Portal:Mathematics

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The Mathematics Portal

Pythagoras

Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

Featured articles –

  Featured articles are displayed here, which represent some of the best content on English Wikipedia.

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  Archimedes Thoughtful by Fetti (1620)

  
  Archimedes of Syracuse (/ˌɑːrkɪˈmiːdiːz/ AR-kim-EE-deez; c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.
  
  Archimedes' other mathematical achievements include deriving an approximation of pi (π), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy known as Archimedes' principle. In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe. He is also said to have built a planetarium device that demonstrated the movements of the known celestial bodies, and may have been a precursor to the Antikythera mechanism. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. (Full article...)
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  Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and in particle physics, for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.
  
  Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams and is widely used. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World, he was ranked the seventh-greatest physicist of all time. (Full article...)
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  Cantor, c. 1910

  
  Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.
  
  Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Konstantin Gutberlet was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). (Full article...)
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  Portrait by Jakob Emanuel Handmann, 1753

  
  Leonhard Euler (/ˈɔɪlər/ ^ⓘ OY-lər; Swiss Standard German: [ˈleːɔnhard ˈɔʏlər]; German: [ˈleːɔnhaʁt ˈɔʏlɐ] ^ⓘ; 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
  
  Euler is credited for popularizing the Greek letter $\pi$ (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation $f(x)$ for the value of a function, the letter $i$ to express the imaginary unit ${\sqrt {-1}}$, the Greek letter $\Sigma$ (capital sigma) to express summations, the Greek letter $\Delta$ (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant $e$, the base of the natural logarithm, now known as Euler's number. Euler made contributions to applied mathematics and engineering, such as his study of ships which helped navigation, his three volumes on optics contributed to the design of microscopes and telescopes, and he studied the bending of beams and the critical load of columns. (Full article...)
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  The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group S̃₃

  
  The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections of permutations (rearrangements) of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in combinatorics and representation theory.
  
  A finite symmetric group consists of all permutations of a finite set. Each affine symmetric group is an infinite extension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as descents and inversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation. (Full article...)
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  Plots of logarithm functions, with three commonly used bases. The special points log_b b = 1 are indicated by dotted lines, and all curves intersect in log_b 1 = 0.

  
  
  In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power: 1000 = 10³ = 10 × 10 × 10. More generally, if x = b^y, then y is the logarithm of x to base b, written log_b x, so log₁₀ 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.
  
  The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base 2 and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written log x. (Full article...)
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  The manipulations of the Rubik's Cube form the Rubik's Cube group.

  
  In mathematics, a group is a set with a binary operation that satisfies: the operation being associative, it having an identity element, and every element of the set having inverse element.
  
  Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation form an infinite group that is generated by a single element called ⁠$1$⁠ (these properties fully characterize the integers). (Full article...)
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  General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever is
  present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations.
  
  Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data. (Full article...)
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  Noether c. 1900–1910

  
  Amalie Emmy Noether (US: /ˈnʌtər/ ^ⓘ, UK: /ˈnɜːtə/; German: [ˈnøːtɐ]; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental in mathematical physics. Noether was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
  
  Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen–Nuremberg, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent. (Full article...)
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  The number π (/paɪ/ ^ⓘ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
  
  The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ${\tfrac {22}{7}}$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found. (Full article...)
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  Portrait by August Köhler, c. 1910, after 1620 original

  
  Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonice Mundi, and Epitome Astronomiae Copernicanae, influencing among others Isaac Newton, providing one of the foundations for his theory of universal gravitation. The variety and impact of his work made Kepler one of the founders and fathers of modern astronomy, the scientific method, natural and modern science. He has been described as the "father of science fiction" for his novel Somnium.
  
  Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein.
  Additionally, he did fundamental work in the field of optics, being named the father of modern optics, in particular for his Astronomiae pars optica. He also invented an improved version of the refracting telescope, the Keplerian telescope, which became the foundation of the modern refracting telescope, while also improving on the telescope design by Galileo Galilei, who mentioned Kepler's discoveries in his work. He is also known for postulating the Kepler conjecture. (Full article...)
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  Figure 1: A solution (in purple) to Apollonius's problem. The given circles are shown in black.

  
  
  In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 BC – c. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts).
  
  In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN. (Full article...)
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  Josiah Willard Gibbs (/ɡɪbz/; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he created modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis.
  
  In 1863, Yale University awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of mathematical physics from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history". In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics". (Full article...)
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  Elementary algebra studies which values solve equations formed using arithmetical operations.

  
  Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
  
  Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. (Full article...)
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  In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
  
  Thus the hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. (Full article...)

Selected image –

graph showing two sets of 4 points, each set perfectly fit by a trend line with positive slope; the set of points on the left is higher and the set on the right lower, so the entire collection of points is best fit by a trend line with negative slope

Simpson's paradox

Credit: Schutz

Simpson's paradox (also known as the Yule–Simpson effect) states that an observed association between two variables can reverse when considered at separate levels of a third variable (or, conversely, that the association can reverse when separate groups are combined). Shown here is an illustration of the paradox for quantitative data. In the graph the overall association between X and Y is negative (as X increases, Y tends to decrease when all of the data is considered, as indicated by the negative slope of the dashed line); but when the blue and red points are considered separately (two levels of a third variable, color), the association between X and Y appears to be positive in each subgroup (positive slopes on the blue and red lines — note that the effect in real-world data is rarely this extreme). Named after British statistician Edward H. Simpson, who first described the paradox in 1951 (in the context of qualitative data), similar effects had been mentioned by Karl Pearson (and coauthors) in 1899, and by Udny Yule in 1903. One famous real-life instance of Simpson's paradox occurred in the UC Berkeley gender-bias case of the 1970s, in which the university was sued for gender discrimination because it had a higher admission rate for male applicants to its graduate schools than for female applicants (and the effect was statistically significant). The effect was reversed, however, when the data was split by department: most departments showed a small but significant bias in favor of women. The explanation was that women tended to apply to competitive departments with low rates of admission even among qualified applicants, whereas men tended to apply to less-competitive departments with high rates of admission among qualified applicants. (Note that splitting by department was a more appropriate way of looking at the data since it is individual departments, not the university as a whole, that admit graduate students.)

Good articles –

  These are Good articles, which meet a core set of high editorial standards.

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  Rosa Margaret Morris (16 July 1914 – 15 October 2011) was a Welsh applied mathematician, working in potential theory and aerodynamics. When she was 23, her research and examination results made national and international news. She received several fellowships and awards before graduating with a PhD in mathematics from the University of Cambridge in 1940. In her later career, she taught at the University College of South Wales and Monmouthshire (now Cardiff University), where she co-authored a successful textbook on Mathematical Methods of Physics with Roy Chisholm and became one of the first female Heads of School of Mathematics in the United Kingdom. (Full article...)
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  Advanced Placement (AP) Statistics (also known as AP Stats) is a college-level high school statistics course offered in the United States through the College Board's Advanced Placement program. This course is equivalent to a one semester, non-calculus-based introductory college statistics course and is normally offered to sophomores, juniors and seniors in high school.
  
  One of the College Board's more recent additions, the AP Statistics exam was first administered in May 1996 to supplement the AP program's math offerings, which had previously consisted of only AP Calculus AB and BC. In the United States, enrollment in AP Statistics classes has increased at a higher rate than in any other AP class. (Full article...)
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  In this graph, an even number of vertices (the four vertices numbered 2, 4, 5, and 6) have odd degrees. The sum of degrees of all six vertices is 2 + 3 + 2 + 3 + 3 + 1 = 14, twice the number of edges.

  
  In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. The handshaking lemma is a consequence of the degree sum formula, also sometimes called the handshaking lemma, according to which the sum of the degrees (the numbers of times each vertex is touched) equals twice the number of edges in the graph. Both results were proven by Leonhard Euler (1736) in his famous paper on the Seven Bridges of Königsberg that began the study of graph theory.
  
  Beyond the Seven Bridges of Königsberg Problem, which subsequently formalized Eulerian Tours, other applications of the degree sum formula include proofs of certain combinatorial structures. For example, in the proofs of Sperner's lemma and the mountain climbing problem the geometric properties of the formula commonly arise. The complexity class PPA encapsulates the difficulty of finding a second odd vertex, given one such vertex in a large implicitly-defined graph. (Full article...)
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  Escher in 1971

  
  Maurits Cornelis Escher (/ˈɛʃər/; Dutch: [ˈmʌurɪts kɔrˈneːlɪs ˈɛɕər]; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithographs, and mezzotints, many of which were inspired by mathematics.
  Despite wide popular interest, for most of his life Escher was neglected in the art world, even in his native Netherlands. He was 70 before a retrospective exhibition was held. In the late twentieth century, he became more widely appreciated, and in the twenty-first century he has been celebrated in exhibitions around the world.
  
  His work features mathematical objects and operations including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, and Donald Coxeter, and the crystallographer Friedrich Haag, and conducted his own research into tessellation. (Full article...)
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  Malfatti circles

  
  In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle.
  
  Malfatti's problem has been used to refer both to the problem of constructing the Malfatti circles and to the problem of finding three area-maximizing circles within a triangle.
  A simple construction of the Malfatti circles was given by Steiner (1826), and many mathematicians have since studied the problem. Malfatti himself supplied a formula for the radii of the three circles, and they may also be used to define two triangle centers, the Ajima–Malfatti points of a triangle. (Full article...)
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  The explanatory indispensability argument is an argument in the philosophy of mathematics for the existence of mathematical objects. It claims that rationally we should believe in mathematical objects such as numbers because they are indispensable to scientific explanations of empirical phenomena. An altered form of the Quine–Putnam indispensability argument, it differs from that argument in its increased focus on specific explanations instead of whole theories and in its shift towards inference to the best explanation as a justification for belief in mathematical objects rather than confirmational holism.
  
  Specific explanations proposed as examples of mathematical explanations in science include why periodical cicadas have prime-numbered life cycles, why bee honeycomb has a hexagonal structure, and the solution to the Seven Bridges of Königsberg problem. Objections to the argument include the idea that mathematics is only used as a representational device, even when it features in scientific explanations; that mathematics does not need to be true to be explanatory because it could be a useful fiction; and that the argument is circular and so begs the question in favour of mathematical objects. (Full article...)
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  In number theory, a Wieferich prime is a prime number p such that p² divides 2^p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2^p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.
  
  Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture. (Full article...)
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  Schwarz lantern on display in the German Museum of Technology, Berlin

  
  
  In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stacked rings of isosceles triangles, arranged within each ring in the same pattern as an antiprism. The resulting shape can be folded from paper, and is named after mathematician Hermann Schwarz and for its resemblance to a cylindrical paper lantern. It is also known as Schwarz's boot, Schwarz's polyhedron, or the Chinese lantern.
  
  As Schwarz showed, for the surface area of a polyhedron to converge to the surface area of a curved surface, it is not sufficient to simply increase the number of rings and the number of isosceles triangles per ring. Depending on the relation of the number of rings to the number of triangles per ring, the area of the lantern can converge to the area of the cylinder, to a limit arbitrarily larger than the area of the cylinder, or to infinity—in other words, the area can diverge. The Schwarz lantern demonstrates that sampling a curved surface by close-together points and connecting them by small triangles is inadequate to ensure an accurate approximation of area, in contrast to the accurate approximation of arc length by inscribed polygonal chains. (Full article...)
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  1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral.
  
  In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. (Full article...)
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  The graph of the 3-3 duoprism (the line graph of $K_{3,3}$) is perfect. Here it is colored with three colors, with one of its 3-vertex maximum cliques highlighted.

  
  In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices.
  
  The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time, despite their greater complexity for non-perfect graphs. In addition, several important minimax theorems in combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, Kőnig's theorem on matchings, and the Erdős–Szekeres theorem on monotonic sequences, can be expressed in terms of the perfection of certain associated graphs. (Full article...)
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  Squaring the circle: the areas of this square and this circle are both equal to π. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.

  
  Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
  
  In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ($\pi$) is a transcendental number.
  That is, $\pi$ is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if $\pi$ were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. (Full article...)
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  In geometry, a square pyramid is a pyramid with a square base and four triangles, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral. It is called an equilateral square pyramid, an example of a Johnson solid.
  
  Square pyramids have appeared throughout the history of architecture, with examples being Egyptian pyramids and many other similar buildings. They also occur in chemistry in square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches. (Full article...)

Did you know (auto-generated) –

• ... that in 1967 two mathematicians published PhD dissertations independently disproving the same thirteen-year-old conjecture?
• ... that Latvian-Soviet artist Karlis Johansons exhibited a skeletal tensegrity form of the Schönhardt polyhedron seven years before Erich Schönhardt's 1928 paper on its mathematics?
• ... that multiple mathematics competitions have made use of Sophie Germain's identity?
• ... that the discovery of Descartes' theorem in geometry came from a too-difficult mathematics problem posed to a princess?
• ... that two members of the French parliament were killed when a delayed-action German bomb exploded in the town hall at Bapaume on 25 March 1917?
• ... that in the aftermath of the American Civil War, the only Black-led organization providing teachers to formerly enslaved people was the African Civilization Society?
• ... that after Archimedes first defined convex curves, mathematicians lost interest in their analysis until the 19th century, more than two millennia later?
• ... that people in Madagascar perform algebra on tree seeds in order to tell the future?

More did you know –

Did you know...

• ... that the Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle?
• ... that the Life without Death cellular automaton, a mathematical model of pattern formation, is a variant of Conway's Game of Life in which cells, once brought to life, never die?
• ... that one can list every positive rational number without repetition by breadth-first traversal of the Calkin–Wilf tree?
• ... that the Hadwiger conjecture implies that the external surface of any three-dimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?
• ... that an equitable coloring of a graph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than a graph coloring without this constraint?
• ... that no matter how biased a coin one uses, flipping a coin to determine whether each edge is present or absent in a countably infinite graph will always produce the same graph, the Rado graph?
• ...that it is possible to stack identical dominoes off the edge of a table to create an arbitrarily large overhang?

Selected article –

Johannes Kepler Image credit: User:ArtMechanic

Johannes Kepler (1571 – 1630) was an Austrian Lutheran mathematician, astronomer and a key figure in the 17th century astronomical revolution. He is best known for his laws of planetary motion, based on his works Astronomia nova and Harmonice Mundi; Kepler's laws provided one of the foundations of Isaac Newton's theory of universal gravitation. Before Kepler, planets' paths were computed by combinations of the circular motions of the celestial orbs; after Kepler astronomers shifted their attention from orbs to orbits—paths that could be represented mathematically as an ellipse.

During his career Kepler was a mathematics teacher at a Graz seminary school (later the University of Graz, Austria), an assistant to Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics teacher in Linz, Austria, and adviser to General Wallenstein. He also did fundamental work in the field of optics and helped to legitimize the telescopic discoveries of his contemporary Galileo Galilei.

Kepler lived in an era when there was no clear distinction between astronomy and astrology, while there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of the more prestigious discipline of philosophy). (Full article...)

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