefool]][[User talk:gracefool|💬]] 22:51, 9 September 2015 (UTC)

== Popular culture ... wtf? ==

The Popular Culture section bit about the Ricky Gervaise show ... is it essential, encyclopaedic, etc? Doesn't really read that way to me. In particular, is there some dialect of English in which this bit makes sense: ''"He then doubled down by using the acumen that ..."'' He what? The what?? Help meeeeeeeeee ... [[Special:Contributions/77.96.249.228|77.96.249.228]] ([[User talk:77.96.249.228|talk]]) 20:23, 13 November 2015 (UTC)
:Yes, that was over the top. Removed, thanks. [[User:Johnuniq|Johnuniq]] ([[User talk:Johnuniq|talk]]) 23:05, 13 November 2015 (UTC)

== Actual immortal monkeys ==

Recently a dubious sentence in the lead, marked <nowiki>{{citation needed}}</nowiki>, was edited to an (in my opinion) even more dubious sentence:
<blockquote>It should also be noted that real monkeys do not produce actually random output, which means that an "actual" monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text - some letters or combinations of letters in Hamlet may have precisely zero probability of being typed.</blockquote>
How do "actual" monkeys differ from actual monkeys? An actual or "actual" monkey hitting keys for an infinite amount of time is apparently immortal. I pronounce with absolute certainty that no such actual monkeys exist. Even if they did, the typewriter would be total loss within 101010 seconds, a negligeable fraction of an infinite amount of time. Therefore the argument about what combinations of letters "actual" monkeys might or might not produce appears to be without merit. I also cannot think of any reasonable argument why "some letters or combinations of letters in Hamlet may have precisely zero probability of being typed". Granted the existence of immortal monkeys with everlasting typewriters, it may take eons before one hits the letter Q, but if it is physically possible, it is bound to happen eventually. I think this sentence is unconvincing original research; I have therefore removed it.  --[[User talk:Lambiam|Lambiam]] 11:22, 22 April 2016 (UTC)

== "Ridiculous" probability ==

{{re|Trovatore|Adrums63|Slawekb}}, Firstly {{diff|Infinite monkey theorem|718344271|718343086|my edit should not have been reverted}} because it contained two unrelated edits and the reverter didn't seem to be opposed to the other edit.

Secondly just because something seems subjective doesn't mean it is. The probabilities involved are '''objectively''' ridiculous. If three hundred and sixty thousand orders of magnitude longer than the estimated total age of the universe isn't ridiculous, ''nothing is''. I can't think of a better word - if you can think of one, please use it, but at this stage, "ridiculous" is more accurate than weaker language like "far more". "Far" doesn't even begin to cover it. ··[[User:Gracefool|gracefool]] [[User talk:gracefool|💬]] 22:49, 3 May 2016 (UTC)

:I agree that "ridiculous" is no good in an encyclopedia article. I think the article does a pretty good job of expressing the size of the numbers involved. And anyway "ridiculous" is subjective. There are numbers so large that the computational complexity needed to express those numbers exceeds the total number of particles in the observable universe. These are numbers that are useless, ''and can never even in principle be used in a mathematical argument'', whose size ''can never be comprehended in any human terms''. The numbers in the article are peanuts by comparison.* [[User:Slawekb|<big>S</big>ławomir]] [[User talk:Slawekb|Biały]] 22:57, 3 May 2016 (UTC) *Actually, some infinitesimally small peanut, whose smallness can never be comprehended in human terms, by comparison.
::In addition to the subjectivity, there's also the question of encyclopedic tone. In informal discourse, "ridiculous" is a fine word for "extreme beyond ordinary conception". But that's not what the word ''means'', in formal writing.
::In formal writing, something that is "ridiculous" is worthy of ridicule. It is not clear why one would ridicule a tiny probability or a very long time, or what good it would do to do so. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 04:03, 4 May 2016 (UTC)
: gracefool: I don't see any objection to the other part of your edit, that is true. So don't get upset; just restore that part of it. You can't expect people reviewing edits to sort through the whole thing, when there are multiple pieces to it. They just revert the whole thing; then you can restore the non-controversial parts. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 23:32, 3 May 2016 (UTC)
Yes of course numbers can't be ridiculous in themselves. What's ridiculous - as in worthy of ridicule - is actually '''using''' that number to say anything about actual probabilities. But okay we'll leave that to the criticisms section.

{{re|Trovatore}} people certainly do expect others reviewing edits to sort through the whole thing. Unless it's very complicated and you intend to revert the vast majority of it (but that means you still had to sort through it to some degree to make the decision). In this case the "multiple pieces" were two pieces. You're supposed to explain your revert; it seems your explanation is "I couldn't be bothered". Why should other people have to do what you're too lazy to do? Also [[Wikipedia:Revert only when necessary|there are a lot of good reasons to keep reverting to a minimum]]. ··[[User:Gracefool|gracefool]] [[User talk:gracefool|💬]] 01:34, 6 May 2016 (UTC)
:No, I'm sorry, I completely disagree with you. Reversion is ordinarily best done atomically. That minimizes the "version hell". Also, it's the new edit that has to justify itself, not the revert. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 04:40, 6 May 2016 (UTC)
::{{re|Trovatore}}: But my edit '''was''' a revert! By your logic, by restoring the original edit, you need to justify it. ··[[User:Gracefool|gracefool]] [[User talk:gracefool|💬]] 21:26, 12 May 2016 (UTC)

== A Universe Full of Monkeys ==

The opening paragraph states the following: ''"However, the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero)"''. I'd love to see the rough math on this, or at least an indication of how many monkeys the author thinks would fit into the universe. Even if we consider a non-expanding universe fixed at it's current dimensions, that's a butt-load of monkeys! — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/50.69.100.186|50.69.100.186]] ([[User talk:50.69.100.186#top|talk]]) 02:34, 4 January 2017 (UTC) 

: The section [[Infinite monkey theorem#Probabilities]] has the details you seek. Of course, notably the section excludes the fact that if you were to fill the universe with monkeys [[event horizon]]s would form all over the place, so that the question of "did the collection of monkeys filling the entire universe type the text of Hamlet" arguably becomes meaningless anyway. In fact, based on [[WP:OR|my own calculation]], any collection of about <math>10^{40}</math> or so monkeys, if they are closely but comfortably gathered together into a roughly spherical region, will produce a horizon. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Sławomir Biały|talk]]) 13:12, 4 January 2017 (UTC)
::Hmm? The universe may be infinite in extent. Of course that's a problem with the text as it stands as well — the common conflation of "universe" with "observable universe" is common, but still sloppy. We should fix that in the article. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:02, 22 April 2017 (UTC)
::: I understand it that the universe is filled with monkeys, but in the age of the universe (that is, in our particle horizon) the probability that a complete work could have been typed is very small. This amounts to the same thing as "observable universe". [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Sławomir Biały|talk]]) 10:44, 23 April 2017 (UTC)

== Transfinite Typing ==

"So the probability of the word banana appearing at some point after an infinite number of keystrokes is equal to one."

I'm confused by this. It seems to assert that the probability of the word banana appearing after an infinite number of keystrokes might not be one initially but if we keep typing it eventually will be. I read the comment that an infinite (ordered) cadre of monkeys yields an infinite sequence of letters with each keystroke, so if they type together a countable infinity (i.e. ω) of keystrokes their output has order type ω². Here we have continued typing after an infinite number of keystrokes. Is this the idea? Would striking the words "at some point" change the meaning of the sentence? [[User:Lewis Goudy|Lewis Goudy]] ([[User talk:Lewis Goudy|talk]]) 19:59, 22 April 2017 (UTC)
:Agreed - the entire article is somewhat confused. To say that it is 'almost certain' that a specified string would be typed in an infinite amount of time is meaningless, and also misleading - bearing in mind that the probability of its happening is negligible in the entire age of the universe. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 16:22, 14 September 2017 (UTC)
::It's neither meaningless nor misleading. It is meaningful and true. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 00:03, 15 September 2017 (UTC)
:::So what is the meaning? [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 15:59, 18 September 2017 (UTC)
:::: From the article: "Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position." [[Almost surely]] means with probability one. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Sławomir Biały|talk]]) 16:59, 18 September 2017 (UTC)
:::::So, given that there is no such thing as an infinite string, what is the meaning of that? [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 18:38, 18 September 2017 (UTC)
::::::There is in fact such a thing as an infinite string. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 18:41, 18 September 2017 (UTC)
::::::An infinite string is a mapping from the set of positive integers to the alphabet of characters. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Sławomir Biały|talk]]) 23:19, 18 September 2017 (UTC)

== External links modified ==

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*Added archive https://web.archive.org/web/20080513012236/http://skylla.wz-berlin.de/pdf/2002/ii02-101.pdf to http://skylla.wz-berlin.de/pdf/2002/ii02-101.pdf

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== Proof ==

An editor is apparently confused about the appropriateness of including proofs in mathematics articles. From [[WP:MSM]]: "we often want to include proofs, as a way of really exposing the meaning of some theorem", and from [[Wikipedia:WikiProject Mathematics/Proofs]]: "Proofs are often discussed in Wikipedia's mathematics articles, just as axioms, definitions, theorems, and lemmas are. Because much of the published professional literature of mathematics consists of the details of proofs, it would be very difficult to write in any depth about mathematics without including at least some proofs or proof sketches." We quite regularly include proofs in articles, when the proof is important to a reader understanding a topic. In this case, it is very clear that a proof, [[WP:MTAA|written at a level and style appropriate to likely readers]], is an essential part of the article. Maintenance templates should not be added without discussion of issues, and should certainly not be restored after they have been removed without first developing [[WP:CON|consensus]] on the discussion page. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Sławomir Biały|talk]]) 12:25, 10 February 2018 (UTC)

{{ec}}
:Wikipedia is [[WP:NOT|not a guidebook, manual or textbook]]. A general outline of a proof in prose, describing the approach a proof takes, is fine. But when you see fragments like this, it's a warning sign that the article is slipping into a textbook-like essay, which is not appropriate for an encyclopedia:
:* ''As an introduction, recall that...'' – this is not a description, this is standard verbiage from the set-up to a proof. Not appropriate language here.
:* ''Suppose...'' – don't suppose. Describe.
:* ''Let X...
:* ''Then, the chance that...'' – Don't follow the proof logic step by step; just describe it.
:* ''From the above...'' – Nope; just describe the proof, don't give it.
:* ''As n grows, Xn gets smaller...'' – this is not description, this is detail of the internals.
:Just describe the proof (if needed), don't go into the actual details of it. Wikipedia is not a mathematical journal. [[User:Mathglot|Mathglot]] ([[User talk:Mathglot|talk]]) 12:37, 10 February 2018 (UTC)

:: The above suggests a lack of familiarity with how mathematical proofs are written in prose. Imperative verbs like "let", "suppose", and "then" are regularly used in proofs. Encyclopedias on mathematics regularly use these words too. See, for instance, the Springer series ''Encyclopedia of the mathematical sciences''. In a Wikipedia article, we are expected to present content in a way that readers will find understandable. That often means describing aspects of the details. There is nothing in this article that is like a "mathematical journal". That also suggests unfamiliarity with how mathematical journals are written. Likewise, the proof would look very different from this if it were presented in a textbook on the subject. Instead, the proof is presented in a manner suitable for a general reader of an encyclopedia, not a student reader of a textbook or a researcher reading a journal article. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Sławomir Biały|talk]]) 12:43, 10 February 2018 (UTC)

:::{{ec}}
:::Reading [[WP:MSM#Proofs]], it appears that the guideline allows for Proofs in some circumstances; whether this one helps is debatable.
:::What's not debatable, is [[WP:WNTRMT]]; and your comment above that ''Maintenance templates should not be added without discussion of issues, and should certainly not be restored after they have been removed..'' has no basis in any guideline that I've seen, and is just your own, personal preference. I haven't removed the proof section itself, *that* would have been subject to [[WP:BRD]], I agree. However, *you* removed the template; that *is* subject to [[WP:WNTRMT]], and you should defend that here, or revert yourself. Cordially, [[User:Mathglot|Mathglot]] ([[User talk:Mathglot|talk]]) 12:48, 10 February 2018 (UTC)
:::: Tags should generally be accompanied by talk page discussion, and are usually for obvious and uncontroversial problems (e.g., an article has ''no references'' or is ''an essay''). Per [[Wikipedia:Template messages/Cleanup]]: '''Tags must be accompanied by a comment on the article's talk page explaining the problem and beginning a discussion on how to fix it, or for simpler and more obvious problems, a remark using the reason parameter as shown below. Tagging editors must be willing to follow through with substantive discussion.''' Furthermore, whether a tag is included in an article or not is still governed by [[WP:CON]] policy. (You've cited a help page, which is not a guideline or policy.) So far you've not convinced anyone that there is actually a problem with the section that warrants tagging. [[User:Sławomir Biały|Sławomir Biały]] ([[User talk:Sławomir Biały|talk]]) 13:12, 10 February 2018 (UTC)

== Sloppy treatment of infinities ==

Loose talk about infinities and 'almost certain' causes more confusion than enlightenment. The mathematically correct way of dealing with these issues is to speak of the approach to a limit, as the variable expands indefinitely. [[User:DaveApter|DaveApter]] ([[User talk:DaveApter|talk]]) 17:18, 5 December 2018 (UTC)
:No, that is incorrect. The modern treatment is to use [[measure theory]], which does not rely on limits per se. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:30, 5 December 2018 (UTC)

== Possible mistake about probabilities ==

In the second paragraph of 'Almost Surely,' the article states that "Equally probable is any other string of four characters allowed by the typewriter, such as "GGGG", "mATh", or "q%8e"." However, this is not true as (for example) to type "GGGG," not only would four 'g's need to be pressed, but the shift key would need to be held down for '''all four of them.''' [[Special:Contributions/2600:6C50:17F:F672:B9B3:58F6:4BF8:9E2D|2600:6C50:17F:F672:B9B3:58F6:4BF8:9E2D]] ([[User talk:2600:6C50:17F:F672:B9B3:58F6:4BF8:9E2D|talk]]) 18:25, 27 March 2020 (UTC)
:The example is supposing that capital letters and lower case letters are considered distinct keys with the same probability for simplicity. Note also that it includes %, which would have similar rules given a normal keyboard. I'm not sure which would be better: adding a note explaining that factoring in shift makes the math more complicated, or just forgetting about it and removing shift-key-only letters from the examples.'''[[User:IntegralPython| Integral Python]]''[[User talk:IntegralPython| click here to argue with me]]''''' 21:39, 30 August 2020 (UTC)

== Archiving request ==

The page [[WP:AATP]] suggests a talk page start to archive content once its over 70,000 bytes for several reasons, and this talk page is currently twice that at about 150,000. I would do it myself, but I lack the technical know-how, so could somebody more versed in this area look into archiving some of the much older sections on this talk page, and/or set up the auto-archive bot? Thanks! '''[[User:IntegralPython| Integral Python]]''[[User talk:IntegralPython| click here to argue with me]]''''' 23:32, 30 August 2020 (UTC)
:If there's no word against it, I'll take the silence as a consensus and figure out how to start an automatic archive, unless someone else who has more experience is willing to do it. '''[[User:IntegralPython| Integral Python]]''[[User talk:IntegralPython| click here to argue with me]]''''' 21:45, 25 March 2021 (UTC)

== Proof flawed ==

The flaw is here:

« Because each block is typed independently, the chance Xn of not typing banana in any of the first n blocks of 6 letters is… »

This is wrong precisely because each « block » is independent of all others. That argument requires that all « blocks » be different. Regardless of how many times you pull 6 letters, the probability remains 1 in 15 billions each time.
(Yes, Banana. Monke need banan-a to return to monk-e. --[[User:Teuf0rt|Teuf0rt]] ([[User talk:Teuf0rt|talk]]) 13:36, 25 March 2021 (UTC)https://return-to-monke.com/)
[[Special:Contributions/2605:B100:E034:D642:9DDC:E402:9D28:67F3|2605:B100:E034:D642:9DDC:E402:9D28:67F3]] ([[User talk:2605:B100:E034:D642:9DDC:E402:9D28:67F3|talk]]) 01:51, 2 October 2020 (UTC)
:No, it doesn't require that the blocks are different; it only requires that all of them are unequal to "banana". [[User:Rp|Rp]] ([[User talk:Rp|talk]]) 15:40, 5 October 2020 (UTC)

Topologist's sine curve

2021-03-23T21:07:59Z

IntegralPython: reflist + plus Munkres' book uses closed version as the namesake

{{Short description|Pathological topological space}}
[[Image:Topologist's sine curve.svg|420px|thumb|Topologist's Sine Curve| As ''x'' approaches zero from the right, the magnitude of the rate of change of 1/''x'' increases. This is why the frequency of the sine wave increases as one moves to the left in the graph.]]

In the branch of [[mathematics]] known as [[topology]], the '''topologist's sine curve''' or '''Warsaw sine curve''' is a [[topological space]] with several interesting properties that make it an important textbook example.

It can be defined as the [[graph of a function|graph]] of the function sin(1/''x'') on the [[half-open interval]] (0, 1], together with the origin, under the topology [[subspace topology|induced]] from the [[Euclidean plane]]:

:<math> T = \left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}. </math>

==Properties==
The topologist's sine curve ''T'' is [[connected space|connected]] but neither [[locally connected space|locally connected]] nor [[connected space#Path connectedness|path connected]]. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a [[path (topology)|path]].

The space ''T'' is the continuous image of a [[locally compact]] space (namely, let ''V'' be the space {−1} ∪ (0, 1<nowiki>]</nowiki>, and use the map ''f'' from ''V'' to ''T'' defined by ''f''(−1) = (0,0) and ''f''(''x'') = (''x'', sin(1/''x'')) for ''x'' > 0), but ''T'' is not locally compact itself.

The [[topological dimension]] of ''T'' is 1.

==Variants==
Two variants of the topologist's sine curve have other interesting properties.

The '''closed topologist's sine curve''' can be defined by taking the topologist's sine curve and adding its set of [[limit point]]s, <math>\{(0,y)\mid y\in[-1,1]\}</math>; some texts define the topologist's sine curve itself as just this closed version.<ref>{{cite book |last=Munkres |first=James R |date=1979 |title=Topology; a First Course |publisher=Englewood Cliffs |page=158 |isbn=9780139254956}}</ref> This space is closed and bounded and so [[compact space|compact]] by the [[Heine–Borel theorem]], but has similar properties to the topologist's sine curve—it too is connected but neither locally connected nor path-connected.

The '''extended topologist's sine curve''' can be defined by taking the closed topologist's sine curve and adding to it the set <math>\{(x,1) \mid x\in[0,1]\}</math>. It is [[arc connected]] but not [[Locally connected space|locally connected]].

== See also ==

* [[List of topologies]]
* [[Warsaw circle]]

==References==
{{reflist}}
*{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=Dover Publications, Inc. | location=Mineola, NY | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 |mr=1382863 | year=1995 | pages=137–138}}
*{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}}

[[Category:Topological spaces]]

Topologist's sine curve

2021-03-23T20:55:09Z

IntegralPython: There is no reason the image should be its own section; it seems to be a relic from a long time ago

{{Short description|Pathological topological space}}
[[Image:Topologist's sine curve.svg|420px|thumb|Topologist's Sine Curve| As ''x'' approaches zero from the right, the magnitude of the rate of change of 1/''x'' increases. This is why the frequency of the sine wave increases as one moves to the left in the graph.]]

In the branch of [[mathematics]] known as [[topology]], the '''topologist's sine curve''' or '''Warsaw sine curve''' is a [[topological space]] with several interesting properties that make it an important textbook example.

It can be defined as the [[graph of a function|graph]] of the function sin(1/''x'') on the [[half-open interval]] (0, 1], together with the origin, under the topology [[subspace topology|induced]] from the [[Euclidean plane]]:

:<math> T = \left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}. </math>

==Properties==
The topologist's sine curve ''T'' is [[connected space|connected]] but neither [[locally connected space|locally connected]] nor [[connected space#Path connectedness|path connected]]. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a [[path (topology)|path]].

The space ''T'' is the continuous image of a [[locally compact]] space (namely, let ''V'' be the space {−1} ∪ (0, 1<nowiki>]</nowiki>, and use the map ''f'' from ''V'' to ''T'' defined by ''f''(−1) = (0,0) and ''f''(''x'') = (''x'', sin(1/''x'')) for ''x'' > 0), but ''T'' is not locally compact itself.

The [[topological dimension]] of ''T'' is 1.

==Variants==
Two variants of the topologist's sine curve have other interesting properties.

The '''closed topologist's sine curve''' can be defined by taking the topologist's sine curve and adding its set of [[limit point]]s, <math>\{(0,y)\mid y\in[-1,1]\}</math>. This space is closed and bounded and so [[compact space|compact]] by the [[Heine–Borel theorem]], but has similar properties to the topologist's sine curve—it too is connected but neither locally connected nor path-connected.

The '''extended topologist's sine curve''' can be defined by taking the closed topologist's sine curve and adding to it the set <math>\{(x,1) \mid x\in[0,1]\}</math>. It is [[arc connected]] but not [[Locally connected space|locally connected]].

== See also ==

* [[List of topologies]]
* [[Warsaw circle]]

==References==
*{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=Dover Publications, Inc. | location=Mineola, NY | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 |mr=1382863 | year=1995 | pages=137–138}}
*{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}}

[[Category:Topological spaces]]

Topologist's sine curve

2021-03-23T20:46:48Z

IntegralPython: Importing Wikidata short description: "Pathological topological space" (Shortdesc helper)

{{Short description|Pathological topological space}}
In the branch of [[mathematics]] known as [[topology]], the '''topologist's sine curve''' or '''Warsaw sine curve''' is a [[topological space]] with several interesting properties that make it an important textbook example.

It can be defined as the [[graph of a function|graph]] of the function sin(1/''x'') on the [[half-open interval]] (0, 1], together with the origin, under the topology [[subspace topology|induced]] from the [[Euclidean plane]]:

:<math> T = \left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}. </math>

==Image of the curve==
[[Image:Topologist's sine curve.svg|420px|Topologist's Sine Curve]]

As ''x'' approaches zero from the right, the magnitude of the rate of change of 1/''x'' increases. This is why the frequency of the sine wave increases as one moves to the left in the graph.

==Properties==
The topologist's sine curve ''T'' is [[connected space|connected]] but neither [[locally connected space|locally connected]] nor [[connected space#Path connectedness|path connected]]. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a [[path (topology)|path]].

The space ''T'' is the continuous image of a [[locally compact]] space (namely, let ''V'' be the space {−1} ∪ (0, 1<nowiki>]</nowiki>, and use the map ''f'' from ''V'' to ''T'' defined by ''f''(−1) = (0,0) and ''f''(''x'') = (''x'', sin(1/''x'')) for ''x'' > 0), but ''T'' is not locally compact itself.

The [[topological dimension]] of ''T'' is 1.

==Variants==
Two variants of the topologist's sine curve have other interesting properties.

The '''closed topologist's sine curve''' can be defined by taking the topologist's sine curve and adding its set of [[limit point]]s, <math>\{(0,y)\mid y\in[-1,1]\}</math>. This space is closed and bounded and so [[compact space|compact]] by the [[Heine–Borel theorem]], but has similar properties to the topologist's sine curve—it too is connected but neither locally connected nor path-connected.

The '''extended topologist's sine curve''' can be defined by taking the closed topologist's sine curve and adding to it the set <math>\{(x,1) \mid x\in[0,1]\}</math>. It is [[arc connected]] but not [[Locally connected space|locally connected]].

== See also ==

* [[List of topologies]]
* [[Warsaw circle]]

==References==
*{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=Dover Publications, Inc. | location=Mineola, NY | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 |mr=1382863 | year=1995 | pages=137–138}}
*{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}}

[[Category:Topological spaces]]

Lower limit topology

2021-03-22T23:29:22Z

IntegralPython: Adding short description: "Topology on the real numbers" (Shortdesc helper)

{{Short description|Topology on the real numbers}}
In [[mathematics]], the '''lower limit topology''' or '''right half-open interval topology''' is a [[topological space|topology]] defined on the set <math>\mathbb{R}</math> of [[real numbers]]; it is different from the standard topology on <math>\mathbb{R}</math> (generated by the [[open interval]]s) and has a number of interesting properties. It is the topology generated by the [[basis (topology)|basis]] of all [[half-open interval]]s <nowiki>[</nowiki>''a'',''b''<nowiki>)</nowiki>, where ''a'' and ''b'' are real numbers.

The resulting [[topological space]] is called the '''Sorgenfrey line''' after [[Robert Sorgenfrey]] or the '''arrow''' and is sometimes written <math>\mathbb{R}_l</math>. Like the [[Cantor set]] and the [[long line (topology)|long line]], the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in [[general topology]].
The [[product space|product]] of <math>\mathbb{R}_l</math> with itself is also a useful counterexample, known as the [[Sorgenfrey plane]].

In complete analogy, one can also define the '''upper limit topology''', or '''left half-open interval topology'''.

== Properties ==
* The lower limit topology is [[finer topology|finer]] (has more open sets) than the standard topology on the real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a (countably infinite) union of half-open intervals.
* For any real <math>a</math> and <math>b</math>, the interval <math>[a,b)</math> is [[clopen set|clopen]] in <math>\mathbb{R}_l</math> (i.e., both [[open set|open]] and [[closed set|closed]]). Furthermore, for all real <math>a</math>, the sets <math>\{x\in\mathbb{R} : x < a\}</math> and <math>\{x \in\mathbb{R} : x \geq a\}</math> are also clopen. This shows that the Sorgenfrey line is [[totally disconnected]].
* Any [[compact space|compact subset]] of <math>\mathbb{R}_l</math> must be an at most [[countable set]]. To see this, consider a non-empty compact subset <math>C\subseteq\mathbb{R}_l</math>. Fix an <math>x \in C</math>, consider the following open cover of <math>C</math>:
::<math> \bigl\{ [x, +\infty) \bigr\} \cup \Bigl\{ \bigl(-\infty, x - \tfrac{1}{n} \bigr) \,\Big|\, n \in \mathbb{N} \Bigr\}.</math>
:Since <math>C</math> is compact, this cover has a finite subcover, and hence there exists a real number <math>a(x)</math> such that the interval <math>(a(x), x]</math> contains no point of <math>C</math> apart from <math>x</math>. This is true for all <math>x\in C</math>. Now choose a rational number <math>q(x) \in (a(x), x]\cap\mathbb{Q}</math>. Since the intervals <math>(a(x), x]</math>, parametrized by <math>x \in C</math>, are pairwise disjoint, the function <math>q: C \to \mathbb{Q}</math> is injective, and so <math>C</math> is at most countable.

* The name "lower limit topology" comes from the following fact: a sequence (or [[net (topology)|net]]) <math>(x_\alpha)</math> in <math>\mathbb{R}_l</math> converges to the limit <math>L</math> [[if and only if]] it "approaches <math>L</math> from the right", meaning for every <math>\epsilon>0</math> there exists an index <math>\alpha_0</math> such that <math>\forall\alpha \geq \alpha_0 : L \leq x_\alpha < L+\epsilon</math>. The Sorgenfrey line can thus be used to study [[right-sided limit]]s: if <math>f: \mathbb{R} \to \mathbb{R}</math> is a [[function (mathematics)|function]], then the ordinary right-sided limit of <math>f</math> at <math>x</math> (when the codomain carries the standard topology) is the same as the usual limit of <math>f</math> at <math>x</math> when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
* In terms of [[separation axioms]], <math>\mathbb{R}_l</math> is a [[perfectly normal Hausdorff space]].
* In terms of [[axiom of countability|countability axioms]], <math>\mathbb{R}_l</math> is [[first-countable space|first-countable]] and [[separable space|separable]], but not [[second-countable space|second-countable]].
* In terms of compactness properties, <math>\mathbb{R}_l</math> is [[Lindelöf space|Lindelöf]] and [[paracompact]], but not [[σ-compact space|σ-compact]] nor [[locally compact]].
* <math>\mathbb{R}_l</math> is not [[metrizable]], since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a [[Metric (mathematics)#Quasimetrics|quasimetric]].
* <math>\mathbb{R}_l</math> is a [[Baire space]] [http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2003&task=show_msg&msg=0878.0001.0001].

== See also ==

* [[List of topologies]]

== References ==
* {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}}

[[Category:Topological spaces]]

Large numbers

2021-03-10T15:11:39Z

IntegralPython: MOS:FIRST; it makes no sense to "define" large numbers in bold the way it was when large numbers is a description

{{other uses|Large number (disambiguation)}}
{{Multiple issues|
{{More citations needed|date=November 2011}}
{{overly detailed|date=January 2019}}
}}
{{short description|Numbers that are significantly larger than those used regularly}}

Numbers that are significantly larger than those typically used in everyday life, for instance in simple counting or in monetary transactions, appear frequently in fields such as [[mathematics]], [[physical cosmology|cosmology]], [[cryptography]], and [[statistical mechanics]]. The term typically refers to large positive [[integer]]s, or more generally, large positive [[real number]]s, but it may also be used in other contexts. The study of nomenclature and properties of large numbers is sometimes called googology.<ref>[http://www.mediafire.com/file/45j4oovzgleux3r/One_Million_Things_-_A_Visual_Encyclopedia.pdf/file One Million Things: A Visual Encyclopedia]</ref><ref>[https://books.google.com/books?id=Y87bDgAAQBAJ&pg=PA220 «The study of large numbers is called googology»]</ref>

Sometimes people refer to large numbers as being "astronomically large;" however, it is easy to mathematically define numbers that are much larger even than those used in astronomy.

== In the everyday world ==
{{See also|scientific notation|logarithmic scale|orders of magnitude}}

[[Scientific notation]] was created to handle the wide range of values that occur in scientific study. 1.0 × 109, for example, means one [[1000000000 (number)|billion]], a 1 followed by nine zeros: 1 000 000 000, and 1.0 × 10−9 means one billionth, or 0.000 000 001. Writing 109 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is.

Examples of large numbers describing everyday real-world objects include:
* The number of [[Cell (biology)|cells]] in the human body (estimated at 3.72 × 1013)<ref>{{Cite journal|last=Bianconi|first=Eva|last2=Piovesan|first2=Allison|last3=Facchin|first3=Federica|last4=Beraudi|first4=Alina|last5=Casadei|first5=Raffaella|last6=Frabetti|first6=Flavia|last7=Vitale|first7=Lorenza|last8=Pelleri|first8=Maria Chiara|last9=Tassani|first9=Simone|date=Nov–Dec 2013|title=An estimation of the number of cells in the human body|journal=Annals of Human Biology|volume=40|issue=6|pages=463–471|doi=10.3109/03014460.2013.807878|issn=1464-5033|pmid=23829164}}</ref>
* The number of [[bit]]s on a computer [[hard disk]] ({{as of|2020|lc=true}}, typically about 1013, 1–2 [[Terabyte|TB]])
* The number of [[Neuron|neuronal connections]] in the human brain (estimated at 1014)
* The [[Avogadro constant]] is the number of “elementary entities” (usually atoms or molecules) in one [[Mole (unit)|mole]]; the number of atoms in 12 grams of [[carbon-12]]{{Snd}} approximately {{val|6.022|e=23}}.
* The total number of [[DNA]] [[base pair]]s within the entire [[Biomass (ecology)|biomass]] on Earth, as a possible approximation of global [[biodiversity]], is estimated at (5.3±3.6)×1037<ref>{{cite journal | vauthors = Landenmark HK, Forgan DH, Cockell CS | title = An Estimate of the Total DNA in the Biosphere | journal = PLOS Biology | volume = 13 | issue = 6 | pages = e1002168 | date = June 2015 | pmid = 26066900 | pmc = 4466264 | doi = 10.1371/journal.pbio.1002168 }}</ref><ref name="NYT-20150718-rn">{{cite news |last=Nuwer |first=Rachel | name-list-style = vanc |date=18 July 2015 |title=Counting All the DNA on Earth |url=https://www.nytimes.com/2015/07/21/science/counting-all-the-dna-on-earth.html |work=The New York Times |location=New York |publisher=The New York Times Company |issn=0362-4331 |access-date=2015-07-18}}</ref>
* The mass of Earth consists of about 4x1051 [[nucleon]]s
* The estimated number of [[atom]]s in the [[observable universe]] (1080)
* The lower bound on the game-tree complexity of chess, also known as the “[[Shannon number]]” (estimated at around 10120)<ref>{{cite journal | author = Shannon, Claude | title = XXII. Programming a Computer for Playing Chess | journal = Philosophical Magazine | series = Series 7 | volume = 41 | issue = 314 | date = March 1950 | url = http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf | author-link = Claude Shannon | access-date = 2019-01-25 | archive-url = https://www.webcitation.org/5oFLE7Mgx?url=http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf | archive-date = 2010-03-15 | url-status = dead }}</ref>

== Astronomical ==
Other large numbers, as regards length and time, are found in [[astronomy]] and [[cosmology]]. For example, the current [[Big Bang model]] suggests that the universe is 13.8 billion years (4.355 × 1017 seconds) old, and that the [[observable universe]] is 93 billion [[light years]] across (8.8 × 1026 metres), and contains about 5 × 1022 stars, organized into around 125 billion (1.25 × 1011) galaxies, according to Hubble Space Telescope observations. There are about 1080 atoms in the [[observable universe]], by rough estimation.<ref>[http://www.universetoday.com/36302/atoms-in-the-universe/#gsc.tab=0 Atoms in the Universe]. Universe Today. 30-07-2009. Retrieved 02-03-13.</ref>

According to [[Don Page (physicist)|Don Page]], physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is

::::<math>10^{10^{10^{10^{10^{1.1}}}}} \mbox{ years}</math>

which corresponds to the scale of an estimated [[Poincaré recurrence theorem|Poincaré recurrence time]] for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain [[inflation (cosmology)|inflationary]] model with an [[inflaton]] whose mass is 10−6 [[Planck mass]]es.<ref name=page95>Information Loss in Black Holes and/or Conscious Beings?, Don N. Page, ''Heat Kernel Techniques and Quantum Gravity'' (1995), S. A. Fulling (ed), p. 461. Discourses in Mathematics and its Applications, No. 4, Texas A&M University Department of Mathematics. {{arxiv|hep-th/9411193}}. {{isbn|0-9630728-3-8}}.</ref><ref>[http://www.fpx.de/fp/Fun/Googolplex/GetAGoogol.html How to Get A Googolplex]</ref> This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history [[Loschmidt's paradox|repeats itself]] arbitrarily many times due to [[Ergodic hypothesis|properties of statistical mechanics]]; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.

[[Combinatorial]] processes rapidly generate even larger numbers. The [[factorial]] function, which defines the number of [[permutation]]s on a set of fixed objects, grows very rapidly with the number of objects. [[Stirling's formula]] gives a precise asymptotic expression for this rate of growth.

Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their [[logarithm]]s.

[[Gödel number]]s, and similar numbers used to represent bit-strings in [[algorithmic information theory]], are very large, even for mathematical statements of reasonable length. However, some [[pathological (mathematics)|pathological]] numbers are even larger than the Gödel numbers of typical mathematical propositions.

Logician [[Harvey Friedman]] has done work related to very large numbers, such as with [[Kruskal's tree theorem]] and the [[Robertson–Seymour theorem]].

==="Billions and billions"===
To help viewers of ''[[Cosmos: A Personal Voyage|Cosmos]]'' distinguish between "millions" and "billions", astronomer [[Carl Sagan]] stressed the "b". Sagan never did, however, say "[[billions and billions]]". The public's association of the phrase and Sagan came from a ''[[The Tonight Show|Tonight Show]]'' skit. Parodying Sagan's affect, [[Johnny Carson]] quipped "billions and billions".<ref>[http://www.csicop.org/si/show/carl_sagan_takes_questions Carl Sagan takes questions more from his 'Wonder and Skepticism' CSICOP 1994 keynote, Skeptical Inquirer] {{webarchive |url=https://web.archive.org/web/20161221054208/http://www.csicop.org/si/show/carl_sagan_takes_questions |date=December 21, 2016 }}</ref> The phrase has, however, now become a humorous fictitious number—the [[Indefinite and fictitious numbers#Sagan's number|Sagan]]. ''Cf.'', [[Carl Sagan#Sagan units|Sagan Unit]].

== Examples ==
{{See also|#Examples of numbers in numerical order}}
*[[googol]] = <math>10^{100}</math>
*[[centillion]] = <math>10^{303}</math> or <math>10^{600}</math>, depending on number naming system
*[[millinillion]] = <math>10^{3003}</math> or <math>10^{6000}</math>, depending on number naming system
*[[micrillion]] (millinillinillion) = <math>10^{3000003}</math> or <math>10^{6000000}</math>, depending on number naming system
*The largest known [[Smith number]] = (101031−1) × (104594 + 3{{e|2297}} + 1)1476 {{e|3913210}}
*The largest known [[Mersenne prime]] = <math>2^{82,589,933}-1</math> [https://www.mersenne.org/primes/?press=M82589933 (''as of December 21, 2018'')]
*[[googolplex]] = <math>10^{\text{googol}}=10^{10^{100}}</math>
*[[Skewes' number]]s: the first is approximately <math>10^{10^{10^{34}}}</math>, the second <math>10^{10^{10^{964}}}</math>
*[[Graham's number]], larger than what can be represented even using power towers ([[tetration]]). However, it can be represented using [[Knuth's up-arrow notation]]
*[[Rayo's number]] is a large number named after Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at [[Massachusetts Institute of Technology|MIT]] on 26 January 2007

== Standardized system of writing ==

A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one.

To compare numbers in scientific notation, say 5×104 and 2×105, compare the exponents first, in this case 5 > 4, so 2×105 > 5×104. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×104 > 2×104 because 5 > 2.

Tetration with base 10 gives the sequence <math>10 \uparrow \uparrow n=10 \to n \to 2=(10\uparrow)^n 1</math>, the power towers of numbers 10, where <math>(10\uparrow)^n</math> denotes a [[functional power]] of the function <math>f(n)=10^n</math> (the function also expressed by the suffix "-plex" as in googolplex, see [[Names of large numbers#The googol family|the Googol family]]).

These are very round numbers, each representing an [[order of magnitude]] in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.

More precisely, numbers in between can be expressed in the form <math>(10\uparrow)^n a</math>, i.e., with a power tower of 10s and a number at the top, possibly in scientific notation, e.g. <math>10^{10^{10^{10^{10^{4.829}}}}} = (10\uparrow)^5 4.829</math>, a number between <math>10\uparrow\uparrow 5</math> and <math>10\uparrow\uparrow 6</math> (note that <math>10 \uparrow\uparrow n < (10\uparrow)^n a < 10 \uparrow\uparrow (n+1)</math> if <math> 1 < a < 10</math>). (See also [[Tetration#Extension to real heights|extension of tetration to real heights]].)

Thus googolplex is <math>10^{10^{100}} = (10\uparrow)^2 100 = (10\uparrow)^3 2</math>

Another example:
:<math>2 \uparrow\uparrow\uparrow 4 =
\begin{matrix}
\underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\
\qquad\quad\ \ \ 65,536\mbox{ copies of }2 \end{matrix}
\approx (10\uparrow)^{65,531}(6 \times 10^{19,728}) \approx (10\uparrow)^{65,533} 4.3
</math> (between <math>10\uparrow\uparrow 65,533</math> and <math>10\uparrow\uparrow 65,534</math>)

Thus the "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (''n'') one has to take the <math>log_{10}</math> to get a number between 1 and 10. Thus, the number is between <math>10\uparrow\uparrow n</math> and <math>10\uparrow\uparrow (n+1)</math>. As explained, a more precise description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 1010, or the next, between 0 and 1.

Note that
:<math>10^{(10\uparrow)^{n}x}=(10\uparrow)^{n}10^x</math>
I.e., if a number ''x'' is too large for a representation <math>(10\uparrow)^{n}x</math> we can make the power tower one higher, replacing ''x'' by log10''x'', or find ''x'' from the lower-tower representation of the log10 of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course, similar remarks apply if the whole power tower consists of copies of the same number, different from 10).

If the height of the tower is large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so we can use the double-arrow notation, e.g. <math>10\uparrow\uparrow(7.21\times 10^8)</math>. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.

Examples:
:<math>10\uparrow\uparrow 10^{\,\!10^{10^{3.81\times 10^{17}}}}</math> (between <math>10\uparrow\uparrow\uparrow 2</math> and <math>10\uparrow\uparrow\uparrow 3</math>)
:<math>10\uparrow\uparrow 10\uparrow\uparrow (10\uparrow)^{497}(9.73\times 10^{32})=(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})</math> (between <math>10\uparrow\uparrow\uparrow 4</math> and <math>10\uparrow\uparrow\uparrow 5</math>)

Similarly to the above, if the exponent of <math>(10\uparrow)</math> is not exactly given then giving a value at the right does not make sense, and we can, instead of using the power notation of <math>(10\uparrow)</math>, add 1 to the exponent of <math>(10\uparrow\uparrow)</math>, so we get e.g. <math>(10\uparrow\uparrow)^{3} (2.8\times 10^{12})</math>.

If the exponent of <math>(10\uparrow \uparrow)</math> is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and we can, instead of using the power notation of <math>(10\uparrow \uparrow)</math>, use the triple arrow operator, e.g. <math>10\uparrow\uparrow\uparrow(7.3\times 10^{6})</math>.

If the right-hand argument of the triple arrow operator is large the above applies to it, so we have e.g. <math>10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})</math> (between <math>10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 4</math> and <math>10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 5</math>). This can be done recursively, so we can have a power of the triple arrow operator.

We can proceed with operators with higher numbers of arrows, written <math>\uparrow^n</math>.

Compare this notation with the [[hyper operator]] and the [[Conway chained arrow notation]]:
:<math>a\uparrow^n b</math> = ( ''a'' → ''b'' → ''n'' ) = hyper(''a'', ''n'' + 2, ''b'')
An advantage of the first is that when considered as function of ''b'', there is a natural notation for powers of this function (just like when writing out the ''n'' arrows): <math>(a\uparrow^n)^k b</math>. For example:

:<math>(10\uparrow^2)^3 b</math> = ( 10 → ( 10 → ( 10 → ''b'' → 2 ) → 2 ) → 2 )
and only in special cases the long nested chain notation is reduced; for ''b'' = 1 we get:
:<math>10\uparrow^3 3 = (10\uparrow^2)^3 1</math> = ( 10 → 3 → 3 )

Since the ''b'' can also be very large, in general we write a number with a sequence of powers <math>(10 \uparrow^n)^{k_n}</math> with decreasing values of ''n'' (with exactly given integer exponents <math>{k_n}</math>) with at the end a number in ordinary scientific notation. Whenever a <math>{k_n}</math> is too large to be given exactly, the value of <math>{k_{n+1}}</math> is increased by 1 and everything to the right of <math>({n+1})^{k_{n+1}}</math> is rewritten.

For describing numbers approximately, deviations from the decreasing order of values of ''n'' are not needed. For example, <math>10 \uparrow (10 \uparrow \uparrow)^5 a=(10 \uparrow \uparrow)^6 a</math>, and <math>10 \uparrow (10 \uparrow \uparrow \uparrow 3)=10 \uparrow \uparrow (10 \uparrow \uparrow 10 + 1)\approx 10 \uparrow \uparrow \uparrow 3</math>. Thus we have the somewhat counterintuitive result that a number ''x'' can be so large that, in a way, ''x'' and 10x are "almost equal" (for arithmetic of large numbers see also below).

If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it acts. We can simply use a standard value at the right, say 10, and the expression reduces to <math>10 \uparrow^n 10=(10 \to 10 \to n)</math> with an approximate ''n''. For such numbers the advantage of using the upward arrow notation no longer applies, and we can also use the chain notation.

The above can be applied recursively for this ''n'', so we get the notation <math>\uparrow^n</math> in the superscript of the first arrow, etc., or we have a nested chain notation, e.g.:

:(10 → 10 → (10 → 10 → <math>3 \times 10^5</math>) ) = <math>10 \uparrow ^{10 \uparrow ^{3 \times 10^5} 10} 10 </math>

If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number (like using the superscript of the arrow instead of writing many arrows). Introducing a function <math>f(n)=10 \uparrow^{n} 10</math> = (10 → 10 → ''n''), these levels become functional powers of ''f'', allowing us to write a number in the form <math>f^m(n)</math> where ''m'' is given exactly and n is an integer which may or may not be given exactly (for example: <math>f^2(3 \times 10^5)</math>). If ''n'' is large we can use any of the above for expressing it. The "roundest" of these numbers are those of the form ''f''''m''(1) = (10→10→''m''→2). For example, <math>(10 \to 10 \to 3\to 2) = 10 \uparrow ^{10 \uparrow ^{10^{10}} 10} 10 </math>

Compare the definition of Graham's number: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus <math> G < 3\rightarrow 3\rightarrow 65\rightarrow 2 <(10 \to 10 \to 65\to 2)=f^{65}(1)</math>, but also <math> G < f^{64}(4)<f^{65}(1)</math>.

If ''m'' in <math>f^m(n)</math> is too large to give exactly we can use a fixed ''n'', e.g. ''n'' = 1, and apply the above recursively to ''m'', i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation of ''f'' this gives multiple levels of ''f''. Introducing a function <math>g(n)=f^{n}(1)</math> these levels become functional powers of ''g'', allowing us to write a number in the form <math>g^m(n)</math> where ''m'' is given exactly and n is an integer which may or may not be given exactly. We have (10→10→''m''→3) = ''g''''m''(1). If ''n'' is large we can use any of the above for expressing it. Similarly we can introduce a function ''h'', etc. If we need many such functions we can better number them instead of using a new letter every time, e.g. as a subscript, so we get numbers of the form <math>f_k^m(n)</math> where ''k'' and ''m'' are given exactly and n is an integer which may or may not be given exactly. Using ''k''=1 for the ''f'' above, ''k''=2 for ''g'', etc., we have (10→10→''n''→''k'') = <math>f_k(n)=f_{k-1}^n(1)</math>. If ''n'' is large we can use any of the above for expressing it. Thus we get a nesting of forms <math>{f_k}^{m_k}</math> where going inward the ''k'' decreases, and with as inner argument a sequence of powers <math>(10 \uparrow^n)^{p_n}</math> with decreasing values of ''n'' (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.

When ''k'' is too large to be given exactly, the number concerned can be expressed as <math>{f_n}(10)</math>=(10→10→10→''n'') with an approximate ''n''. Note that the process of going from the sequence <math>10^{n}</math>=(10→''n'') to the sequence <math>10 \uparrow^n 10</math>=(10→10→''n'') is very similar to going from the latter to the sequence <math>{f_n}(10)</math>=(10→10→10→''n''): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). Numbering the subsequent versions of this function a number can be described using functions <math>{f_{qk}}^{m_{qk}}</math>, nested in [[lexicographical order]] with ''q'' the most significant number, but with decreasing order for ''q'' and for ''k''; as inner argument we have a sequence of powers <math>(10 \uparrow^n)^{p_n}</math> with decreasing values of ''n'' (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.

For a number too large to write down in the Conway chained arrow notation we can describe how large it is by the length of that chain, for example only using elements 10 in the chain; in other words, we specify its position in the sequence 10, 10→10, 10→10→10, .. If even the position in the sequence is a large number we can apply the same techniques again for that.

===Examples===
Numbers expressible in decimal notation:
*22 = 4
*222 = 2 ↑↑ 3 = 16
*33 = 27
*44 = 256
*55 = 3,125
*66 = 46,656
*<math>2^{2^{2^{2}}}</math> = 2 ↑↑ 4 = 2↑↑↑3 = 65,536
*77 = 823,543
*106 = 1,000,000 = 1 million
*88 = 16,777,216
*99 = 387,420,489
*109 = 1,000,000,000 = 1 billion
*1010 = 10,000,000,000
*1012 = 1,000,000,000,000 = 1 trillion
*333 = 3 ↑↑ 3 = 7,625,597,484,987 ≈ 7.63 × 1012
*1015 = 1,000,000,000,000,000 = 1 million billion = 1 quadrillion

Numbers expressible in scientific notation:
*Approximate [[Observable universe#Matter content – number of atoms|number of atoms in the observable universe]] = 1080 = 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
*googol = 10100 = 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
*444 = 4 ↑↑ 3 = 2512 ≈ 1.34 × 10154 ≈ (10 ↑)2 2.2
*Approximate number of [[Planck length|Planck volumes]] composing the volume of the observable [[universe]] = 8.5 × 10184
*555 = 5 ↑↑ 3 = 53125 ≈ 1.91 × 102184 ≈ (10 ↑)2 3.3
*<math>2^{2^{2^{2^2}}} = 2 \uparrow \uparrow 5 = 2^{65,536} \approx 2.0 \times 10^{19,728} \approx (10 \uparrow)^2 4.3</math>
*666 = 6 ↑↑ 3 ≈ 2.66 × 1036,305 ≈ (10 ↑)2 4.6
*777 = 7 ↑↑ 3 ≈ 3.76 × 10695,974 ≈ (10 ↑)2 5.8
*888 = 8 ↑↑ 3 ≈ 6.01 × 1015,151,335 ≈ (10 ↑)2 7.2
*<math>M_{77,232,917} \approx 4.67 \times 10^{23,249,424} \approx 10^{10^{7.37}} = (10 \uparrow)^2 \ 7.37</math>, the 50th and {{As of|2018|01|lc=y}} the largest known [[Mersenne prime]].
*999 = 9 ↑↑ 3 ≈ 4.28 × 10369,693,099 ≈ (10 ↑)2 8.6
*101010 =10 ↑↑ 3 = 1010,000,000,000 = (10 ↑)3 1
*<math>3^{3^{3^{3}}} = 3 \uparrow \uparrow 4 \approx 1.26 \times 10^{3,638,334,640,024} \approx (10 \uparrow)^3 1.10</math>

Numbers expressible in (10 ↑)''n'' ''k'' notation:
*googolplex = <math>10^{10^{100}} = (10 \uparrow)^3 2</math>
*<math>2^{2^{2^{2^{2^2}}}} = 2 \uparrow \uparrow 6 = 2^{2^{65,536}} \approx 2^{(10 \uparrow)^2 4.3} \approx 10^{(10 \uparrow)^2 4.3} = (10 \uparrow)^3 4.3</math>
*<math>10^{10^{10^{10}}}=10 \uparrow \uparrow 4=(10 \uparrow)^4 1</math>
*<math>3^{3^{3^{3^3}}} = 3 \uparrow \uparrow 5 \approx 3^{10^{3.6 \times 10^{12}}} \approx (10 \uparrow)^4 1.10</math>
*<math>2^{2^{2^{2^{2^{2^2}}}}} = 2 \uparrow \uparrow 7 \approx (10 \uparrow)^4 4.3</math>
*10 ↑↑ 5 = (10 ↑)5 1
*3 ↑↑ 6 ≈ (10 ↑)5 1.10
*2 ↑↑ 8 ≈ (10 ↑)5 4.3
*10 ↑↑ 6 = (10 ↑)6 1
*10 ↑↑↑ 2 = 10 ↑↑ 10 = (10 ↑)10 1
*2 ↑↑↑↑ 3 = 2 ↑↑↑ 4 = 2 ↑↑ 65,536 ≈ (10 ↑)65,533 4.3 is between 10 ↑↑ 65,533 and 10 ↑↑ 65,534

Bigger numbers:
*3 ↑↑↑ 3 = 3 ↑↑ (3 ↑↑ 3) ≈ 3 ↑↑ 7.6 × 1012 ≈ 10 ↑↑ 7.6 × 1012 is between (10 ↑↑)2 2 and (10 ↑↑)2 3
*<math>10\uparrow\uparrow\uparrow 3=(10 \uparrow \uparrow)^3 1</math> = ( 10 → 3 → 3 )
*<math>(10\uparrow\uparrow)^2 11</math>
*<math>(10\uparrow\uparrow)^2 10^{\,\!10^{10^{3.81\times 10^{17}}}}</math>
*<math>10\uparrow\uparrow\uparrow 4=(10 \uparrow \uparrow)^4 1</math> = ( 10 → 4 → 3 )
*<math>(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})</math>
*<math>10\uparrow\uparrow\uparrow 5=(10 \uparrow \uparrow)^5 1</math> = ( 10 → 5 → 3 )
*<math>10\uparrow\uparrow\uparrow 6=(10 \uparrow \uparrow)^6 1</math> = ( 10 → 6 → 3 )
*<math>10\uparrow\uparrow\uparrow 7=(10 \uparrow \uparrow)^7 1</math> = ( 10 → 7 → 3 )
*<math>10\uparrow\uparrow\uparrow 8=(10 \uparrow \uparrow)^8 1</math> = ( 10 → 8 → 3 )
*<math>10\uparrow\uparrow\uparrow 9=(10 \uparrow \uparrow)^9 1</math> = ( 10 → 9 → 3 )
*<math>10 \uparrow \uparrow \uparrow \uparrow 2 = 10\uparrow\uparrow\uparrow 10=(10 \uparrow \uparrow)^{10} 1</math> = ( 10 → 2 → 4 ) = ( 10 → 10 → 3 )
*The first term in the definition of Graham's number, ''g''1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) ≈ 3 ↑↑↑ (10 ↑↑ 7.6 × 1012) ≈ 10 ↑↑↑ (10 ↑↑ 7.6 × 1012) is between (10 ↑↑↑)2 2 and (10 ↑↑↑)2 3 (See [[Graham's number#Magnitude]])
*<math>10\uparrow\uparrow\uparrow\uparrow 3=(10 \uparrow \uparrow\uparrow)^3 1</math> = (10 → 3 → 4)
*<math>4 \uparrow \uparrow \uparrow \uparrow 4</math> = ( 4 → 4 → 4 ) <math>\approx (10 \uparrow \uparrow \uparrow)^2 (10 \uparrow \uparrow)^3 154</math>
*<math>10\uparrow\uparrow\uparrow\uparrow 4=(10 \uparrow \uparrow\uparrow)^4 1</math> = ( 10 → 4 → 4 )
*<math>10\uparrow\uparrow\uparrow\uparrow 5=(10 \uparrow \uparrow\uparrow)^5 1</math> = ( 10 → 5 → 4 )
*<math>10\uparrow\uparrow\uparrow\uparrow 6=(10 \uparrow \uparrow\uparrow)^6 1</math> = ( 10 → 6 → 4 )
*<math>10\uparrow\uparrow\uparrow\uparrow 7=(10 \uparrow \uparrow\uparrow)^7 1=</math> = ( 10 → 7 → 4 )
*<math>10\uparrow\uparrow\uparrow\uparrow 8=(10 \uparrow \uparrow\uparrow)^8 1=</math> = ( 10 → 8 → 4 )
*<math>10\uparrow\uparrow\uparrow\uparrow 9=(10 \uparrow \uparrow\uparrow)^9 1=</math> = ( 10 → 9 → 4 )
*<math>10 \uparrow \uparrow \uparrow \uparrow \uparrow 2 = 10\uparrow\uparrow\uparrow\uparrow 10=(10 \uparrow \uparrow\uparrow)^{10} 1</math> = ( 10 → 2 → 5 ) = ( 10 → 10 → 4 )
*( 2 → 3 → 2 → 2 ) = ( 2 → 3 → 8 )
*( 3 → 2 → 2 → 2 ) = ( 3 → 2 → 9 ) = ( 3 → 3 → 8 )
*( 10 → 10 → 10 ) = ( 10 → 2 → 11 )
*( 10 → 2 → 2 → 2 ) = ( 10 → 2 → 100 )
*( 10 → 10 → 2 → 2 ) = ( 10 → 2 → <math>10^{10}</math> ) = <math>10 \uparrow ^{10^{10}} 10 </math>
*The second term in the definition of Graham's number, ''g''2 = 3 ↑''g''1 3 > 10 ↑''g''1 – 1 10.
*( 10 → 10 → 3 → 2 ) = (10 → 10 → (10 → 10 → <math>10^{10}</math>) ) = <math>10 \uparrow ^{10 \uparrow ^{10^{10}} 10} 10 </math>
*''g''3 = (3 → 3 → ''g''2) > (10 → 10 → ''g''2 – 1) > (10 → 10 → 3 → 2)
*''g''4 = (3 → 3 → ''g''3) > (10 → 10 → ''g''3 – 1) > (10 → 10 → 4 → 2)
*...
*''g''9 = (3 → 3 → ''g''8) is between (10 → 10 → 9 → 2) and (10 → 10 → 10 → 2)
*( 10 → 10 → 10 → 2 )
*''g''10 = (3 → 3 → ''g''9) is between (10 → 10 → 10 → 2) and (10 → 10 → 11 → 2)
*...
*''g''63 = (3 → 3 → ''g''62) is between (10 → 10 → 63 → 2) and (10 → 10 → 64 → 2)
*( 10 → 10 → 64 → 2 )
*Graham's number, ''g''64<ref>Regarding the comparison with the previous value: <math>10\uparrow ^n 10 < 3 \uparrow ^{n+1} 3</math>, so starting the 64 steps with 1 instead of 4 more than compensates for replacing the numbers 3 by 10</ref>
*( 10 → 10 → 65 → 2 )
*( 10 → 10 → 10 → 3 )
*( 10 → 10 → 10 → 4 )
*( 10 → 10 → 10 → 10 )
*( 10 → 10 → 10 → 10 → 10 )
*( 10 → 10 → 10 → 10 → 10 → 10 )
*( 10 → 10 → 10 → 10 → 10 → 10 → 10 → ... → 10 → 10 → 10 → 10 → 10 → 10 → 10 → 10 ) where there are ( 10 → 10 → 10 ) "10"s

=== Other notations ===
Some notations for extremely large numbers:
*[[Knuth's up-arrow notation]]/[[hyperoperator]]s/[[Ackermann function]], including tetration
*[[Conway chained arrow notation]]
*[[Steinhaus-Moser notation]]; apart from the method of construction of large numbers, this also involves a graphical notation with [[polygon]]s. Alternative notations, like a more conventional function notation, can also be used with the same functions.
*[[Fast-growing hierarchy]]
*[[Bashicu Matrix System]]
These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever-faster-increasing functions can easily be constructed recursively by applying these functions with large integers as argument.

A function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the reciprocal.

== Comparison of base values ==
The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers and the arithmetic.

<math>100^{12}=10^{24}</math>, with base 10 the exponent is doubled.

<math>100^{100^{12}}=10^{2*10^{24}}</math>, ditto.

<math>100^{100^{100^{12}}} \approx 10^{10^{2*10^{24}+0.30103}}</math>, the highest exponent is very little more than doubled (increased by log102).

*<math>100\uparrow\uparrow 2=10^ {200} </math>
*<math>100\uparrow\uparrow 3=10^ {2 \times 10^ {200}}</math>
*<math>100\uparrow\uparrow 4=(10\uparrow)^2 (2 \times 10^ {200}+0.3)=(10\uparrow)^2 (2\times 10^ {200})=(10\uparrow)^3 200.3=(10\uparrow)^4 2.3</math>
*<math>100\uparrow\uparrow n=(10\uparrow)^{n-2} (2 \times 10^ {200})=(10\uparrow)^{n-1} 200.3=(10\uparrow)^{n}2.3<10\uparrow\uparrow (n+1)</math> (thus if ''n'' is large it seems fair to say that <math>100\uparrow\uparrow n</math> is "approximately equal to" <math>10\uparrow\uparrow n</math>)
*<math>100\uparrow\uparrow\uparrow 2=(10\uparrow)^{98} (2 \times 10^ {200})=(10\uparrow)^{100} 2.3</math>
*<math>100\uparrow\uparrow\uparrow 3=10\uparrow\uparrow(10\uparrow)^{98} (2 \times 10^ {200})=10\uparrow\uparrow(10\uparrow)^{100} 2.3</math>
*<math>100\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{98} (2 \times 10^ {200})=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{100} 2.3<10\uparrow\uparrow\uparrow (n+1)</math> (compare <math>10\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{10}1<10\uparrow\uparrow\uparrow (n+1)</math>; thus if ''n'' is large it seems fair to say that <math>100\uparrow\uparrow\uparrow n</math> is "approximately equal to" <math>10\uparrow\uparrow\uparrow n</math>)
*<math>100\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3</math> (compare <math>10\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1</math>)
*<math>100\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3</math> (compare <math>10\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1</math>)
*<math>100\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3</math> (compare <math>10\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1</math>; if ''n'' is large this is "approximately" equal)

== Accuracy ==

For a number <math>10^n</math>, one unit change in ''n'' changes the result by a factor 10. In a number like <math>10^{\,\!6.2 \times 10^3}</math>, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor <math>10^{50}</math> too large or too small. This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be "relatively small" and therefore acceptable).

=== For very large numbers ===

In the case of an approximation of an extremely large number, the [[relative error]] may be large, yet there may still be a sense in which we want to consider the numbers as "close in magnitude". For example, consider

:<math>10^{10}</math> and <math>10^9</math>

The relative error is

:<math>1 - \frac{10^9}{10^{10}} = 1 - \frac{1}{10} = 90\%</math>

a large relative error. However, we can also consider the relative error in the logarithms; in this case, the logarithms (to base 10) are 10 and 9, so the relative error in the logarithms is only 10%.

The point is that [[exponential function]]s magnify relative errors greatly – if ''a'' and ''b'' have a small relative error,

:<math>10^a</math> and <math>10^b</math>

the relative error is larger, and

:<math>10^{10^a}</math> and <math>10^{10^b}</math>

will have an even larger relative error. The question then becomes: on which level of iterated logarithms do we wish to compare two numbers? There is a sense in which we may want to consider

:<math>10^{10^{10}}</math> and <math>10^{10^9}</math>

to be "close in magnitude". The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small:

:<math>\log_{10}(\log_{10}(10^{10^{10}})) = 10</math> and <math>\log_{10}(\log_{10}(10^{10^9})) = 9</math>

Such comparisons of iterated logarithms are common, e.g., in [[analytic number theory]].

===Approximate arithmetic===

There are some general rules relating to the usual arithmetic operations performed on very large numbers:

*The sum and the product of two very large numbers are both "approximately" equal to the larger one.
*<math>(10^a)^{\,\!10^b}=10^{a 10^b}=10^{10^{b+\log _{10} a}}</math>
Hence:
*A very large number raised to a very large power is "approximately" equal to the larger of the following two values: the first value and 10 to the power the second. For example, for very large n we have <math>n^n\approx 10^n</math> (see e.g. [[Steinhaus-Moser notation#Mega|the computation of mega]]) and also <math>2^n\approx 10^n</math>. Thus <math>2\uparrow\uparrow 65536 \approx 10\uparrow\uparrow 65533</math>, see [[Knuth's up-arrow notation#Tables of values|table]].

==Systematically creating ever-faster-increasing sequences==
{{Main|fast-growing hierarchy}}
Given a strictly increasing integer sequence/function <math>f_0(n)</math> (''n''≥1) we can produce a faster-growing sequence <math>f_1(n) = f_0^n(n)</math> (where the superscript ''n'' denotes the ''n''th [[functional power]]). This can be repeated any number of times by letting <math>f_k(n) = f_{k-1}^n(n)</math>, each sequence growing much faster than the one before it. Then we could define <math>f_\omega(n) = f_n(n)</math>, which grows much faster than any <math>f_k</math> for finite ''k'' (here ω is the first infinite [[ordinal number]], representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.

For example, starting with ''f''0(''n'') = ''n'' + 1:

* ''f''1(''n'') = ''f''0''n''(''n'') = ''n'' + ''n'' = 2''n''
* ''f''2(''n'') = ''f''1''n''(''n'') = 2''n''''n'' > (2 ↑) ''n'' for n ≥ 2 (using [[Knuth up-arrow notation]])
* ''f''3(''n'') = ''f''2''n''(''n'') > (2 ↑)''n'' ''n'' ≥ 2 ↑2 ''n'' for ''n'' ≥ 2
* ''f''''k''+1(''n'') > 2 ↑''k'' ''n'' for ''n'' ≥ 2, ''k'' < ω
* ''f''ω(''n'') = ''f''''n''(''n'') > 2 ↑''n'' – 1 ''n'' > 2 ↑''n'' − 2 (''n'' + 3) − 3 = ''A''(''n'', ''n'') for ''n'' ≥ 2, where ''A'' is the [[Ackermann function]] (of which ''f''ω is a unary version)
* ''f''ω+1(64) > ''f''ω64(6) > [[Graham's number#Definition|Graham's number]] (= ''g''64 in the sequence defined by ''g''0 = 4, ''g''''k''+1 = 3 ↑''g''''k'' 3)
**This follows by noting ''f''ω(''n'') > 2 ↑''n'' – 1 ''n'' > 3 ↑''n'' – 2 3 + 2, and hence ''f''ω(''g''''k'' + 2) > ''g''''k''+1 + 2
* ''f''ω(''n'') > 2 ↑''n'' – 1 ''n'' = (2 → ''n'' → ''n''-1) = (2 → ''n'' → ''n''-1 → 1) (using [[Conway chained arrow notation]])
* ''f''ω+1(''n'') = ''f''ω''n''(''n'') > (2 → ''n'' → ''n''-1 → 2) (because if ''g''''k''(''n'') = X → ''n'' → ''k'' then X → ''n'' → ''k''+1 = ''g''''k''''n''(1))
* ''f''ω+''k''(''n'') > (2 → ''n'' → ''n''-1 → ''k''+1) > (''n'' → ''n'' → ''k'')
* ''f''ω2(''n'') = ''f''ω+''n''(''n'') > (''n'' → ''n'' → ''n'') = (''n'' → ''n'' → ''n''→ 1)
* ''f''ω2+''k''(''n'') > (''n'' → ''n'' → ''n'' → ''k'')
* ''f''ω3(''n'') > (''n'' → ''n'' → ''n'' → ''n'')
* ''f''ω''k''(''n'') > (''n'' → ''n'' → ... → ''n'' → ''n'') (Chain of ''k''+1 ''n'''s)
* ''f''ω2(''n'') = ''f''ω''n''(''n'') > (''n'' → ''n'' → ... → ''n'' → ''n'') (Chain of ''n''+1 ''n'''s)
{{mvar|}}

== In some noncomputable sequences ==

The [[busy beaver]] function Σ is an example of a function which grows faster than any [[Computability theory (computer science)|computable]] function. Its value for even relatively small input is huge. The values of Σ(''n'') for ''n'' = 1, 2, 3, 4 are 1, 4, 6, 13 {{OEIS|id=A028444}}. Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 3.5×1018267.

== Infinite numbers ==
{{Main|cardinal number}}
{{See also|large cardinal|Mahlo cardinal|totally indescribable cardinal}}
Although all the numbers discussed above are very large, they are all still decidedly [[finite set|finite]]. Certain fields of mathematics define [[Infinity|infinite]] and [[transfinite number]]s. For example, [[aleph-null]] is the [[cardinality]] of the [[infinite set]] of [[natural number]]s, and [[aleph-one]] is the next greatest cardinal number. <math>\mathfrak{c}</math> is the [[cardinality of the continuum|cardinality of the reals]]. The proposition that <math>\mathfrak{c} = \aleph_1</math> is known as the [[continuum hypothesis]].

== See also ==
{{colbegin|colwidth=25em}}
* [[Arbitrary-precision arithmetic]]
* [[List of arbitrary-precision arithmetic software]]
* [[Dirac large numbers hypothesis]]
* [[Exponential growth]]
* [[History of large numbers]]
* [[Human scale]]
* [[Largest number]]
* [[Law of large numbers]]
* [[Myriad#East Asia|Myriads (10,000) in East Asia]]
* [[Names of large numbers]]
* [[Power of two]]
* [[Power of 10]]
* [[Tetration]]
{{colend}}

==References==
{{Reflist}}

{{Large numbers}}
{{Hyperoperations}}

[[Category:Large numbers| ]]
[[Category:Mathematical notation]]

Hahn embedding theorem

2021-03-10T01:12:12Z

IntegralPython: Adding short description: "Description of linearly ordered groups" (Shortdesc helper)

{{Short description|Description of linearly ordered groups}}
{{No footnotes|date=November 2020}}
In [[mathematics]] – especially in the area of [[abstract algebra]] dealing with ordered structures on [[abelian group]]s – the '''Hahn embedding theorem''' gives a simple description of all [[linearly ordered group|linearly ordered abelian group]]s. It is named after [[Hans Hahn (mathematician)|Hans Hahn]].<ref>{{Cite web|title=lo.logic - Hahn's Embedding Theorem and the oldest open question in set theory|url=https://mathoverflow.net/questions/128935/hahns-embedding-theorem-and-the-oldest-open-question-in-set-theory|access-date=2021-01-28|website=MathOverflow}}</ref>

==Overview==
The theorem states that every [[linearly ordered group|linearly ordered abelian group]] ''G'' can be embedded as an ordered subgroup of the additive group ℝΩ endowed with a [[lexicographical order]], where ℝ is the additive group of [[real numbers]] (with its standard order), Ω is the set of ''Archimedean equivalence classes'' of ''G'', and ℝΩ is the set of all functions from Ω to ℝ which vanish outside a well-ordered set.

Let 0 denote the identity element of ''G''. For any nonzero element ''g'' of ''G'', exactly one of the elements ''g'' or −''g'' is greater than 0; denote this element by |''g''|. Two nonzero elements ''g'' and ''h'' of ''G'' are ''Archimedean equivalent'' if there exist [[natural number]]s ''N'' and ''M'' such that ''N''|''g''| > |h| and ''M''|''h''| > |g|. Intuitively, this means that neither ''g'' nor ''h'' is "infinitesimal" with respect to the other. The group ''G'' is [[Archimedean group|Archimedean]] if ''all'' nonzero elements are Archimedean-equivalent. In this case, Ω is a singleton, so ℝΩ is just the group of real numbers. Then Hahn's Embedding Theorem reduces to [[Otto Hölder|Hölder]]'s theorem (which states that a linearly ordered abelian group is [[Archimedean group|Archimedean]] if and only if it is a subgroup of the ordered additive group of the real numbers).

{{harvtxt|Gravett|1956}} gives a clear statement and proof of the theorem. The papers of {{harvtxt|Clifford|1954}} and {{harvtxt|Hausner|Wendel|1952}} together provide another proof. See also {{harvtxt|Fuchs|Salce|2001|p=62}}.

==See also==
* [[Archimedean group]]

==References==
*{{Citation | last1=Fuchs | first1=László | last2=Salce | first2=Luigi | title=Modules over non-Noetherian domains | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1963-0 |mr=1794715 | year=2001 | volume=84}}
* {{Citation | last1=Ehrlich | first1=Philip | url=http://www.ohio.edu/people/ehrlich/HahnNew.pdf | chapter=Hahn’s “Über die nichtarchimedischen Grössensysteme” and the Origins of the Modern Theory of Magnitudes and Numbers to Measure Them | title=From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics |editor-first=Jaakko |editor-last=Hintikka | publisher= Kluwer Academic Publishers |year=1995|pages=165–213}}
* {{Citation | last1=Hahn | first1=H. | author1-link=Hans Hahn (mathematician) | title=Über die nichtarchimedischen Größensysteme. | language=German | year=1907 | journal=Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.) | volume=116 | pages=601–655}}
* {{Citation | doi=10.1093/qmath/7.1.57 | last1=Gravett | first1=K. A. H.| title=Ordered Abelian Groups| journal=The Quarterly Journal of Mathematics. Oxford. Second Series| volume=7| year=1956| pages=57–63}}
* {{Citation | last1=Clifford | first1=A.H. | title= Note on Hahn's Theorem on Ordered Abelian Groups| journal=Proceedings of the American Mathematical Society| volume=5| issue=6| year=1954| pages=860–863 | doi=10.2307/2032549}}
* {{Citation | doi=10.1090/S0002-9939-1952-0052045-1 | last1=Hausner| first1=M. | last2=Wendel| first2=J.G.| title=Ordered vector spaces| journal=Proceedings of the American Mathematical Society| volume=3| year=1952| pages=977–982| doi-access=free}}

[[Category:Ordered groups]]
[[Category:Theorems in group theory]]

Internet meme

2021-03-09T01:02:41Z

IntegralPython: /* Modern memes */ ce

{{short description|Concept that spreads from person to person via the Internet}}
{{Pp-vandalism|small=yes}}
{{Use mdy dates|date=April 2020}}
{{Use American English|date=April 2020}}
{{Internet}}
An '''Internet meme''', more commonly known simply as a '''meme''' ({{IPAc-en|m|iː|m}} {{respell|MEEM}}), is a type of idea, behaviour, or style ([[meme]]) that is spread via the [[Internet]], often through [[social media platforms]] and especially for humorous purposes. Memes can spread from person to person via [[social network]]s, [[blog]]s, direct [[email]], or news sources. They may relate to various existing [[Internet culture]]s or [[subculture]]s, often created or spread on various websites. One hallmark of Internet memes is the appropriation of a part of broader culture, for instance by giving words and phrases intentional misspellings (such as [[lolcat]]s) or using incorrect grammar (such as [[Doge (meme)|doge]]). In particular, many memes utilize [[popular culture]] (especially in [[image macros]] of other media), which sometimes can lead to issues with [[copyright]].

Instant communication on the Internet facilitates [[word of mouth]] transmission, resulting in [[fad]]s and sensations that tend to grow rapidly. An example of such a fad is that of [[Planking (fad)|planking]] (lying down in public places); posting a photo of someone planking online brings attention to the fad and allows it to reach many people in little time. The internet also facilitates the rapid evolution of memes. “Dank” memes have emerged as a new form of image-macros, and many modern memes take on inclusion of [[Surrealism|surreal]], nonsensical, and non-sequitur themes.

Colloquially, the terms ''meme'' and ''Internet meme'' may refer to pieces of media that are designed in the format of true Internet memes, but which are not themselves intended to spread or evolve, and which have recently become umbrella terms referring to any piece of quickly-consumed comedic or relatable content. What is considered a meme may vary across different communities on the Internet and is subject to change over time: traditionally, memes consisted of a combination of [[image macros]] and a concept or catchphrase, but the concept has since become broader and more multi-faceted, evolving to include more elaborate structures such as challenges, [[GIF]]s, videos, and [[Viral phenomenon|viral sensations]].

== Characteristics ==
There are two central attributes of Internet memes: creative reproduction of materials and [[intertextuality]]. Creative reproduction refers to "parodies, remixes, or mashups," and include notable examples such as "''Hitler's Downfall'' Parodies",<ref name=":3">{{Cite book|last=Shifman|first=Limor|title=Memes in Digital Culture|publisher=CRC Press|year=2015|isbn=978-1-4619-4733-2|oclc=926526630}}</ref> and "''[[Nyan Cat]]''", among others. [[Intertextuality]] may be demonstrated through memes that combine different cultures; for example, a meme may combine United States politician [[Mitt Romney]]'s assertion of the phrase "[[binders full of women]]" from a 2012 US presidential debate with the Korean pop song "[[Gangnam Style in popular culture|Gangnam style]]" by overlaying the politician's quote onto a frame from [[Psy]]'s music video where paper blows around him. The intertextuality in the example gives new meaning to the paper blowing around Psy, the meme indexes intertextual practices in political and cultural discourses of two nations.<ref name=":3" />

The spread of Internet memes has been described as occurring via two mechanisms: [[mimicry]] and [[remix]]. Remix occurs when the original meme is altered in some way, while mimicry occurs when the meme is recreated in a different fashion to the original.<ref name=":032">{{Cite book|last=Shifman|first=Limor|url=https://books.google.com/books?id=cZI9AQAAQBAJ&q=memes&pg=PP6|title=Memes in Digital Culture|date=2014|publisher=MIT Press|isbn=978-0-262-52543-5|language=en}}</ref><ref>{{Cite web|last=Madison|date=2019-04-09|title=Meme-ology: Studying Patterns in Viral Media|url=https://medium.com/@madisonicole/meme-ology-studying-patterns-in-viral-media-f1931b3d1c7e|access-date=2020-11-26|website=Medium|language=en}}</ref> The results in the study of ''Online Memes, Affinities, and Cultural Production,'' show that the internet directly adds some longevity in a meme's lifespan.<ref>{{Cite journal|last1=Knobel|first1=Michele|last2=Lankshear|first2=Colin|date=2018|title=Online memes, affinities and cultural production (2018 update to our 2007 chapter) To appear as: Knobel, M. and Lankshear, C. (forthcoming). Memes online, afinidades e produção cultural (2007 – 2018). In Chagas, Viktor (ed.). Estudos sobre Memes: história, política e novas experiências de letramento. 2019.|url=http://rgdoi.net/10.13140/RG.2.2.34717.77280|language=en|doi=10.13140/RG.2.2.34717.77280}}</ref>

There is no single format that memes must follow. Photographs of people or animals, especially stock photos, can be turned into memes by superimposing text, such as in [[Overly Attached Girlfriend]]. [[Rage comic]]s are a subcategory of memes which depict a series of human emotions and conclude with a satirical punchline;<ref>{{citation |url=https://www.nytimes.com/2012/05/10/technology/personaltech/rage-comics-turn-everyday-stress-into-laughs.html |date=May 9, 2012 |last=Boutin |first=Paul |title=Put Your Rage Into a Cartoon and Exit Laughing |work=[[The New York Times]] }}</ref> the sources for these memes often come from [[webcomic]]s. Other memes are purely viral sensations such as in [[Keyboard Cat]].

== Evolution and propagation ==
[[File:Typical internet meme image format.svg|upright=0.9|thumb|right|Typical format for image macros.]]
An Internet meme may stay the same or may evolve over time, by chance or through commentary, imitations, [[parody]], or by incorporating news accounts about itself. Internet memes can evolve and spread extremely rapidly, sometimes reaching worldwide popularity within a few days. Consequently, an internet meme can also rapidly become 'unfashionable', losing its humorous qualities to certain audiences, often even most prevalently by its creator(s). Internet memes usually are formed from some social interaction, pop culture reference, or situations people often find themselves in. Their rapid growth and impact has caught the attention of both researchers and industry.<ref name="researchSN">{{Cite conference |first1=David |last1=Kempe |first2=Jon |last2=Kleinberg |first3=Éva |last3=Tardos |title=Maximizing the spread of influence through a social network |publisher=ACM Press |book-title=Int. Conf. on Knowledge Discovery and Data Mining |year=2003 |doi=10.1145/956750.956769 |url=http://www.cs.cmu.edu/~aladdin/workshops/wsa/papers/spread.pdf}}</ref> Academically, researchers model how they evolve and predict which memes will survive and spread throughout the [[World Wide Web|Web]]. Commercially, they are used in [[viral marketing]] where they are an inexpensive form of mass advertising.

One empirical approach studied meme characteristics and behavior independently from the networks in which they propagated, and reached a set of conclusions concerning successful meme propagation.<ref name="CosciaHarvardCID2013">{{Cite arXiv|eprint=1304.1712|class=physics.soc-ph|first=Michele|last=Coscia|title=Competition and Success in the Meme Pool: a Case Study on Quickmeme.com|date=April 5, 2013}} Paper explained for laymen by {{Cite web|last=Mims|first=Christopher|date=June 28, 2013|title=Why you'll share this story: The new science of memes|url=http://qz.com/98677/why-youll-share-this-story-the-new-science-of-memes/|url-status=dead|archive-url=https://web.archive.org/web/20130703055640/http://qz.com/98677/why-youll-share-this-story-the-new-science-of-memes/|archive-date=July 3, 2013|website=[[Quartz (publication)|Quartz]]}}</ref> For example, the study asserted that Internet memes not only ''compete'' for viewer attention generally resulting in a shorter life, but also, through user creativity, memes can ''collaborate'' with each other and achieve greater survival.<ref name="CosciaHarvardCID2013" /> Also, paradoxically, an individual meme that experiences a popularity peak significantly higher than its average popularity is not generally expected to survive unless it is unique, whereas a meme with no such popularity peak keeps being used together with other memes and thus has greater survivability.<ref name="CosciaHarvardCID2013" />

Multiple opposing studies on media psychology and communication have aimed to characterize and analyze the concept and representations in order to make it accessible for the academic research.<ref name="Characterising IM">{{Cite journal |last1=Castaño |first1=Carlos |title=Defining and Characterising the Concept of Internet Meme |journal=Revista CES Psicología |date=2013 |volume=6 |issue=2 |pages=82–104 |issn=2011-3080 |url=http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S2011-30802013000200007|access-date=April 23, 2015|archive-url=https://web.archive.org/web/20150713085630/http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S2011-30802013000200007|archive-date=July 13, 2015|url-status=live}}</ref><ref>{{Cite journal |last=Julien |first=Chris |date=June 30, 2014 |title=Bourdieu, Social Capital and Online Interaction |journal=Sociology |volume=49 |issue=2 |pages=356–373 |doi=10.1177/0038038514535862 |s2cid=144559268 |url=https://zenodo.org/record/894539|access-date=August 28, 2019|archive-url=https://web.archive.org/web/20191013163908/https://zenodo.org/record/894539|archive-date=October 13, 2019|url-status=live}}</ref> Thus, Internet memes can be regarded as a unit of information which replicates via the Internet. This unit can replicate or mutate. This mutation instead of being generational<ref name="cream">{{cite book|last=Dawkins|first=Richard|url=https://books.google.com/books?id=WkHO9HI7koEC&pg=PA192|title=The Selfish Gene|publisher=Oxford University Press|year=1989|isbn=978-0-19-286092-7|edition=2|page=192|quote=We need a name for the new replicator, a noun that conveys the idea of a unit of cultural transmission, or a unit of ''imitation''. 'Mimeme' comes from a suitable Greek root, but I want a monosyllable that sounds a bit like 'gene'. I hope my classicist friends will forgive me if I abbreviate mimeme to ''meme''. If it is any consolation, it could alternatively be thought of as being related to 'memory', or to the French word ''même''. It should be pronounced to rhyme with 'cream'.|author-link=Richard Dawkins|access-date=June 16, 2015|archive-url=https://web.archive.org/web/20150316114026/http://books.google.com/books?id=WkHO9HI7koEC&pg=PA192|archive-date=March 16, 2015|url-status=live}}</ref> follows more a viral pattern,<ref name= "Wired temes">{{Cite news |title=Humans Are Just Machines for Propagating Memes |url=https://www.wired.com/science/discoveries/news/2008/02/ted_blackmore?currentPage=all |last=Zetter |first=K. |date=February 29, 2008 |work=[[Wired (magazine)|Wired]] |access-date= March 7, 2017 |archive-url= https://web.archive.org/web/20140202123609/http://www.wired.com/science/discoveries/news/2008/02/ted_blackmore?currentPage=all |archive-date= February 2, 2014 |url-status= live}}</ref> giving the Internet memes generally a short life. Other theoretical problems with the Internet memes are their behavior, their type of change, and their teleology.<ref name="Characterising IM" />

Internet memes have been examined by Dancygier and Vandelanotte in 2017 for aspects of cognitive linguistic and construction grammar. The authors analyzed some selective popular image macros like, Said no one ever, One does not simply, But that's none of my business, and Good Girl Gina to draw attention to the constructionally, multimodality, viewpoint and intersubjectivity of these memes. They further argued that with the combination of text and images, the Internet memes can add to the functioning linguistic construction frame as well as create new linguistic constructions.<ref>{{Cite journal|last1=Dancygier|first1=Barbara|last2=Vandelanotte|first2=Lieven|date=2017-08-28|title=Internet memes as multimodal constructions|url=https://www.degruyter.com/view/journals/cogl/28/3/article-p565.xml|journal=Cognitive Linguistics|volume=28|issue=3|pages=565–598|doi=10.1515/cog-2017-0074|s2cid=149309447|issn=0936-5907}}</ref>

Writing for ''[[The Washington Post]]'' in 2013, Dominic Basulto asserted that with the growth of the Internet and the practices of the marketing and advertising industries, memes have come to transmit fewer snippets of human culture that could survive for centuries as originally envisioned by Dawkins, and instead transmit banality at the expense of big ideas.<ref name=WashingtonPost20130705>{{Cite web |url=https://www.washingtonpost.com/blogs/innovations/wp/2013/07/05/have-internet-memes-lost-their-meaning/ |title=Have Internet memes lost their meaning? |last=Basulto |first=Dominic |date=July 5, 2013 |website=The Washington Post |archive-url=https://web.archive.org/web/20130705202602/http://www.washingtonpost.com/blogs/innovations/wp/2013/07/05/have-internet-memes-lost-their-meaning/ |archive-date=July 5, 2013 |url-status=dead}}</ref>

==History==

=== Origins and early memes ===
[[File:Wikipedia meme vector version.svg|thumb|An example of an [[image macro]], a common type of Internet meme in the [[2000s]].]]
The word ''[[meme]]'' was coined by [[Richard Dawkins]] in his 1976 book ''[[The Selfish Gene]]'' as an attempt to explain how ideas replicate, mutate, and evolve ([[memetics]]).<ref name="cream" /> The concept of the Internet meme was first proposed by [[Mike Godwin]] in the June 1993 issue of ''[[Wired (magazine)|Wired]]''. In 2013, Dawkins characterized an Internet meme as being a meme deliberately altered by human creativity—distinguished from biological genes and his own pre-Internet concept of a meme, which involved mutation by random change and spreading through accurate replication as in Darwinian selection.<ref name="Wired20130620">{{Cite journal|last=Solon|first=Olivia|date=June 20, 2013|title=Richard Dawkins on The Internet's hijacking of the word 'meme'|url=https://www.wired.co.uk/news/archive/2013-06/20/richard-dawkins-memes|url-status=dead|journal=Wired UK|archive-url=https://web.archive.org/web/20130709152558/http://www.wired.co.uk/news/archive/2013-06/20/richard-dawkins-memes|archive-date=July 9, 2013}}</ref> Dawkins explained that Internet memes are thus a "hijacking of the original idea", the very idea of a meme having mutated and evolved in this new direction.<ref name="Saatchi20130622">{{Cite web|last=Dawkins|first=Richard|date=June 22, 2013|title=Just for Hits|url=https://www.youtube.com/watch?v=GFn-ixX9edg|url-status=live|archive-url=https://web.archive.org/web/20150617051511/https://www.youtube.com/watch?v=GFn-ixX9edg|archive-date=June 17, 2015|access-date=June 16, 2015|website=The Saatchi & Saatchi New Directors' Showcase}} (video of speech)</ref> Furthermore, Internet memes carry an additional property that ordinary memes do not: Internet memes leave a footprint in the media through which they propagate (for example, social networks) that renders them traceable and analyzable.<ref name="CosciaHarvardCID2013" />

Internet memes grew as a concept in the mid-1990s. At the time, memes were just short clips that were shared between people in [[Usenet]] forums.{{Citation needed|date=November 2020}} As the Internet evolved, so did memes. When [[YouTube]] was released in 2005, video memes became popular. Around this time, [[rickrolling]] became popular and the link to this video was sent around via email or other messaging sites. Video sharing also created memes such as "[[Turn Down for What]]" and the "[[Harlem Shake (meme)|Harlem Shake]]". As social media websites such as [[Twitter]] and [[Facebook]] started appearing, it was now easy to share [[GIF]]s and [[image macro]]s to a large audience. Meme generator websites were created to let users create their own memes out of existing templates. Memes during this time could remain popular for a long time, from a few months to a decade, which contrasts with the fast lifespan of modern memes.<ref>{{Cite magazine |url=https://www.wired.com/story/guide-memes/ |title=The WIRED Guide to Memes |magazine=Wired|access-date=November 30, 2018 |first1=Angela |last1=Watercutter |first2=Emma |last2=Grey Ellisby |date=April 1, 2018|archive-url=https://web.archive.org/web/20190201164321/https://www.wired.com/story/guide-memes/|archive-date=February 1, 2019|url-status=live}}</ref> Over the years, many memes have originated on the [[4chan]] website, which have been described as "the cradle of memes, [[Internet troll|trolling]] and [[alterculture]]"; major memes popularized by that site include [[lolcat]]s as well as the [[pedobear]].<ref name="JemielniakPrzegalinska202023">{{cite book|author1=Dariusz Jemielniak|url=https://books.google.com/books?id=yLDMDwAAQBAJ|title=Collaborative Society|author2=Aleksandra Przegalinska|date=18 February 2020|publisher=MIT Press|isbn=978-0-262-35645-9}}</ref>{{Rp|74}}

Early in the Internet's history, memes were primarily spread via [[email]] or [[Usenet]] discussion communities. [[Internet forum|Messageboards]] and [[Usenet newsgroup|newsgroups]] were also popular because they allowed a simple method for people to share information or memes with a diverse population of Internet users in a short period. They encourage communication between people, and thus between meme sets, that do not normally come in contact. Furthermore, they actively promote meme-sharing within the messageboard or newsgroup population by asking for feedback, comments, opinions, etc. This format is what gave rise to early Internet memes, like the [[Hampster Dance]].<ref>{{Cite web|url=https://www.grunge.com/184711/the-oldest-memes-on-the-internet/|title=The oldest memes on the internet|last=Cantrell|first=Asher|date=2020-01-22|website=Grunge.com|access-date=2020-04-16}}</ref> Another factor in the increased meme transmission observed over the Internet is its interactive nature. [[Publishing|Print]] matter, [[radio]], and [[television]] are all essentially passive experiences requiring the reader, listener, or viewer to perform all necessary cognitive processing; in contrast, the social nature of the Internet allows phenomena to propagate more readily. Many phenomena are also spread via [[web search engine]]s, [[Internet forum]]s, [[social networking service]]s, [[Social news website|social news]] sites, and [[video hosting service]]s. Much of the Internet's ability to spread information is assisted from results found through search engines, which can allow users to find memes even with obscure information.<ref>{{Cite web |title=Memes On the Internet |url=http://library.thinkquest.org/C004367/ce7.shtml |publisher=[[Oracle Thinkquest]] |access-date=November 30, 2012 |url-status=dead |archive-url=https://web.archive.org/web/20130511193449/http://library.thinkquest.org/C004367/ce7.shtml |archive-date=May 11, 2013}}</ref><ref>{{Cite web |last=Marshall |first=Garry |title=The Internet and Memetics |url=http://pespmc1.vub.ac.be/Conf/MemePap/Marshall.html |publisher=School of Computing Science, [[Middlesex University]]|access-date=November 30, 2012|archive-url=https://web.archive.org/web/20190119134239/http://pespmc1.vub.ac.be/Conf/MemePap/Marshall.html|archive-date=January 19, 2019|url-status=dead}}</ref>

The earlier forms of image based memes include the demotivator, image macro, [[Photo manipulation|photoshopped]] image, LOLCats, advice animal, and comic.<ref name=":0222">{{Cite book|last=Denisova |first=Anastasia|title=Internet Memes and Society: Social, Cultural, and Political Contexts|isbn=978-0-429-46940-4|location=New York, NY|oclc=1090540034}}</ref> The Demotivator image includes a black background with white, capitalized text, often in [[Times New Roman]]. The objective of using this format was to parodize inspirational and motivational posters, where the name "demotivator" is derived from.<ref name=":0222"/> Image macro consists of an image with white Impact font within a black border. The text/context of the meme is at the top and bottom of the image itself.<ref name=":0222"/> The photoshopped image is closely related to the macro image, but often is created without the use of text, mostly edited with another image.<ref name=":0222"/> Advice animals contain a photoshopped image of an animal's head on top of a rainbow/color wheel background. It includes the image macro of the top and bottom text with Impact font.<ref name=":0222"/> LOLCats incorporate the design of image macro and advice animals, but instead of just the cat's head, it is the entire picture unedited with top and bottom text, often with the usage of Internet slang.<ref name=":0222" /> Comics follow a typical newspaper comic strip format; there are a variety of different ways to create one, as multiple images and texts can be used to create the overall meme.
[[Rage comic]]s such as [[Trollface]] were often used to create comic memes.<ref>{{Cite web|title=The Maker Of The Trollface Meme Is Counting His Money|url=https://kotaku.com/the-maker-of-the-trollface-meme-is-counting-his-money-1696228810|last=Klepek|first=Patrick|date=April 8, 2015|website=Kotaku|url-status=live|archive-url=https://web.archive.org/web/20200221101337/https://kotaku.com/the-maker-of-the-trollface-meme-is-counting-his-money-1696228810|archive-date=February 21, 2020|access-date=2020-05-28}}</ref><ref>{{cite news |url=https://www.nytimes.com/2012/05/10/technology/personaltech/rage-comics-turn-everyday-stress-into-laughs.html |date=May 9, 2012 |last=Boutin |first=Paul |title=Put Your Rage Into a Cartoon and Exit Laughing |work=[[The New York Times]] |access-date=2020-11-07}}</ref>

=== Modern memes ===
[[File:Example of modern internet meme.jpg|thumb|Modern Internet meme on the subject of Wikipedia and pages breaking when certain characters are removed. Internet memes sometimes represent everyday problems.]]
Modern memes can generally be described as more visually (rather than contextually) humorous, absurd, niche, diverse and [[Self-referential humour|self-referential]] than earlier forms. As a result, they are less intuitive and are less likely to be fully understood by a wider audience. By the mid-2010s, they began to arise first in the form of "dank" memes, <ref>https://knowyourmeme.com/memes/dank-memes</ref>a sub-genre of memes usually involving meme formats in a different way to the image macros that were in large use before. The term "[[wiktionary:dank|dank]]", which means "a cold, damp place", was later adapted by [[Cannabis (drug)|marijuana]] smokers to refer to high-quality marijuana, and then became an ironic term for a type of meme, also becoming synonymous for "[[wiktionary:cool|cool]]".<ref>{{Cite web |last1=Hoffman |first1=Ashley |title=Donald Trump Jr. Just Became a Dank Meme, Literally |url=http://time.com/5130384/donald-trump-jr-dankness-tweet/ |website=Time|access-date=May 19, 2018 |date=February 2, 2018|archive-url=https://web.archive.org/web/20180501012644/http://time.com/5130384/donald-trump-jr-dankness-tweet/|archive-date=May 1, 2018|url-status=live}}</ref> This term originally meant a meme that was significantly different from the norm but is now used mainly to differentiate these modern types of memes from other, older types such as image macros.{{Citation needed|date=November 2018}} Dank memes can also refer to those which are "exceptionally unique or odd".<ref>{{Cite web |url=https://www.dictionary.com/e/slang/dank-meme/ |title=Dank Memes — What does dank meme mean? |website=[[Dictionary.com]]|access-date=November 30, 2018|archive-url=https://web.archive.org/web/20181130071447/https://www.dictionary.com/e/slang/dank-meme/|archive-date=November 30, 2018|url-status=live}}</ref> They have been described as "Internet in-jokes" that are "so played out that they become funny again" or are "so nonsensical that they are hilarious".<ref>{{Cite web |last1=Griffin |first1=Annaliese |title=What does "dank" mean? A definition of everyone's new favourite adjective |url=https://quartzy.qz.com/1221995/dank-is-the-new-umami/ |website=Quartzy|access-date=May 19, 2018 |date=March 9, 2018|archive-url=https://web.archive.org/web/20180519121617/https://quartzy.qz.com/1221995/dank-is-the-new-umami/|archive-date=May 19, 2018|url-status=live}}</ref> A highly prevalent meme at the beginning of this era was the 'montage parody'. These were videos which sought to ironically exaggerate the visual effects used by video game content creators; <ref>{{Cite web|url=https://youtube.fandom.com/wiki/MLG_parodies|title=MLG parodies|website=Wikitubia}}</ref> some elements of these memes have carried on for years after their initial popularity.

The formats are usually from popular television shows, movies, or video games and users then add humorous text and images over it.{{Citation needed|date=May 2018}} The culture surrounding memes, especially dank memes, grew to the point of the creation of many subcultures surrounding them. For instance, a "meme market", satirizing on the kind of talks and stocks found normally on Wall Street, was created in September 2016. Originally started on [[Reddit]] as r/MemeEconomy, people would only jokingly "buy" or "sell" shares in a meme to indicate how popular a meme was thought to be. The market is seen as a way to show how people assign value to commonplace and otherwise valueless things such as memes.<ref>{{Cite web |url=https://www.theverge.com/2017/1/10/14223264/meme-economy-reddit-stock-market |title=How a group of Redditors is creating a fake stock market to figure out the value of memes |last=Plaugic |first=Lizzie |date=January 10, 2017 |website=The Verge|access-date=December 10, 2018|archive-url=https://web.archive.org/web/20181211010109/https://www.theverge.com/2017/1/10/14223264/meme-economy-reddit-stock-market|archive-date=December 11, 2018|url-status=live}}</ref>

One example of a dank meme is "''Who Killed Hannibal''", which is made of two frames from a 2013 episode of ''[[The Eric Andre Show]]''. The meme features the host Andre shooting his co-host Buress in the first frame and then lamenting that his co-host has been shot in the next, with Andre often depicted blaming someone else for the shot. This was then adapted to other situations, such as [[baby boomers]] blaming [[millennials]] for problems that they allegedly caused.<ref>{{Cite web |author=Mary von Aue |title=Meme About 'Who Killed Hannibal' Is Reddit's Current Obsession |website=Inverse|access-date=May 19, 2018 |date=April 19, 2018 |url=https://www.inverse.com/article/43903-who-killed-hannibal-reddit-meme|archive-url=https://web.archive.org/web/20180519120857/https://www.inverse.com/article/43903-who-killed-hannibal-reddit-meme|archive-date=May 19, 2018|url-status=live}}</ref>

{{Anchor|Moth memes}}
Dank memes also stem from interesting real-life images that are shared or remixed many times. So-called "moth" memes (often stylized as "möth") came about after a Reddit user posted a close up picture of a [[moth]] that they had found outside their window onto the r/creepy [[subreddit]].<ref>{{Cite web |url=https://www.reddit.com/r/creepy/comments/8ys873/close_up_of_moth_outside_my_window/ |title=Close up of moth outside my window |author=u/No_Reason27 |date=July 2018|access-date=November 30, 2018 |website=[[Reddit]]|archive-url=https://web.archive.org/web/20181006145655/https://www.reddit.com/r/creepy/comments/8ys873/close_up_of_moth_outside_my_window/|archive-date=October 6, 2018|url-status=live}}</ref> The image became popular and began to be used in memes; according to Chris Grinter, a [[lepidopterist]] from the [[California Academy of Sciences]], moth memes gained recognition because of the inexplicability surrounding moths' attraction to lamps.<ref>{{Cite web |url=https://www.iflscience.com/editors-blog/the-latest-viral-meme-trend-is-possibly-not-as-stupid-as-you-think/ |website=IFL Science |access-date=November 30, 2018 |title=The Latest Viral Meme Trend Is (Possibly) Not As Stupid As You Think |first=Katie |last=Spalding |date=October 2, 2018 |archive-url=https://web.archive.org/web/20181002222940/https://www.iflscience.com/editors-blog/the-latest-viral-meme-trend-is-possibly-not-as-stupid-as-you-think/ |archive-date=October 2, 2018 |url-status=live}}</ref>

====Irony and absurdism====
[[File:Deepfyed meme "W".jpg|thumb|Example of a deepfried meme without any context. Surrealist and nonsensical themes are typical of modern memes]]
{{Anchor|They did surgery on a grape}}
Many modern memes stem from nonsense or otherwise unrelated phrases that are repeated and placed onto other formats. One example of this is "they did surgery on a grape," from a video of a [[da Vinci Surgical System]] performing test surgery on a grape.<ref>{{Cite web |url=https://www.youtube.com/watch?v=KNHgeykDXFw |title=da Vinci Surgical System: Surgery on a grape |last=EdwardHospital |date=August 11, 2010|access-date=November 30, 2018 |via=YouTube|archive-url=https://web.archive.org/web/20181127050657/https://www.youtube.com/watch?v=KNHgeykDXFw|archive-date=November 27, 2018|url-status=live}}</ref> People sharing the post tended to add the same caption to it ("they did surgery on a grape"), and eventually created a satirical image with several layers of captions on it. Memes such as this one continue to propagate as people start to include the phrase in different, otherwise unrelated memes.<ref>{{Cite web |last1=Feldman |first1=Brian |title=They Did Surgery on a Grape |url=https://nymag.com/intelligencer/2018/11/why-is-everyone-saying-they-did-surgery-on-a-grape.html |work=Intelligencer |publisher=NYMag |date=November 26, 2018|access-date=November 27, 2018 |archive-url=https://web.archive.org/web/20181126201420/http://nymag.com/intelligencer/2018/11/why-is-everyone-saying-they-did-surgery-on-a-grape.html |archive-date=November 26, 2018 |url-status=live}}</ref><ref>{{Cite web |last1=Hess |first1=Peter |title="They Did Surgery on a Grape" Meme Began With Legally Suspect Medical Tool |url=https://www.inverse.com/article/51209-they-did-surgery-on-a-grape-but-is-the-grape-okay |website=Inverse |date=November 27, 2018|access-date=November 27, 2018 |archive-url=https://web.archive.org/web/20181127110402/https://www.inverse.com/article/51209-they-did-surgery-on-a-grape-but-is-the-grape-okay |archive-date=November 27, 2018 |url-status=live}}</ref><ref>{{Cite web |last=Santiago |first=Amanda Luz Henning |title='They did surgery on a grape' is the weird meme that's your new obsession |url=https://mashable.com/article/they-did-surgery-on-a-grape-meme/ |website=Mashable |date=November 26, 2018|access-date=November 27, 2018 |archive-url=https://web.archive.org/web/20181126200029/https://mashable.com/article/they-did-surgery-on-a-grape-meme/ |archive-date=November 26, 2018 |url-status=live}}</ref>

The increasing trend towards irony in meme culture has resulted in absurdist memes not unlike [[postmodern art]]. Many Internet memes have several layers of meaning built off of other memes, not being understandable unless the viewer has seen all previous memes. "Deep-fried" memes, memes that have been distorted and run through several filters, are often strange to one not familiar with them.<ref>{{cite web|website=Know Your Meme |url=https://knowyourmeme.com/memes/deep-fried-memes |title=Deep-fried memes |archive-url=https://web.archive.org/web/20190327091650/https://knowyourmeme.com/memes/deep-fried-memes |archive-date=March 27, 2019|access-date=March 26, 2019}}</ref> An example of these memes is the "E" meme, a picture of [[Markiplier]] photoshopped onto [[Lord Farquaad]] from the film ''[[Shrek]]'', photoshopped into a scene from [[Mark Zuckerberg]]'s hearing in Congress.<ref>{{cite web|last=Hathaway|first=Jay |website=The Daily Dot |date=November 5, 2018 |url=https://www.dailydot.com/unclick/lord-farquaad-e-meme/ |title=The 'E' meme shows just how weird memes can get|archive-url=https://web.archive.org/web/20190326134655/https://www.dailydot.com/unclick/lord-farquaad-e-meme/ |archive-date=March 26, 2019|access-date=March 26, 2019}}</ref> "Surreal" memes are based on the idea of increasing layers of irony so that they are not understandable by popular culture or corporations.<ref>{{cite web|last=Bryan|first=Chloe |website=Mashable |date=February 6, 2019 |url=https://mashable.com/article/surreal-memes/#dBMHaD8Josqt |title=Surreal memes deserve their own internet dimension |archive-url=https://web.archive.org/web/20190327090359/https://mashable.com/article/surreal-memes/#dBMHaD8Josqt |archive-date=March 27, 2019 |access-date=March 26, 2019}}</ref> This strange irony was discussed in the ''Washington Post'' article "Why is millennial humor so weird?" to show the disconnect from how millennials and other generations conceive of humor;<ref>{{cite news|last=Bruenig|first=Elizabeth |work=The Washington Post |date=August 11, 2017 |url=https://www.washingtonpost.com/outlook/why-is-millennial-humor-so-weird/2017/08/11/64af9cae-7dd5-11e7-83c7-5bd5460f0d7e_story.html |title=Why is millennial humor so weird? |archive-url=https://web.archive.org/web/20190507081349/https://www.washingtonpost.com/outlook/why-is-millennial-humor-so-weird/2017/08/11/64af9cae-7dd5-11e7-83c7-5bd5460f0d7e_story.html |archive-date=May 7, 2019|access-date=March 26, 2019}}</ref> the article itself also became a meme where people photoshopped examples of deep-fried and surreal memes onto the article to make fun of the point of the article and the abstraction of meme culture.<ref>{{cite web|website=Know Your Meme |url=https://knowyourmeme.com/memes/why-is-millennial-humor-so-weird |title=Why Is Millennial Humor So Weird? |archive-url=https://web.archive.org/web/20190326134706/https://knowyourmeme.com/memes/why-is-millennial-humor-so-weird |archive-date=March 26, 2019|access-date=March 26, 2019}}</ref>

====Short-form video ====
{{See also|Vine (service)|TikTok}} After the success of the application Vine, a format of memes emerged in the form of short videos and scripted sketches.<ref>{{cite web|last=Dry|first=Jude |url=https://www.indiewire.com/2016/10/vine-dead-twitter-snapchat-video-legacy-1201740985/ |title=Vine Is Gone, But Not Forgotten: Why Twitter's Defunct Platform Was an Incubator for Digital Creatives |website=IndieWire|date=October 27, 2016 |archive-url=https://web.archive.org/web/20191112182428/https://www.indiewire.com/2016/10/vine-dead-twitter-snapchat-video-legacy-1201740985/ |archive-date=November 12, 2019|access-date=September 19, 2019}}</ref> Vine, in spite of its closure in early 2017, has still retained relevance through uploads of viral vines in compilations onto other sharing social media sites such as [[Twitter]] and [[YouTube]].<ref>{{cite magazine|last=Glum|first=Julia |url=http://money.com/money/longform/vine-compilations-youtube-collab-payouts/ |title=Millions Are Obsessed With Vine Compilations on YouTube. Now There's a Battle Brewing Over Who Should Get Paid |magazine=Money |date=April 10, 2019|archive-url=https://web.archive.org/web/20190916222751/http://money.com/money/longform/vine-compilations-youtube-collab-payouts/ |archive-date=September 16, 2019 |access-date=September 19, 2019}}</ref> Since Vine's shutdown, the service [[TikTok]] has been described as a better version of Vine and many comparisons have been made between the two platforms;<ref>{{cite web|last=Esposito|first=Brad |url=https://www.pedestrian.tv/news/tik-tok-is-winning-because-it-finally-gives-us-what-we-want/ |title=Tik Tok Is Winning Because It Finally Gives Us What We Want |website=Pedestrian |date=May 22, 2019 |archive-url=https://web.archive.org/web/20190524232636/https://www.pedestrian.tv/news/tik-tok-is-winning-because-it-finally-gives-us-what-we-want/ |archive-date=May 24, 2019|access-date=September 19, 2019}}</ref> also based on the upload of short-form videos, TikTok, however, allows videos and memes up to a minute in length rather than six seconds.<ref>{{cite web|last=Alexander|first=Julia |date=April 2, 2019|url=https://www.theverge.com/2019/4/2/18201898/tiktok-guide-for-you-challenge-creator-trend-algorithm-privacy |title=Your guide to using TikTok |website=The Verge |archive-url=https://web.archive.org/web/20190819135207/https://www.theverge.com/2019/4/2/18201898/tiktok-guide-for-you-challenge-creator-trend-algorithm-privacy |archive-date=August 19, 2019|access-date=September 19, 2019}}</ref> {{See also|Reaction video}}

The short-form videos created on sites like Vine and TikTok found use in being posted on other social media sites, such as Twitter, as a form of reacting and responding to other posts. These videos become replicated into other contexts and often become part of [[Internet culture]]. An example of a TikTok meme is the cosplay by Nyannyancosplay juxtaposed to the musical track "[[Mia Khalifa (song)|Mia Khalifa]]" by iLoveFriday. This meme became known as Hit or Miss.<ref>{{Cite web|url=https://knowyourmeme.com/memes/people/nyannyancosplay-hit-or-miss|title=Nyannyancosplay / Hit or Miss|website=Know Your Meme|access-date=2020-04-28}}</ref> Hit or Miss has been referenced multiple times, including [[PewDiePie]]'s 2018 Rewind as one of the most influential memes of the year alongside numerous other influential memes of the year.<ref>{{cite AV media|author=PewDiePie |date=December 27, 2018 |via=YouTube|title=YouTube Rewind 2018, but it's actually good|url=https://www.youtube.com/watch?v=By_Cn5ixYLg|access-date=2020-05-02}}</ref> PewDiePie's 2018 rewind video has been viewed over 70 million times and has 8.9 million likes as of April 28, 2020. Hit or Miss has been remixed as well, including by other social media influencers such as [[Belle Delphine]]. SirKibbs' YouTube has uploaded a video of Belle Delphine and Kat (Nyannyancosplay) side-by-side comparison and has garnered 2.7 million views as of April 28, 2020.<ref>{{cite AV media|author=SirKibbs|date=November 19, 2018|via=YouTube |title=Hit or miss - Belle Delphine vs Kat|url=https://www.youtube.com/watch?v=2tXmsxvCOGc|access-date=2020-05-02}}</ref>

==Marketing==
Public relations, advertising, and marketing professionals have embraced Internet memes as a form of [[viral marketing]] and [[guerrilla marketing]] to create marketing "[[Marketing buzz|buzz]]" for their product or service. The practice of using memes to market products or services is known as [[Memetics|memetic]] marketing.<ref name="memetic marketing">{{Cite news |first=Nick |last=Flor |title=Memetic Marketing |publisher=InformIT |date=December 11, 2000 |access-date=July 29, 2011 |url=http://www.informit.com/articles/article.aspx?p=19996 |archive-url=https://web.archive.org/web/20120114180653/http://www.informit.com/articles/article.aspx?p=19996 |archive-date=January 14, 2012 |url-status=live}}</ref> Internet memes are seen as cost-effective, and because they are a (sometimes self-conscious) [[fad]], they are therefore used as a way to create an image of awareness or trendiness. To this end, businesses have taken to attempting two methods of using memes to increase publicity and sales of their company; either creating a meme or attempting to adapt or perpetuate an existing one.<ref>{{Cite web |url=https://www.forbes.com/sites/forbescommunicationscouncil/2017/05/08/meme-marketing-how-brands-are-speaking-a-new-consumer-language/ |title=Meme Marketing: How Brands Are Speaking A New Consumer Language |last=McCrae|first=James |website=Forbes |date=May 8, 2017|access-date=December 10, 2018|archive-url=https://web.archive.org/web/20180315134855/https://www.forbes.com/sites/forbescommunicationscouncil/2017/05/08/meme-marketing-how-brands-are-speaking-a-new-consumer-language/|archive-date=March 15, 2018|url-status=live}}</ref> Examples of memetic marketing include the [[FreeCreditReport.com]] singing ad campaign,<ref>{{Cite news |last=McAlone |first=Nathan |url=http://pigeonsandplanes.com/in-depth/2014/03/freecreditreport-com-band |title=We Found The FreeCreditReport.Com Band, and They Aren't Who You Thought They Were |work=PigeonsandPlanes|date=March 4, 2014|access-date=April 19, 2017 |archive-url=https://web.archive.org/web/20170419201842/http://pigeonsandplanes.com/in-depth/2014/03/freecreditreport-com-band|archive-date=April 19, 2017|url-status=live}}</ref> the "Nope, Chuck Testa" meme from an advertisement for taxidermist [[Chuck Testa]], [[Wilford Brimley]] saying "Diabeetus" from [[Liberty Medical]]{{citation needed|date=June 2020}} and the [[Dumb Ways to Die]] public announcement ad campaign by [[Metro Trains Melbourne]].

Marketers, for example, use Internet memes to create interest in films that would otherwise not generate positive publicity among critics. The 2006 film ''[[Snakes on a Plane]]'' generated much publicity via this method.<ref>{{Cite news |url=https://www.nytimes.com/2006/05/29/business/worldbusiness/29iht-carr.1839216.html |work=The New York Times |first=David |last=Carr |title=Hollywood bypassing critics and print as digital gets hotter |date=May 29, 2006 |access-date=October 16, 2012 |archive-url=https://web.archive.org/web/20120703071008/http://www.nytimes.com/2006/05/29/business/worldbusiness/29iht-carr.1839216.html |archive-date=July 3, 2012 |url-status=live}}</ref> Used in the context of public relations, the term would be more of an advertising [[buzzword]] than a proper Internet meme, although there is still an implication that the interest in the content is for purposes of trivia, ephemera, or frivolity rather than straightforward advertising and news.

Brands' use of memes has disadvantages when considering people's perception of a brand. While effective use of a meme can lead to increased sales and attention, seemingly forced, unoriginal, or unfunny usage of memes can negatively impact the brand as a whole.<ref>{{cite web|url=https://jacobinmag.com/2021/02/memes-never-monetized-corporate-advertising|title=Why Memes Will Never Be Monetized|website=[[Jacobin (magazine)|Jacobin]]|last1=Pegolo|first1=Valentina|last2=Carpenter|first2=Lucie|date=February 6, 2021|accessdate=February 7, 2021}}</ref> For instance, the fast food company [[Wendy's]] began a social media approach in 2017 that heavily featured memes and was initially met with success, resulting in an almost 50% profit growth that year;<ref>{{cite book |last1=Kao |first1=Griffin |last2=Perusse |first2=Michael |last3=Sheng |first3=Weizhen |last4=Hong |first4=Jessica |date=February 2020 |title=Turning Silicon into Gold |url=https://www.researchgate.net/publication/339544830 |location=Research Gate |publisher=Apress |pages=99–107 |isbn=978-1484256299}}</ref> however, the strategy has also backfired when sharing memes that are controversial or otherwise negatively perceived by consumers.<ref>{{cite news |first=Sarah |last=Whitten |title=A Wendy's tweet just went viral for all the wrong reasons |publisher=CNBC |date=January 4, 2017 |access-date=August 19, 2020 |url=https://www.cnbc.com/2017/01/04/wendys-saucy-tweets-are-hit-and-miss-on-social-media.html}}</ref><ref>{{Cite web |url=https://stuntandgimmicks.com/2018/03/09/memes-politics-and-snarky-web-content-marketing-when-brands-break-bad/ |title=Memes, Politics, and Snarky Web Content Marketing: When #Brands Break #Bad |last=Mouravskiy |first=Alex |website=Stunt and Gimmicks |date=March 9, 2018 |access-date=August 19, 2020}}</ref>

== By context ==

=== Politics ===
As internet memes become a common means of online expression, they become quickly used by those seeking to express political opinions or to actively campaign for (or against) a political entity.<ref>{{Cite journal|last1=Seiffert-Brockmann|first1=Jens|last2=Diehl|first2=Trevor|last3=Dobusch|first3=Leonhard|date=August 2018|title=Memes as games: The evolution of a digital discourse online|journal=New Media & Society|volume=20|issue=8|pages=2862–2879|doi=10.1177/1461444817735334|issn=1461-4448|s2cid=206729243}}</ref> In some ways, they can be seen as a modern form of the [[political cartoon]], offering up a way to democratize political commentary.<ref>{{cite web |url=https://theconversation.com/political-cartoonists-are-out-of-touch-its-time-to-make-way-for-memes-116471 |title=Political cartoonists are out of touch – it’s time to make way for memes |last=Grygiel |first=Jennifer |date=May 17, 2019 |website=theconversation.com |publisher=The Conversation |access-date=March 6, 2021}}</ref>

====Elections====
Early examples of political memes can be seen from those resulting from the [[Dean Scream]]. Another example can be seen from MyDavidCameron.com, a website that allowed users to change the text of a [[Conservative Party (UK)|British Conservative]] election campaign poster featuring [[David Cameron]] from the [[2010 United Kingdom general election|2010 general election]]. This website was often used to produce memes that replaced the original slogan with a series of exaggerated claims or sarcastic fake campaign promises along with derision of David Cameron's airbrushed appearance.

Within each subsequent election, and the growing importance of visual communications due to the Internet and social media, memes have become a more important element within political campaigns as fringe communities have shaped broader discourse through the use of Internet memes.<ref>{{Cite journal|last=MacLeod|first=Alan|date=2018-10-12|title=Book review: Kill all normies: Online culture wars from 4chan and Tumblr to Trump and the alt-right|journal=New Media & Society|volume=21|issue=2|pages=535–537|doi=10.1177/1461444818804143|issn=1461-4448|s2cid=67774146}}</ref> For example, [[Ted Cruz]]'s 2016 Republican presidential bid was damaged by [[Ted Cruz–Zodiac Killer meme|Internet memes]] that speculated he was the [[Zodiac Killer]].<ref>{{Cite web|last1=Stuart|first1=Tessa|date=2016-02-26|title=Is Ted Cruz the Zodiac Killer? Maybe, Say Florida Voters|url=https://www.rollingstone.com/politics/politics-news/is-ted-cruz-the-zodiac-killer-maybe-say-38-percent-of-florida-voters-89135/|access-date=2020-07-22|website=Rolling Stone}}</ref>

Another internet meme was created from the 2012 US presidential debate surrounding United States politician [[Mitt Romney]]'s usage of the phrase "[[binders full of women]]". Internet meme creators quickly created "My Binders Full of Women Exploded", referencing the Korean pop song "[[Gangnam Style in popular culture|Gangnam style]]" by overlaying the politician's quote onto a frame from [[Psy]]'s music video where paper blows around him. This internet meme specifically indexes the central attribute of intertextuality by blending together pop culture with politics.<ref name=":032"/>

There has further been academic research that provides evidence that the use of memes during elections has a role to play in informing the public. In a study of 378 Internet memes posted across Facebook during the [[2017 United Kingdom general election|2017 general election]], McLoughlin and Southern found memes were a widely shared conduit for basic political information to audiences who often did not seek it out.<ref name="auto2">{{Cite journal|last1=McLoughlin|first1=Liam|last2=Southern|first2=Rosalynd|date=2020-07-14|title=By any memes necessary? Small political acts, incidental exposure and memes during the 2017 UK general election|journal=The British Journal of Politics and International Relations|pages=136914812093059|doi=10.1177/1369148120930594|issn=1369-1481|doi-access=free}}</ref> Indeed, a fifth of all political memes posted during the election referenced a political policy which was part of a political parties mandate, while messages promoting people to vote were shared more than 160,000 times, suggesting memes have a small role to play in increasing [[voter turnout]].<ref name="auto2"/> Satirical memes that express political opinions are effective in not only informing others but also driving political debate and engagement with politics by offering an easy and even fun way to talk about important issues.<ref>{{Cite journal|last=Plevriti|first=Vasiliki|date=2014|title=Satirical User-Generated Memes as an Effective Source of Political Criticism, Extending Debate and Enhancing Civic Engagement|journal=University of Warwick}}</ref>

Some political campaigns have begun to explicitly taken advantage of the increasing influence of memes; as part of the [[2020 United States presidential election|2020 US presidential campaign]], [[Michael Bloomberg]] sponsored a number of Instagram accounts with over 60 million collective followers to post memes related to the Bloomberg campaign.<ref>{{Cite news|last=Lorenz|first=Taylor|date=February 13, 2020|title=Michael Bloomberg's Campaign Suddenly Drops Memes Everywhere|work=The New York Times|url=https://www.nytimes.com/2020/02/13/style/michael-bloomberg-memes-jerry-media.html|access-date=July 30, 2020}}</ref> Similar to criticisms against corporations who use meme marketing, the campaign was faulted for treating meme culture as an advertisement or something that can be bought.<ref>{{Cite web|last=Tiffany|first=Kaitlyn|date=February 28, 2020|title=You Can't Buy Memes|url=https://www.theatlantic.com/technology/archive/2020/02/bloomberg-memes-instagram-ads/607219/|access-date=July 30, 2020|work=The Atlantic}}</ref>

The 2020 Presidential Campaign of [[Kanye West]] quickly became a meme, following its announcement on [[Twitter]], with numerous celebrities and influencers endorsing the rapper out of [[irony]]. Other personalities began announcing their own satirical presidential campaigns, parodying West.

====Social movements====
Memes can play a significant role in various forms of activism.<ref name="JemielniakPrzegalinska202023"/>{{Rp|75}}<ref>{{Cite journal|last=Lenhardt|first=Corinna|date=2016-01-01|title="Free Peltier Now!" The Use of Internet Memes in American Indian Activism|url=https://meridian.allenpress.com/aicrj/article/40/3/67/211544/Free-Peltier-Now-The-Use-of-Internet-Memes-in|journal=American Indian Culture and Research Journal|language=en|volume=40|issue=3|pages=67–84|doi=10.17953/aicrj.40.3.lenhardt|issn=0161-6463}}</ref><ref>{{Cite journal|last=Huntington|first=Heidi E.|date=2016-01-01|title=Pepper Spray Cop and the American Dream: Using Synecdoche and Metaphor to Unlock Internet Memes' Visual Political Rhetoric|url=https://doi.org/10.1080/10510974.2015.1087414|journal=Communication Studies|volume=67|issue=1|pages=77–93|doi=10.1080/10510974.2015.1087414|s2cid=146429531|issn=1051-0974}}</ref><ref>{{Cite book|last1=Wimmer|first1=Jeffrey|url=https://books.google.com/books?id=czZDDwAAQBAJ&q=internet+memes+activism&pg=PT150|title=(Mis)Understanding Political Participation: Digital Practices, New Forms of Participation and the Renewal of Democracy|last2=Wallner|first2=Cornelia|last3=Winter|first3=Rainer|last4=Oelsner|first4=Karoline|date=2017-12-15|publisher=Routledge|isbn=978-1-317-21741-1|language=en}}</ref>

The [[Occupy Wall Street|Occupy Wall Street (OWS)]] protest movement saw a rise in internet memes after gaining attention on social media. All internet memes that were created and shared during the movement were very important in mediated discussions surrounding the OWS. Typical phrases such as "[[We are the 99%|We Are the 99]]%" and "This is what democracy looks like", were remixed into memes and subsequently posted in the discussion board of OWS on popular social media sites such as [[Reddit]], Tumblr, and [[4chan]]. Those who actively participated in the movement conversed through these visuals.<ref>{{Cite journal|last=Milner|first=Ryan M.|date=2013-10-30|title=Pop Polyvocality: Internet Memes, Public Participation, and the Occupy Wall Street Movement|url=https://ijoc.org/index.php/ijoc/article/view/1949|journal=International Journal of Communication|language=en|volume=7|pages=34|issn=1932-8036}}</ref>

=== Culture ===
==== Gender ====
Internet memes have been used in the context of gender and the LGBT issues on both sides of the issue.<ref>{{Cite journal|last1=Gal|first1=Noam|last2=Shifman|first2=Limor|last3=Kampf|first3=Zohar|date=2016-09-01|title="It Gets Better": Internet memes and the construction of collective identity|url=https://doi.org/10.1177/1461444814568784|journal=New Media & Society|language=en|volume=18|issue=8|pages=1698–1714|doi=10.1177/1461444814568784|s2cid=206728484|issn=1461-4448}}</ref><ref>{{Cite journal|last1=Drakett|first1=Jessica|last2=Rickett|first2=Bridgette|last3=Day|first3=Katy|last4=Milnes|first4=Kate|date=2018-02-01|title=Old jokes, new media – Online sexism and constructions of gender in Internet memes|url=https://doi.org/10.1177/0959353517727560|journal=Feminism & Psychology|language=en|volume=28|issue=1|pages=109–127|doi=10.1177/0959353517727560|s2cid=55756135|issn=0959-3535}}</ref> For example, the phrase, "I sexually identify as an attack helicopter," is a meme used to mock the concept of non-binary genders.<ref>Blake, K., Godwin, M., & Whyte, S. (2020). [https://journals.uic.edu/ojs/index.php/fm/article/view/10601 “I sexually identify as an Attack Helicopter”: Incels, trolls, and non-binary gender politics online.] ''First Monday'', ''25''(9). [[doi:10.5210/fm.v25i9.10601]]</ref> In contrast, memes supporting the LGBT community also exist. Memes about the [[incel]] community deal with issues of feminism and toxic masculinity;<ref>{{cite web |url=https://slate.com/human-interest/2018/07/incel-memes-like-millimeters-of-bone-and-virgin-vs-chad-mask-a-dangerous-and-toxic-culture.html |title=Incel Memes Aren’t a Joke |last=Cauterucci |first=Christina |date=July 19, 2018 |website=Slate |publisher=The Slate Group |access-date=March 6, 2021}}</ref> In this way, they often serve as ways to marginalize or draw attention to social problems depending on the punchline.

==== Religion ====
Internet memes have also been used in the context of religion.<ref>{{Cite journal|last1=Aguilar|first1=Gabrielle K.|last2=Campbell|first2=Heidi A.|last3=Stanley|first3=Mariah|last4=Taylor|first4=Ellen|date=2017-10-03|title=Communicating mixed messages about religion through internet memes|url=https://doi.org/10.1080/1369118X.2016.1229004|journal=Information, Communication & Society|volume=20|issue=10|pages=1498–1520|doi=10.1080/1369118X.2016.1229004|s2cid=151721706|issn=1369-118X}}</ref><ref>{{Cite journal|last1=Church|first1=Scott Haden|last2=Feller|first2=Gavin|date=2020-01-02|title=Synecdoche, Aesthetics, and the Sublime Online: Or, What's a Religious Internet Meme?|url=https://doi.org/10.1080/15348423.2020.1728188|journal=Journal of Media and Religion|volume=19|issue=1|pages=12–23|doi=10.1080/15348423.2020.1728188|s2cid=213540194|issn=1534-8423}}</ref>

== Copyright ==
{{Multiple issues|section=yes|
{{Off topic|date=October 2020}}
{{Original research|section|date=October 2020}}
}}
The eligibility of any memes to get copyright protection depends on the copyright law of the country in which such protection is sought. Some of the most popular formats of memes include cinematographic stills, personal or stock photographs, [[rage comic]]s, and illustrations meant to be a meme,<ref name="auto1">{{Cite journal |last1=S Iyer |first1=Aishwaria |last2=Mehrotra |first2=Raghav |title=A Critical Analysis of Memes and Fair Use |url=http://dspace.jgu.edu.in:8080/jspui/bitstream/10739/1443/1/A%20Critcal%20analysis%20of%20meme.pdf |journal=Rostrum Law Review}}</ref> and the copyright implications differ for each of these different formats. There is precedent both for memes to be in violation of copyright and in other memes having copyrights of their own.

If it is found that the meme has made use of a copyrighted work, such as the movie still or photograph without due permission from the original owner, it would amount to [[copyright infringement]]. Rage comics and memes created for the sole purpose of becoming memes would normally be original works of the creator and therefore, the question of infringing other copyright work does not arise.<ref name="lawandarts.org">{{Cite journal |last1=Offsay |first1=Max |title="What Do You Meme?": A Fair Use Analysis |url=https://lawandarts.org/2018/04/02/critical-corner-what-do-you-meme-a-fair-use-analysis/ |journal=Columbia Journal of Law and Arts|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190422040922/https://lawandarts.org/2018/04/02/critical-corner-what-do-you-meme-a-fair-use-analysis/|archive-date=April 22, 2019|url-status=dead}}</ref> In a cinematographic still, part of the entire end product is taken out of context and presented solely for its face value. The still is generally accompanied by a superimposed text of which conveys a distinctive idea or comment, such as the [[Boromir]] meme<ref>{{Cite web |url=https://knowyourmeme.com/memes/one-does-not-simply-walk-into-mordor |title=One Does Not Simply Walk into Mordor |website=Know Your Meme|access-date=April 20, 2019|archive-url=https://web.archive.org/web/20190425064033/https://knowyourmeme.com/memes/one-does-not-simply-walk-into-mordor|archive-date=April 25, 2019|url-status=live}}</ref> or "Gru's Plan".<ref>{{Cite web |url=https://knowyourmeme.com/memes/grus-plan |title=Gru's Plan |website=Know Your Meme|access-date=April 20, 2019|archive-url=https://web.archive.org/web/20190419192012/https://knowyourmeme.com/memes/grus-plan|archive-date=April 19, 2019|url-status=live}}</ref> This does not mean that all memes made from movie still or photographs are infringing copyright. There are defenses available for such use in various jurisdictions which could exempt the meme from attracting liability for the infringement.

=== United States ===
{{Main|Copyright#Obtaining protection}}
Under United States copyright law, a creation receives copyright protection if it satisfies four conditions under 17 U.S.C. § 102.<ref name="17USCode§102">{{Cite web |url=https://www.law.cornell.edu/uscode/text/17/102 |title=17 U.S. Code § 102. Subject matter of copyright: In general|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190517020438/https://www.law.cornell.edu/uscode/text/17/102|archive-date=May 17, 2019|url-status=live}}</ref> For a meme to get [[copyright]] protection, it would have to satisfy four conditions:
#It falls under one of the categories of work which is protected under the law
#It is an "expression"
#It has a modest amount of creativity
#It is "fixed".<ref name=":0">{{Cite web |url=http://www.mondaq.com/india/x/668356/Copyright/MEMES+AND+COPYRIGHT+FAIR+USE+OR+INFRINGEMENT |title=Memes and Copyright: Fair Use or Infringement? |last=Rout |first=Shrabani |website=Mondaq|date=January 30, 2018|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190424230716/http://www.mondaq.com/india/x/668356/Copyright/MEMES+AND+COPYRIGHT+FAIR+USE+OR+INFRINGEMENT|archive-date=April 24, 2019|url-status=live}}</ref>

Memes can be considered pictorial, graphical or motion picture, and so are subject to copyright law<ref name="17USCode§102" /> As such, memes are protected under copyright under the same conditions as these mediums, including concepts such as the low [[threshold of originality]] for what constitutes creativity (as demonstrated by [[Feist Publications, Inc., v. Rural Telephone Service Co.|Feist Publications, Inc., v. Rural Telephone Service Co]]).<ref>{{ussc|name=Feist Publications, Inc., v. Rural Telephone Service Co.|link=|volume=499|page=340|pin=|year=1991}}.</ref> Since a meme is essentially a comment, satire, ridicule or expression of an emotion it constitutes the expression of an idea. Memes are contained in the medium of the Internet and so are fixed expressions by 17 U.S.C. § 101.<ref>{{Cite web |url=https://www.law.cornell.edu/uscode/text/17/101 |title=17 U.S. Code § 101. Definitions|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20160430034315/https://www.law.cornell.edu/uscode/text/17/101|archive-date=April 30, 2016|url-status=live}}</ref>

==== Fair use ====
{{Main|Fair use}}

[[Fair use]] is a defense under US Copyright Law which protects work that has made using other copyrighted works.<ref>{{Cite web |url=https://www.law.cornell.edu/uscode/text/17/107 |title=17 U.S. Code § 107. Limitations on exclusive rights: Fair use|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190507195714/https://www.law.cornell.edu/uscode/text/17/107|archive-date=May 7, 2019|url-status=live}}</ref> The section provides that if a copyrighted work is reproduced "for purposes such as criticism, comment, news reporting, teaching [...], scholarship or research", it would not amount to infringement. Notably, for memes, the use of the term ''"such as"'' in the section denotes that the list is not exhaustive but merely illustrative. Furthermore, the factors mentioned in the section are subjective in nature and the weight of each factor varies on a case to case basis.<ref name=":2">{{Cite journal |last1=Patel |first1=Ronak |title=First World Problems:' A Fair Use Analysis of Internet Memes |url=https://cloudfront.escholarship.org/dist/prd/content/qt96h003jt/qt96h003jt.pdf?t=nndf5x&v=lg |journal=UCLA Entertainment Law Review |volume=20 |issue=2}}</ref>

The four factors are:
#The purpose or character of use,
#The nature of the copyrighted work,
#The amount and substantiality of the portion used, and
#Effect on the market.

Many memes are [[Transformation (law)|transformative]] in nature as they have no relation to the original work and the motive behind the communication of the meme is personal, in terms of disseminating humor to the public; such memes, being transformative, would be covered by fair use.<ref name=":2" /> However, copying memes that are made for the sole purpose of being memes would not enjoy this protection as there is no transformation{{em dash}}the copying has the same purpose as the original meme which is to communicate humorous or entertaining anecdotes.<ref name="barandbench.com">{{Cite web |url=https://barandbench.com/viewpoint-memes-copyright-got |title=The Viewpoint – Game of Thrones Memes: Potential Copyright Infringement or Fair Use? |last1=Mishra |first1=Meghna |last2=Nigam |first2=Anusuya |website=Bar and Bench|date=September 25, 2017|access-date=April 20, 2019}}</ref> Purpose and character of use weigh in against memes which have been used for commercial purposes because in those cases, the work has not been created for the communication of humor but for economic gain. For example, [[Grumpy Cat]] won $710,001 in a copyright lawsuit against the beverage company Grenade which used the Grumpy Cat image on its roasted coffee line and t-shirts.<ref>{{Cite news |url=https://www.thewrap.com/grumpy-cat-wins-710001-in-copyright-lawsuit-memes-have-rights-too/ |title=Grumpy Cat Wins $710,001 in Copyright Lawsuit: 'Memes Have Rights Too' |last1=Nakamura |first1=Reid |publisher=The Wrap|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190422040921/https://www.thewrap.com/grumpy-cat-wins-710001-in-copyright-lawsuit-memes-have-rights-too/|archive-date=April 22, 2019|url-status=live}}</ref>

The nature of the copyrighted work asks what the differences between the meme and the other material are. This factor applies to many types of memes because the original work is an artistic creation that has been published and thus the latter enjoys protection under copyright which the memes are violating. However, as memes are transformative, this factor does not have much weight.<ref name="lawandarts.org"/>

The amount and substantiality of the portion used tests not only the quantity of the work copied but the quality that is copied as well.<ref name="SCOTUS">{{Ussc|name=Harper & Row v. Nation Enterprises|471|539|1985}}. {{Usgovpd}}</ref> Memes copy only a small portion of a complete film, whereas for rage comics and personal photographs, the entire portion has been used to create the meme. Despite this, all categories of memes could fall under fair use because the text that is added to those images adds value, without which it would just be pictures.<ref name=":2" /> Moreover, the heart of the work is not affected because the still/picture is taken out of context and portrays something entirely different from what the image originally wanted to depict.<ref>{{Cite journal |last1=M. Lantagne |first1=Stacey |title=Famous on The Internet: The Spectrum of Internet Memes and The Legal Challenge of Evolving Methods of Communication |url=https://lawreview.richmond.edu/files/2018/01/Lantagne-522.pdf |journal=University of Richmond Law Review|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20191127185203/https://lawreview.richmond.edu/files/2018/01/Lantagne-522.pdf|archive-date=November 27, 2019|url-status=live}}</ref>

Lastly, the effect on the market offers court analysis on whether the meme would cause harm to the actual market of the original copyright work and also the harm it could cause to the potential market.<ref>{{Cite web |url=https://law.justia.com/cases/federal/appellate-courts/ca2/15-3885/15-3885-2018-02-27.html |title=Fox News Network, LLC v. TVEyes, Inc, Nos. 15-3885, 15-3886 (2d Cir. Feb. 27, 2018)|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190422040921/https://law.justia.com/cases/federal/appellate-courts/ca2/15-3885/15-3885-2018-02-27.html|archive-date=April 22, 2019|url-status=live}}</ref> The target audience for the original work and meme is entirely different as the latter is taken out of the context of the original and created for use and dissemination on social media.<ref name="lawandarts.org"/> Rage comics and memes created for the purpose of being memes are an exception to this because the target audience for both is the same and copied work could infringe on the potential market of the original. Warner Brothers was sued for infringing the [[Nyan Cat]] meme by using it its game Scribblenauts.<ref>{{Cite news |url=https://www.forbes.com/sites/emmawoollacott/2013/05/03/warner-brothers-sued-for-infringing-cat-meme-copyright/#24bc370d2e34 |title=Warner Brothers Sued For Infringing Cat Meme Copyright |last=Woollacott |first=Emma |work=Forbes|date=May 3, 2013|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190422040933/https://www.forbes.com/sites/emmawoollacott/2013/05/03/warner-brothers-sued-for-infringing-cat-meme-copyright/#24bc370d2e34|archive-date=April 22, 2019|url-status=live}}</ref>

=== India ===
Under Section 2(c)<ref>{{Cite web |url=https://indiankanoon.org/doc/121334999/ |title=Section 2(c) in the Copyright Act, 1957 |access-date=May 7, 2019 |archive-url=https://web.archive.org/web/20190420163506/https://indiankanoon.org/doc/121334999/ |archive-date=April 20, 2019 |url-status=live}}</ref> of the Indian Copyright Act, 1957, a meme could be classified as an 'artistic work' which states that an artistic work includes painting, sculpture, drawing (including a diagram, map, chart or plan), an engraving or a photograph, whether or not any such work possesses artistic quality.<ref name=":0" /> The section uses the phrase ''"whether or not possessing artistic quality"'', the memes that are rage comics or those such as [[Keyboard Cat]] would enjoy protection as they are original creations in the form a painting, drawing, photograph or short video clip, despite not having artistic quality.<ref name=":1">{{Cite web |url=https://spicyip.com/2013/12/guest-post-keep-calm-and-share-copyright-not-being-infringed.html |title=Keep Calm and Share – Copyright Not Being Infringed |last=Barooah |first= Swaraj Paul |website=SpicyIP|date=December 13, 2013 |access-date=May 7, 2019|archive-url=https://web.archive.org/web/20131216115115/https://spicyip.com/2013/12/guest-post-keep-calm-and-share-copyright-not-being-infringed.html|archive-date=December 16, 2013|url-status=live}}</ref> Memes that made from cinematograph still or photographs, the original image in the background for the meme would also be protected as the picture or the still from the series/movie is an 'artistic work'.<ref name="auto1"/> These memes are a modification of that already existing artistic work with some little amount of creativity and therefore, they would also enjoy copyright protection.

==== Fair dealing ====
{{Main|Fair dealing}}

India follows a fair dealing approach as an exception to copyright infringement under Section 52(1)(a) for the purposes of private or personal use, criticism or review.<ref name="auto">{{Cite journal|url=http://indiacode.nic.in/handle/123456789/1367|title=Copyright Act, 1957|date=June 4, 1957|via=indiacode.nic.in}}</ref> The analysis requires three steps: the amount and substantiality of dealing, the purpose of copying, and the effect on potential markets.

The amount of sustainability of dealing asks about how much of the original work is used in the meme, or how the meme transforms the original content. A meme makes use to existing copyright work whether it is a cinematograph still, rage comic, personal photograph or a meme made for the purpose of being a meme. However, since a meme is made for comedic purposes, taken out of context of the original work, they are transforming the work and creating a new work.<ref name=":0" />

The purpose of copying factors in the purpose of the meme compared to the purpose of the original work. Under Section 52(1)(a), the purpose is restricted to criticism or review.<ref name="auto"/> A meme, as long as it is a parody or a criticism of the original work would be protected under the exception, but once an element of commercialization comes in, they would no longer be exempted and because the purpose no longer falls under the those mentioned in the section .<ref name=":1" /> When the Indian comedic group [[All India Bakchod]] (AIB) parodied [[Game of Thrones]] through a series of memes, the primary purpose was to advertise products of companies that have endorsed the group and thus was not fair dealing.<ref name="barandbench.com"/>

Memes generally do not have an effect on the potential market for a work. There must be no intention on part of the infringer to compete with the original owner of the work and derive profits from it.<ref>{{Cite web |url=https://indiankanoon.org/doc/1685540/ |title=Blackwood & Sons Ltd. v. A.N. Parasuraman [AIR 1959 Mad. 410]|access-date=May 7, 2019|archive-url=https://web.archive.org/web/20190422040923/https://indiankanoon.org/doc/1685540/|archive-date=April 22, 2019|url-status=live}}</ref> Since memes are generally meant for comedic value and have no intention to supplant the market of the original creator, they fall within the ambit of this section.<ref name=":1"/>

==See also==
{{Portal|Internet}}
* [[Cliché]]
* [[List of Internet phenomena]]
* [[Pepe the Frog]]
* [[Remix culture]]

==References==
{{Reflist}}

==Further reading==
* {{Cite book |last=Blackmore |first=Susan |title=The Meme Machine |publisher=[[Oxford University Press]], 2000 |isbn=978-0192862129 |page=288 |url=https://books.google.com/books?id=dtkeLWVMlcsC&q=The+Meme+Machine |edition=Volume 25 of Popular Science Series|access-date=November 30, 2012 |date=March 16, 2000}}
* {{Cite book |last=Shifman |first=Limor |title=Memes in Digital Culture |publisher=[[MIT Press]], 2013 |date=November 8, 2013}}
* Wiggins, Bradley E. (September 22, 2014). How the Russia-Ukraine crisis became a magnet for memes. The Conversation. [https://theconversation.com/how-the-russia-ukraine-crisis-became-a-magnet-for-memes-31199 Theconversation.com]
* {{cite journal | last1 = Wiggins | first1 = Bradley E. | last2 = Bowers | first2 = G. Bret | year = 2014 | title = Memes as genre: A Structurational Analysis of the Memescape | journal = New Media & Society | volume = 17| issue = 11| pages = 1886–1906| doi = 10.1177/1461444814535194 | s2cid = 30729349 }}
* {{cite book |last=Distin |first=Kate |year=2005 |title=The Selfish Meme: A Critical Reassessment |place=Cambridge, U.K |publisher=Cambridge}}

==External links==
* {{Commonscatinline}}
* Gary Marshall, [https://web.archive.org/web/20190119134239/http://pespmc1.vub.ac.be/Conf/MemePap/Marshall.html The Internet and Memetics] – academic article about Internet and memes.

{{Internet slang}}

{{DEFAULTSORT:Internet Meme}}
[[Category:Internet memes| ]]
[[Category:1990s neologisms]]