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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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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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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Hilary Putnam

The Quine–Putnam indispensability argument is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. It was named after the philosophers Willard Van Orman Quine and Hilary Putnam, and is one of the most important arguments in the philosophy of mathematics.

Although elements of the indispensability argument may have originated with thinkers such as Gottlob Frege and Kurt Gödel, Quine's development of the argument was unique for introducing to it a number of his philosophical positions such as naturalism, confirmational holism, and the criterion of ontological commitment. Putnam gave Quine's argument its first detailed formulation in his 1971 book Philosophy of Logic. He later came to disagree with various aspects of Quine's thinking, however, and formulated his own indispensability argument based on the no miracles argument in the philosophy of science. A standard form of the argument in contemporary philosophy is credited to Mark Colyvan; whilst being influenced by both Quine and Putnam, it differs in important ways from their formulations. It is presented in the Stanford Encyclopedia of Philosophy: (Full article...)
$Image 4 Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because the remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300). In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, that number is the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...)$

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Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because the remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300).

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).
It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules,
and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.

The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, $21$ is the GCD of $252$ and $105$ (as $252 = 21 \times 12$ and $105 = 21 \times 5)$ , and the same number $21$ is also the GCD of $105$ and $252 - 105 = 147$ . Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, that number is the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, $21 = 5 \times 105 + (-2) \times 252$ ). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (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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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 Constantin 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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Logic studies valid forms of inference like modus ponens.

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like $\land$ (and) or $\to$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. (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 following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation form an infinite group, which is generated by a single element called ⁠ $1$ ⁠ (these properties characterize the integers in a unique way). (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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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.

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...)
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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 classical gravity, 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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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 $3$ rd power: $1000 = 10 3 = 10 \times 10 \times 10$ . More generally, if $x = b y$ , then $y$ is the logarithm of $x$ to base $b$ , written $log b x$ , so $log 10 1000 = 3$ . 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 $e \approx 2.718$ 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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Portrait by Jakob Emanuel Handmann, 1753

Leonhard Euler (/ˈɔɪlər/ OY-lər; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleɔnhard ˈɔʏlər]; 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 π {\displaystyle \pi } (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle \Sigma } (capital sigma) to express summations, the Greek letter Δ {\displaystyle \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 {\displaystyle 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 title page of a 1634 version of Hues' Tractatus de globis in the collection of the Biblioteca Nacional de Portugal

Robert Hues (1553 – 24 May 1632) was an English mathematician and geographer. He attended St. Mary Hall at Oxford, and graduated in 1578. Hues became interested in geography and mathematics, and studied navigation at a school set up by Walter Raleigh. During a trip to Newfoundland, he made observations which caused him to doubt the accepted published values for variations of the compass. Between 1586 and 1588, Hues travelled with Thomas Cavendish on a circumnavigation of the globe, performing astronomical observations and taking the latitudes of places they visited. Beginning in August 1591, Hues and Cavendish again set out on another circumnavigation of the globe. During the voyage, Hues made astronomical observations in the South Atlantic, and continued his observations of the variation of the compass at various latitudes and at the Equator. Cavendish died on the journey in 1592, and Hues returned to England the following year.

In 1594, Hues published his discoveries in the Latin work Tractatus de globis et eorum usu (Treatise on Globes and Their Use) which was written to explain the use of the terrestrial and celestial globes that had been made and published by Emery Molyneux in late 1592 or early 1593, and to encourage English sailors to use practical astronomical navigation. Hues' work subsequently went into at least 12 other printings in Dutch, English, French and Latin. (Full article...)

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Lorenz attractor

Credit: Wikimol

The Lorenz attractor is an iconic example of a strange attractor in chaos theory. This three-dimensional fractal structure, resembling a butterfly or figure eight, reflects the long-term behavior of solutions to the Lorenz system, a set of three differential equations used by mathematician and meteorologist Edward N. Lorenz as a simple description of fluid circulation in a shallow layer (of liquid or gas) uniformly heated from below and cooled from above. To be more specific, the figure is set in a three-dimensional coordinate system whose axes measure the rate of convection in the layer (x), the horizontal temperature variation (y), and the vertical temperature variation (z). As these quantities change over time, a path is traced out within the coordinate system reflecting a particular solution to the differential equations. Lorenz's analysis revealed that while all solutions are completely deterministic, some choices of input parameters and initial conditions result in solutions showing complex, non-repeating patterns that are highly dependent on the exact values chosen. As stated by Lorenz in his 1963 paper Deterministic Nonperiodic Flow: "Two states differing by imperceptible amounts may eventually evolve into two considerably different states". He later coined the term "butterfly effect" to describe the phenomenon. One implication is that computing such chaotic solutions to the Lorenz system (i.e., with a computer program) to arbitrary precision is not possible, as any real-world computer will have a limitation on the precision with which it can represent numerical values. The particular solution plotted in this animation is based on the parameter values used by Lorenz (

σ = 10

,

ρ = 28

, and

β = 8/3

, constants reflecting certain physical attributes of the fluid). Note that the animation repeatedly shows one solution plotted over a specific period of time; as previously mentioned, the true solution never exactly retraces itself. Not all solutions are chaotic, however. Some choices of parameter values result in solutions that tend toward equilibrium at a fixed point (as seen, for example, in this image). Initially developed to describe atmospheric convection, the Lorenz equations also arise in simplified models for lasers, electrical generators and motors, and chemical reactions.

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Image 1
Stars is a wood engraving print created by the Dutch artist M. C. Escher in 1948, depicting two chameleons in a polyhedral cage floating through space.

The compound of three octahedra used for the central cage in Stars had been studied before in mathematics, and Escher likely learned of it from the book Vielecke und Vielflache by Max Brückner. Escher used similar compound polyhedral forms in several other works, including Crystal (1947), Study for Stars (1948), Double Planetoid (1949), and Waterfall (1961). (Full article...)
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Srinivasa Ramanujan Aiyangar (22 December 1887 – 26 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.

Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything in the least like them before", and some recently proven but highly advanced results. (Full article...)
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A well-covered graph, the intersection graph of the nine diagonals of a hexagon. The three red vertices form one of its 14 equal-sized maximal independent sets, and the six blue vertices form the complementary minimal vertex cover.

In graph theory, a well-covered graph is an undirected graph in which the minimal vertex covers all have the same size. Here, a vertex cover is a set of vertices that touches all edges, and it is minimal if removing any vertex from it would leave some edge uncovered. Equivalently, well-covered graphs are the graphs in which all maximal independent sets have equal size. Well-covered graphs were defined and first studied by Michael D. Plummer in 1970.

The well-covered graphs include all complete graphs, balanced complete bipartite graphs, and the rook's graphs whose vertices represent squares of a chessboard and edges represent moves of a chess rook. Known characterizations of the well-covered cubic graphs, well-covered claw-free graphs, and well-covered graphs of high girth allow these graphs to be recognized in polynomial time, but testing whether other kinds of graph are well-covered is a coNP-complete problem. (Full article...)
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The 13 possible strict weak orderings on a set of three elements ${a, b, c$ }

In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the weak orderings on a set of $n$ elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race.

The ordered Bell numbers were studied in the 19th century by Arthur Cayley and William Allen Whitworth. They are named after Eric Temple Bell, who wrote about the Bell numbers, which count the partitions of a set; the ordered Bell numbers count partitions that have been equipped with a total order. Their alternative name, the Fubini numbers, comes from a connection to Guido Fubini and Fubini's theorem on equivalent forms of multiple integrals. Because weak orderings have many names, ordered Bell numbers may also be called by those names, for instance as the numbers of preferential arrangements or the numbers of asymmetric generalized weak orders. (Full article...)
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The convex hull of the red set is the blue and red convex set.

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points.
The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. They can be solved in time $O(n\log n)$ for two or three dimensional point sets, and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions. (Full article...)
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A kite, showing its pairs of equal-length sides and its inscribed circle.

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential. They include as special cases the right kites, with two opposite right angles; the rhombi, with two diagonal axes of symmetry; and the squares, which are also special cases of both right kites and rhombi. (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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In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.

In technical terms, QED can be described as a very accurate way to calculate the probability of the position and movement of particles, even those massless such as photons, and the quantity depending on position (field) of those particles, and described light and matter beyond the wave-particle duality proposed by Albert Einstein in 1905. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen. It is the most precise and stringently tested theory in physics. (Full article...)
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In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

The Mayer–Vietoris sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds for those theories satisfying the Eilenberg–Steenrod axioms, and it has variations for both reduced and relative (co)homology. Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in topology are constructed by piecing together very simple patches. Carefully choosing the two covering subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the Mayer–Vietoris sequence is analogous to the Seifert–van Kampen theorem for the fundamental group, and a precise relation exists for homology of dimension one. (Full article...)
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Berlin, 1959

Andrew Mattei Gleason (1921–2008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in mathematics teaching at all levels. Gleason's theorem in quantum logic and the Greenwood–Gleason graph, an important example in Ramsey theory, are named for him.

As a young World War II naval officer, Gleason broke German and Japanese military codes. After the war he spent his entire academic career at Harvard University, from which he retired in 1992. His numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and the Harvard Society of Fellows, and presidency of the American Mathematical Society. He continued to advise the United States government on cryptographic security, and the Commonwealth of Massachusetts on mathematics education for children, almost until the end of his life. (Full article...)
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Turing in 1951

Alan Mathison Turing (/ˈtjʊərɪŋ/; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science.

Born in London, Turing was raised in southern England. He graduated from King's College, Cambridge, and in 1938, earned a doctorate degree from Princeton University. During World War II, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bomba method, an electromechanical machine that could find settings for the Enigma machine. He played a crucial role in cracking intercepted messages that enabled the Allies to defeat the Axis powers in many engagements, including the Battle of the Atlantic. (Full article...)
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Numbers written with Cistercian numerals. From left to right: 1 in units place, 2 in tens place (20), 3 in hundreds place (300), 4 in thousands place (4,000), then compound numbers 5,555, 6,789, 9,394.

The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single glyph able to indicate any integer from 1 to 9,999.

Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its place value (units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century. (Full article...)

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... that in 1967 two mathematicians published PhD dissertations independently disproving the same thirteen-year-old conjecture?
... that Fairleigh Dickinson's upset victory over Purdue was the biggest upset in terms of point spread in NCAA tournament history, with Purdue being a 23+1⁄2-point favorite?
... that the British National Hospital Service Reserve trained volunteers to carry out first aid in the aftermath of a nuclear or chemical attack?
... that Catechumen, a Christian first-person shooter, was funded only in the aftermath of the Columbine High School massacre?
... that The Math Myth advocates for American high schools to stop requiring advanced algebra?
... that in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?
... that more than 60 scientific papers authored by mathematician Paul Erdős were published posthumously?
... that after Archimedes first defined convex curves, mathematicians lost interest in their analysis until the 19th century, more than two millennia later?

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...that the set of rational numbers is equal in size to the set of integers; that is, they can be put in one-to-one correspondence?
...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
...that it is unknown whether π and e are algebraically independent?
...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area, such as Gabriel's Horn?
... that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
...that the primality of a number can be determined using only a single division using Wilson's Theorem?

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3D illustration of a stereographic projection from the north pole onto a plane below the sphere.
Image credit: Mark Howison

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. (Full article...)

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