Portal:Mathematics

The Mathematics Portal

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...)

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Émile Michel Hyacinthe Lemoine (French: [emil ləmwan]; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most notably, the École Polytechnique. Lemoine taught as a private tutor for a short period after his graduation from the latter school.

Lemoine is best known for his proof of the existence of the Lemoine point (or the symmedian point) of a triangle. Other mathematical work includes a system he called Géométrographie and a method which related algebraic expressions to geometric objects. He has been called a co-founder of modern triangle geometry, as many of its characteristics are present in his work. (Full article...)
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Archimedes Thoughtful by Fetti (1620)

Archimedes of Syracuse (/ˌɑːrkɪˈmiːdiːz/ AR-kih-MEE-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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In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously. (Full article...)
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Kaczynski after his arrest in 1996

Theodore John Kaczynski (/kəˈzɪnski/ kə-ZIN-skee; May 22, 1942 – June 10, 2023), also known as the Unabomber (/ˈjuːnəbɒmər/ YOO-nə-bom-ər), was an American mathematician and domestic terrorist. He was a mathematics prodigy, but abandoned his academic career in 1969 to pursue a reclusive primitive lifestyle and lone wolf terrorism campaign.

Kaczynski murdered three people and injured 23 others between 1978 and 1995 in a nationwide mail bombing campaign against people he believed to be advancing modern technology and the destruction of the natural environment. He authored a roughly 35,000-word manifesto and social critique called Industrial Society and Its Future which opposes all forms of technology, rejects leftism and fascism, advocates cultural primitivism, and ultimately suggests violent revolution. (Full article...)
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High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of spacetime (blue lines) due to the Sun's mass.

General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.

By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion. (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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The weighing pans of this balance scale contain zero objects, divided into two equal groups.

In mathematics, zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": zero is an integer multiple of 2, specifically 0 × 2. As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if $y$ is even then $y + x$ has the same parity as $x$ —indeed, $0 + x$ and $x$ always have the same parity.

Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as even − even = even, require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all. (Full article...)
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Title page of the first edition of Wright's Certaine Errors in Navigation (1599)

Edward Wright (baptised 8 October 1561; died November 1615) was an English mathematician and cartographer noted for his book Certaine Errors in Navigation (1599; 2nd ed., 1610), which for the first time explained the mathematical basis of the Mercator projection by building on the works of Pedro Nunes, and set out a reference table giving the linear scale multiplication factor as a function of latitude, calculated for each minute of arc up to a latitude of 75°. This was in fact a table of values of the integral of the secant function, and was the essential step needed to make practical both the making and the navigational use of Mercator charts.

Wright was born at Garveston in Norfolk and educated at Gonville and Caius College, Cambridge, where he became a fellow from 1587 to 1596. In 1589 the college granted him leave after Elizabeth I requested that he carry out navigational studies with a raiding expedition organised by the Earl of Cumberland to the Azores to capture Spanish galleons. The expedition's route was the subject of the first map to be prepared according to Wright's projection, which was published in Certaine Errors in 1599. The same year, Wright created and published the first world map produced in England and the first to use the Mercator projection since Gerardus Mercator's original 1569 map. (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 [de; it] 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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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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The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group $S̃ 3$

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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Bust of Shen at the Beijing Ancient Observatory

Shen Kuo (Chinese: 沈括; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁), was a Chinese polymath, scientist, and statesman of the Song dynasty (960–1279). Shen was a master in many fields of study including mathematics, optics, and horology. In his career as a civil servant, he became a finance minister, governmental state inspector, head official for the Bureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as an academic chancellor. At court his political allegiance was to the Reformist faction known as the New Policies Group, headed by Chancellor Wang Anshi (1021–1085).

In his Dream Pool Essays or Dream Torrent Essays (夢溪筆談; Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needle compass, which would be used for navigation (first described in Europe by Alexander Neckam in 1187). Shen discovered the concept of true north in terms of magnetic declination towards the north pole, with experimentation of suspended magnetic needles and "the improved meridian determined by Shen's [astronomical] measurement of the distance between the pole star and true north". This was the decisive step in human history to make compasses more useful for navigation, and may have been a concept unknown in Europe for another four hundred years (evidence of German sundials made circa 1450 show markings similar to Chinese geomancers' compasses in regard to declination). (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...)
$Image 14 Portrait by August Köhler, c. 1910, after 1620 original Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonice Mundi, and Epitome Astronomiae Copernicanae, influencing among others Isaac Newton, providing one of the foundations for his theory of universal gravitation. The variety and impact of his work made Kepler one of the founders and fathers of modern astronomy, the scientific method, natural and modern science. He has been described as the "father of science fiction" for his novel Somnium. Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, being named the father of modern optics, in particular for his Astronomiae pars optica. He also invented an improved version of the refracting telescope, the Keplerian telescope, which became the foundation of the modern refracting telescope, while also improving on the telescope design by Galileo Galilei, who mentioned Kepler's discoveries in his work. He is also known for postulating the Kepler conjecture. (Full article...)$

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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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The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... The graph is situated with positive integers to the right and negative integers to the left.

In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
$\sum _{n=1}^{m}n(-1)^{n-1}.$

The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:
$1-2+3-4+\cdots ={\frac {1}{4}}.$ (Full article...)

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Gamma function

Credit: TakuyaMurata (uploader)

This is a hand-drawn graph of the absolute value (or modulus) of the gamma function on the complex plane, as published in the 1909 book Tables of Higher Functions, by Eugene Jahnke and Fritz Emde. Such three-dimensional graphs of complicated functions were rare before the advent of high-resolution computer graphics (even today, tables of values are used in many contexts to look up function values instead of consulting graphs directly). Published even before applications for the complex gamma function were discovered in theoretical physics in the 1930s, Jahnke and Emde's graph "acquired an almost iconic status", according to physicist Michael Berry. See a similar computer-generated image for comparison. When restricted to positive integers, the gamma function generates the factorials through the relation

Γ(n) = (n - 1)!

, which is the product of all positive integers from

n - 1

down to 1 (

0!

is defined to be equal to 1). For real and complex numbers, the function is defined by the improper integral

\textstyle \Gamma (t)=\int _{0}^{\infty }x^{t-1}e^{-x}\,dx

. This integral diverges when t is a negative integer, which is causing the spikes in the left half of the graph (these are simple poles of the function, where its values increase to infinity, analogous to asymptotes in two-dimensional graphs). The gamma function has applications in quantum physics, astrophysics, and fluid dynamics, as well as in number theory and probability.

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Image 1
The complete graph $K 4$ has ten matchings, corresponding to the value $T (4) = 10$ of the fourth telephone number.

In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways $n$ people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on $n$ vertices, the number of permutations on $n$ elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with $n$ cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values (starting from $n = 0$ ) (Full article...)
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A spiral staircase in the Cathedral of St. John the Divine. Several helical curves in the staircase project to hyperbolic spirals in its photograph.

A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals or decreasing angles of Archimedean spirals. As this curve widens, it approaches an asymptotic line. It can be found in the view up a spiral staircase and the starting arrangement of certain footraces, and is used to model spiral galaxies and architectural volutes.

As a plane curve, a hyperbolic spiral can be described in polar coordinates $(r,\varphi )$ by the equation
$r={\frac {a}{\varphi }},$
for an arbitrary choice of the scale factor $a.$ (Full article...)
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Equilateral form of the Cairo tiling

In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called MacMahon's net after Percy Alexander MacMahon, who depicted it in his 1921 publication New Mathematical Pastimes. John Horton Conway called it a 4-fold pentille.

Infinitely many different pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings. Another, overlaying two flattened tilings by regular hexagons, is the form used in Cairo and has the property that every edge is collinear with infinitely many other edges. (Full article...)
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Title page of the second (1523) edition

Summa de arithmetica, geometria, proportioni et proportionalita (Summary of arithmetic, geometry, proportions and proportionality) is a book on mathematics written by Luca Pacioli and first published in 1494. It contains a comprehensive summary of Renaissance mathematics, including practical arithmetic, basic algebra, basic geometry and accounting, written for use as a textbook and reference work.

Written in vernacular Italian, the Summa is the first printed work on algebra, and it contains the first published description of the double-entry bookkeeping system. It set a new standard for writing and argumentation about algebra, and its impact upon the subsequent development and standardization of professional accounting methods was so great that Pacioli is sometimes referred to as the "father of accounting". (Full article...)
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L6a4

In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.

The Borromean rings are named after the Italian House of Borromeo, who used the circular form of these rings as an element of their coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan. They have been used in Christian symbolism as a sign of the Trinity, and in modern commerce as the logo of Ballantine beer, giving them the alternative name Ballantine rings. Physical instances of the Borromean rings have been made from linked DNA or other molecules, and they have analogues in the Efimov state and Borromean nuclei, both of which have three components bound to each other although no two of them are bound. (Full article...)
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Euclidean minimum spanning tree of 25 random points in the plane

A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. In it, any two points can reach each other along a path through the line segments. It can be found as the minimum spanning tree of a complete graph with the points as vertices and the Euclidean distances between points as edge weights.

The edges of the minimum spanning tree meet at angles of at least 60°, at most six to a vertex. In higher dimensions, the number of edges per vertex is bounded by the kissing number of tangent unit spheres. The total length of the edges, for points in a unit square, is at most proportional to the square root of the number of points. Each edge lies in an empty region of the plane, and these regions can be used to prove that the Euclidean minimum spanning tree is a subgraph of other geometric graphs including the relative neighborhood graph and Delaunay triangulation. By constructing the Delaunay triangulation and then applying a graph minimum spanning tree algorithm, the minimum spanning tree of $n$ given planar points may be found in time $O(n\log n)$ , as expressed in big O notation. This is optimal in some models of computation, although faster randomized algorithms exist for points with integer coordinates. For points in higher dimensions, finding an optimal algorithm remains an open problem. (Full article...)
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Einstein in 1947

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence formula $E = mc 2$ , which arises from special relativity, has been called "the world's most famous equation". He received the 1921 Nobel Prize in Physics for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect.

Born in the German Empire, Einstein moved to Switzerland in 1895, forsaking his German citizenship (as a subject of the Kingdom of Württemberg) the following year. In 1897, at the age of seventeen, he enrolled in the mathematics and physics teaching diploma program at the Swiss federal polytechnic school in Zurich, graduating in 1900. He acquired Swiss citizenship a year later, which he kept for the rest of his life, and afterwards secured a permanent position at the Swiss Patent Office in Bern. In 1905, he submitted a successful PhD dissertation to the University of Zurich. In 1914, he moved to Berlin to join the Prussian Academy of Sciences and the Humboldt University of Berlin, becoming director of the Kaiser Wilhelm Institute for Physics in 1917; he also became a German citizen again, this time as a subject of the Kingdom of Prussia. In 1933, while Einstein was visiting the United States, Adolf Hitler came to power in Germany. Horrified by the Nazi persecution of his fellow Jews, he decided to remain in the US, and was granted American citizenship in 1940. On the eve of World War II, he endorsed a letter to President Franklin D. Roosevelt alerting him to the potential German nuclear weapons program and recommending that the US begin similar research. (Full article...)
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The graph of a function, drawn in black, and a tangent line to that graph, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.

There are multiple different notations for differentiation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances. (Full article...)
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Alexandrov's theorem on polyhedra is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s. (Full article...)
Image 10
A binary tiling in the Poincaré disk model of the hyperbolic plane. Each tile edge lies on a horocycle (shown as circles interior to the disk) or a hyperbolic line (arcs perpendicular to the disk boundary). The horocycles and lines are asymptotic to an ideal point located at the right side of the Poincaré disk.

In geometry, a binary tiling (sometimes called a Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. The tiles are congruent, each adjoining five others. They may be convex pentagons, or non-convex shapes with four sides, alternatingly line segments and horocyclic arcs, meeting at four right angles.

There are uncountably many distinct binary tilings for a given shape of tile. They are all weakly aperiodic, which means that they can have a one-dimensional symmetry group but not a two-dimensional family of symmetries. There exist binary tilings with tiles of arbitrarily small area. (Full article...)
Image 11
A set of 20 points in a 10 × 10 grid, with no three points in a line.

The no-three-in-line problem in discrete geometry asks how many points can be placed in the $n\times n$ grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied geometric questions concerning lattice points".

At most $2n$ points can be placed, because $2n+1$ points in a grid would include a row of three or more points, by the pigeonhole principle. Although the problem can be solved with $2n$ points for every $n$ up to $46$ , it is conjectured that fewer than $2n$ points can be placed in grids of large size. Known methods can place linearly many points in grids of arbitrary size, but the best of these methods place slightly fewer than $1.5n$ points, not $2n$ . (Full article...)
Image 12
As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows $sin x$ and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at $x = 0$ .

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

The partial sum formed by the first $n + 1$ terms of a Taylor series is a polynomial of degree $n$ that is called the $n$ th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as $n$ increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point $x$ if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing $x$ . This implies that the function is analytic at every point of the interval (or disk). (Full article...)

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The graph of a real-valued quadratic function of a real variable x, is a parabola.
Image credit: Enoch Lau

A quadratic equation is a polynomial equation of degree two. The general form is

ax^{2}+bx+c=0,\,\!

where a ≠ 0 (if a = 0, then the equation becomes a linear equation). The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x², the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term.

Quadratic equations are known by that name because quadratus is Latin for "square"; in the leading term the variable is squared.

A quadratic equation has two (not necessarily distinct) solutions, which may be real or complex, given by the quadratic formula:

x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},

If the discriminant $b^{2}-4ac>0$ , then the quadratic equation has two distinct real solutions; if $b^{2}-4ac=0$ , the equation has two real solutions which are equal; if $b^{2}-4ac<0$ , the equation has two complex solutions.

These solutions are roots of the corresponding quadratic function

f(x)=ax^{2}+bx+c.\,

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