Portal:Mathematics

• 
  Abacus, a ancient hand-operated calculating.
• 
  Portrait of Emmy Noether, around 1900.
• 
  Euler's identity

Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). (Full article...)

Featured articles

• 
  
  

  

  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...)
• 
  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...)
• 
  
  

  

  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...)
• 
  
  

  

  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 $\pi$ (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...)
• 
  
  

  

  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, he is considered one of the leading scientists in classical antiquity. Regarded as the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include 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 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...)
• 

  

  
  Josiah Willard Gibbs (/ɡɪbz/; February 11, 1839 – April 28, 1903) was an American scientist who made significant 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...)
• 
  
  

  

  Rejewski, c. 1932

  
  Marian Adam Rejewski (Polish: [ˈmarjan rɛˈjɛfskʲi] ^ⓘ; 16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in late 1932 reconstructed the sight-unseen German military Enigma cipher machine, aided by limited documents obtained by French military intelligence.
  
  Over the next nearly seven years, Rejewski and fellow mathematician-cryptologists Jerzy Różycki and Henryk Zygalski, working at the Polish General Staff's Cipher Bureau, developed techniques and equipment for decrypting the Enigma ciphers, even as the Germans introduced modifications to their Enigma machines and encryption procedures. Rejewski's contributions included the cryptologic card catalog and the cryptologic bomb. (Full article...)
• 

  

  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...)
• 
  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...)
• 
  
  

  

  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...)
• 

  

  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...)
• 
  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 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...)
• 

  

  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...)
• 

  

  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...)
• 

  

  The manipulations of the Rubik's Cube form the Rubik's Cube group.

  
  In mathematics, a group is a set with an operation that associates every pair of elements of the set to an element of the set (as does every binary operation) and 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...)

Good articles

• 

  

  A homomorphism from the flower snark J₅ into the cycle graph C₅.
  It is also a retraction onto the subgraph on the central five vertices. Thus J₅ is in fact homo­mor­phi­cally equivalent to the core C₅.

  
  In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices.
  
  Homomorphisms generalize various notions of graph colorings and allow the expression of an important class of constraint satisfaction problems, such as certain scheduling or frequency assignment problems.
  The fact that homomorphisms can be composed leads to rich algebraic structures: a preorder on graphs, a distributive lattice, and a category (one for undirected graphs and one for directed graphs).
  The computational complexity of finding a homomorphism between given graphs is prohibitive in general, but a lot is known about special cases that are solvable in polynomial time. Boundaries between tractable and intractable cases have been an active area of research. (Full article...)
• 

  

  Four opaque sets for a unit square. Upper left: its boundary, length 4. Upper right: Three sides, length 3. Lower left: a Steiner tree of the vertices, length $1+{\sqrt {3}}\approx 2.732$. Lower right: the conjectured optimal solution, length ${\sqrt {2}}+{\tfrac {1}{2}}{\sqrt {6}}\approx 2.639$.

  
  In discrete geometry, an opaque set is a system of curves or other set in the plane that blocks all lines of sight across a polygon, circle, or other shape. Opaque sets have also been called barriers, beam detectors, opaque covers, or (in cases where they have the form of a forest of line segments or other curves) opaque forests. Opaque sets were introduced by Stefan Mazurkiewicz in 1916, and the problem of minimizing their total length was posed by Frederick Bagemihl in 1959.
  
  For instance, visibility through a unit square can be blocked by its four boundary edges, with length 4, but a shorter opaque forest blocks visibility across the square with length ${\sqrt {2}}+{\tfrac {1}{2}}{\sqrt {6}}\approx 2.639$. It is unproven whether this is the shortest possible opaque set for the square, and for most other shapes this problem similarly remains unsolved. The shortest opaque set for any bounded convex set in the plane has length at most the perimeter of the set, and at least half the perimeter. For the square, a slightly stronger lower bound than half the perimeter is known. Another convex set whose opaque sets are commonly studied is the unit circle, for which the shortest connected opaque set has length $2+\pi$. Without the assumption of connectivity, the shortest opaque set for the circle has length at least $\pi$ and at most $4.7998$. (Full article...)
• 
  In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) systems to multiple-input and multiple-output (MIMO) systems. The matrix relates the outputs of the system to its inputs. It is a particularly useful construction for linear time-invariant (LTI) systems because it can be expressed in terms of the s-plane.
  
  In some systems, especially ones consisting entirely of passive components, it can be ambiguous which variables are inputs and which are outputs. In electrical engineering, a common scheme is to gather all the voltage variables on one side and all the current variables on the other regardless of which are inputs or outputs. This results in all the elements of the transfer matrix being in units of impedance. The concept of impedance (and hence impedance matrices) has been borrowed into other energy domains by analogy, especially mechanics and acoustics. (Full article...)
• 
  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...)
• 

  

  A Möbius strip made with paper and adhesive tape

  
  In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains a Möbius strip.
  
  As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a knotted centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. It has only a single boundary curve. (Full article...)
• 
  The infinity symbol (∞) is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate, after the lemniscate curves of a similar shape studied in algebraic geometry, or "lazy eight", in the terminology of livestock branding.
  
  This symbol was first used mathematically by John Wallis in the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes (potential infinity) rather than infinite values (actual infinity). It has other related technical meanings, such as the use of long-lasting paper in bookbinding, and has been used for its symbolic value of the infinite in modern mysticism and literature. It is a common element of graphic design, for instance in corporate logos as well as in older designs such as the Métis flag. (Full article...)
• 
  The small set expansion hypothesis or small set expansion conjecture in computational complexity theory is an unproven computational hardness assumption. Under the small set expansion hypothesis it is assumed to be computationally infeasible to distinguish between a certain class of expander graphs called "small set expanders" and other graphs that are very far from being small set expanders. This assumption implies the hardness of several other computational problems, and the optimality of certain known approximation algorithms.
  
  The small set expansion hypothesis is related to the unique games conjecture, another unproven computational hardness assumption according to which accurately approximating the value of certain games is computationally infeasible. If the small set expansion hypothesis is true, then so is the unique games conjecture. (Full article...)
• 

  

  
  The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron, composite polyhedron, and Johnson solid.
  
  The edges and vertices of the triaugmented triangular prism form a maximal planar graph with 9 vertices and 21 edges, called the Fritsch graph. It was used by Rudolf and Gerda Fritsch to show that Alfred Kempe's attempted proof of the four color theorem was incorrect. The Fritsch graph is one of only six graphs in which every neighborhood is a 4- or 5-vertex cycle. (Full article...)
• 

  

  The cover page of Ars Conjectandi

  
  
  Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the twelvefold way and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, Abraham de Moivre.
  
  Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. He incorporated fundamental combinatorial topics such as his theory of permutations and combinations (the aforementioned problems from the twelvefold way) as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance. Core topics from probability, such as expected value, were also a significant portion of this important work. (Full article...)
• 
  In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Unless θ is a rational multiple of π, there will also be at least two distinct distances.
  
  This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi, and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square. (Full article...)
• 
  In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
  $$\sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots .$$
  
  The first $n$ terms of the series sum to approximately $\ln n+\gamma$, where $\ln$ is the natural logarithm and $\gamma \approx 0.577$ is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. (Full article...)
• 
  Sunday Osarumwense Iyahen (3 October 1937 – 28 January 2018) was a Nigerian mathematician and politician, recognised for his contributions to the field of topological vector spaces and his service as a senator representing Bendel Central Senatorial District. Born in Benin City, Edo State, Nigeria, Iyahen was the eldest of at least seventeen children and embarked on an academic journey that led him to earn a first-class honours degree in mathematics from the University of Ibadan and later a Ph.D. and D.Sc. from the University of Keele.
  
  Iyahen's academic career was marked by his tenure as a professor of mathematics at several universities in Nigeria and abroad. He served as the Head of the Department of Mathematics and Dean of the Faculty of Science at the University of Ibadan before joining the Institute of Technology, Benin (now known as the University of Benin), where he became the founding dean of the Faculty of Physical Sciences. His scholarly work includes over 100 published papers and contributions as editor-in-chief for mathematical journals. He was honoured with fellowships from the Nigerian Academy of Science and the Mathematical Association of Nigeria. As a politician, he was elected as a senator, where he contributed to national policy and development. (Full article...)

Did you know

• ... that in 1967 two mathematicians published PhD dissertations independently disproving the same thirteen-year-old conjecture?
• ... that mathematician Daniel Larsen was the youngest contributor to the New York Times crossword puzzle?
• ... that people in Madagascar perform algebra on tree seeds in order to tell the future?
• ... 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 ten-sided gaming dice have kite-shaped faces?
• ... that in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?
• ... that The Math Myth advocates for American high schools to stop requiring advanced algebra?
• ... that Catechumen, a Christian first-person shooter, was funded only in the aftermath of the Columbine High School massacre?

Did you know...

• ...that, for all prime numbers p, the pth Perrin number is divisible by p?
• ...that it is impossible to trisect a general angle using only a ruler and a compass?
• ...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?
• ...that the 1966 publication disproving Euler's sum of powers conjecture, proposed nearly 200 years earlier, consisted of only two sentences?
• ...the hyperbolic trigonometric functions of the natural logarithm can be represented by rational algebraic fractions?
• ... that economists blame market failures on non-convexity?
• ... that, according to the pizza theorem, a circular pizza that is sliced off-center into eight equal-angled wedges can still be divided equally between two people?

Featured pictures

• 
  Mandelbulb, by Ondřej Karlík (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Breeder, by Protious/Hyperdeath (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 8, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 6, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 4, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Missing square puzzle, by Fibonacci (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Lorenz attractor at Chaos theory, by Wikimol (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Line integral of scalar field, by Lucas V. Barbosa (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 10, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Radian, by Lucas V. Barbosa (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 7, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 13, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Cellular automata at Reflector (cellular automaton), by Simpsons contributor (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Hypotrochoid, by Sam Derbyshire (edited by Anevrisme and Perhelion) (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Topological knots, by Jkasd (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 2, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Lyapunov fractal, by BernardH (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Proof of the Pythagorean theorem, by Joaquim Alves Gaspar (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Fields Medal, back, by Stefan Zachow (edited by King of Hearts) (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Anscombe's quartet, by Schutz (edited by Avenue) (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Helicatenoid, by Wickerprints (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Desargues' theorem, by Dynablast (edited by Jujutacular and Julia W) (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 9, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 11, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, by Simpsons contributor (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 5, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 1, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 3, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Tetrahedral group at Symmetry group, by Debivort (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Non-uniform rational B-spline, by Greg L (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 12, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Fields Medal, front, by Stefan Zachow (edited by King of Hearts) (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, step 14, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
• 
  Mandelbrot set, start, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)

Get involved

For editor resources and to collaborate with other editors on improving Wikipedia's Mathematics-related articles, visit WikiProject Mathematics.

Categories

» Category:Mathematics, Category:People

Topics

» Lists of mathematics topics, Glossary of areas of mathematics, History of mathematics

Index of articles

ARTICLE INDEX:	0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
MATHEMATICIANS:	A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Vital articles

» subpages: Level 4 Mathematics articles, Level 5 Mathematics articles

Discover Wikipedia using portals

• 

  List of all portals

• 

  The arts portal

• 

  Biography portal

• 

  Current events portal

• 

  Geography portal

• 

  History portal

• 

  Mathematics portal

• 

  Science portal

• 

  Society portal

• 

  Technology portal

• 

  Random portal

• 

  WikiProject Portals