Portal:Mathematics

The Mathematics Portal

Euler's Identity

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

Featured articles –

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

---

• 
  
  

  

  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...)
• 
  
  

  

  A stamp of Zhang Heng issued by China Post in 1955

  
  Zhang Heng (Chinese: 張衡; AD 78–139), formerly romanized Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, mathematician, seismologist, hydraulic engineer, inventor, geographer, cartographer, ethnographer, artist, poet, philosopher, politician, and literary scholar.
  
  Zhang Heng began his career as a minor civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages, and then Palace Attendant at the imperial court. His uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator of Hejian Kingdom in present-day Hebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139. (Full article...)
• 

  

  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...)
• 

  

  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...)
• 

  

  
  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...)
• 

  

  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...)
• 
  
  

  

  Archimedes Thoughtful
  by Domenico 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 regarded as 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, 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. 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...)
• 
  
  

  

  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 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...)
• 
  
  

  

  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.
  
  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 Industrial Society and Its Future, a 35,000-word manifesto and social critique which opposes all forms of technology, rejecting leftism, and advocating a nature-centered form of anarchism. (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 an element of the set to every pair of elements 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...)
• 
  
  

  

  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 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...)
• 
  The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ${\tfrac {22}{7}}$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.
  
  For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. (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 mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.
  
  Euler is regarded as one of the greatest, most prolific mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. (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 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...)

Selected image –

animation showing a torus (a doughnut shape) being cut diagonally by a plane, causing the appearance of two interlocking circles on the cut surface

Villarceau circles

Credit: User:LucasVB

An animation showing how an obliquely cut torus reveals a pair of intersecting circles known as Villarceau circles, named after the French astronomer and mathematician Yvon Villarceau. These are two of the four circles that can be drawn through any given point on the torus. (The other two are oriented horizontally and vertically, and are the analogs of lines of latitude and longitude drawn through the given point.) The circles have no known practical application and seem to be merely a curious characteristic of the torus. However, Villarceau circles appear as the fibers in the Hopf fibration of the 3-sphere over the ordinary 2-sphere, and the Hopf fibration itself has interesting connections to fluid dynamics, particle physics, and quantum theory.

Good articles –

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

---

• 

  

  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...)
• 

  

  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...)
• 
  
  

  

  8x8 Rook's graph

  
  In graph theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and there is an edge between any two squares sharing a row (rank) or column (file), the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape. Although rook's graphs have only minor significance in chess lore, they are more important in the abstract mathematics of graphs through their alternative constructions: rook's graphs are the Cartesian product of two complete graphs, and are the line graphs of complete bipartite graphs. The square rook's graphs constitute the two-dimensional Hamming graphs.
  
  Rook's graphs are highly symmetric, having symmetries taking every vertex to every other vertex. In rook's graphs defined from square chessboards, more strongly, every two edges are symmetric, and every pair of vertices is symmetric to every other pair at the same distance in moves (making the graph distance-transitive). For rectangular chessboards whose width and height are relatively prime, the rook's graphs are circulant graphs. With one exception, the rook's graphs can be distinguished from all other graphs using only two properties: the numbers of triangles each edge belongs to, and the existence of a unique 4-cycle connecting each nonadjacent pair of vertices. (Full article...)
• 
  
  

  

  

  Two symmetric orthographic projections

  
  In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, simple, and star polyhedron. (Full article...)
• 
  Donkey Kong Jr. Math is an edutainment platform video game developed and published by Nintendo for the Nintendo Entertainment System. It is a spin-off of the 1982 arcade game Donkey Kong Jr. In the game, players control Donkey Kong Jr. as he solves math problems set up by his father Donkey Kong. It was released in Japan in 1983 for the Family Computer, and in North America and the PAL region in 1986.
  
  It is the only game in the Education Series of NES games in North America, owing to the game's lack of success. It was made available in various forms, including in the 2002 GameCube video game Animal Crossing and on the Virtual Console services for Wii and Wii U in 2007 and 2014 respectively, and in 2024 for the Nintendo Switch Online service. Donkey Kong Jr. Math was a critical and commercial failure. It has received criticism from several publications including IGN staff, who called it one of the worst Virtual Console games. (Full article...)
• 

  

  A unit cube with a hole cut through it, large enough to allow Prince Rupert's cube to pass

  
  In geometry, Prince Rupert's cube is the largest cube that can pass through a hole cut through a unit cube without splitting it into separate pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.
  
  Prince Rupert's cube is named after Prince Rupert of the Rhine, who asked whether a cube could be passed through a hole made in another cube of the same size without splitting the cube into two pieces. A positive answer was given by John Wallis. Approximately 100 years later, Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube. (Full article...)
• 
  The Alexandrov uniqueness theorem 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...)
• 

  

  Six points in the unit square, with the smallest triangles (red) having area 1/8, the optimal area for this number of points. Other larger triangles are colored blue. These points are an affine transformation of a regular hexagon, but for larger numbers of points the optimal solution does not form a convex polygon.

  
  In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed in a given area, the smallest triangle area will be at most inversely proportional to the square of the number of points. His conjecture was proven false, but the asymptotic growth rate of the minimum triangle area remains unknown. (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...)
• 
  1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral.
  
  In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. (Full article...)
• 
  
  

  

  Wigner in 1963

  
  Eugene Paul Wigner (Hungarian: Wigner Jenő Pál, pronounced [ˈviɡnɛr ˈjɛnøː ˈpaːl]; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".
  
  A graduate of the Technical Hochschule Berlin (now Technische Universität Berlin), Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen. Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus. In 1930, Princeton University recruited Wigner, along with John von Neumann, and he moved to the United States, where he obtained citizenship in 1937. (Full article...)
• 

  

  The Rado graph, as numbered by Ackermann (1937) and Rado (1964).

  
  In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann (1937). The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
  
  Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a time. The Rado graph is uniquely defined, among countable graphs, by an extension property that guarantees the correctness of this algorithm: no matter which vertices have already been chosen to form part of the induced subgraph, and no matter what pattern of adjacencies is needed to extend the subgraph by one more vertex, there will always exist another vertex with that pattern of adjacencies that the greedy algorithm can choose. (Full article...)

Did you know (auto-generated) –

• ... that mathematician Mathias Metternich was one of the founders of the Jacobin club of the Republic of Mainz?
• ... that owner Matthew Benham influenced both Brentford FC in the UK and FC Midtjylland in Denmark to use mathematical modelling to recruit undervalued football players?
• ... that although the problem of squaring the circle with compass and straightedge goes back to Greek mathematics, it was not proven impossible until 1882?
• ... that ten-sided gaming dice have kite-shaped faces?
• ... 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 circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?
• ... that the British National Hospital Service Reserve trained volunteers to carry out first aid in the aftermath of a nuclear or chemical attack?
• ... that despite published scholarship to the contrary, Andrew Planta neither received a doctorate nor taught mathematics at Erlangen?

More did you know –

Did you know...

• ...that Auction theory was successfully used in 1994 to sell FCC airwave spectrum, in a financial application of game theory?
• ...properties of Pascal's triangle have application in many fields of mathematics including combinatorics, algebra, calculus and geometry?
• ...work in artificial intelligence makes use of swarm intelligence, which has foundations in the behavioral examples found in nature of ants, birds, bees, and fish among others?
• ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
• ...that modular arithmetic has application in at least ten different fields of study, including the arts, computer science, and chemistry in addition to mathematics?
• ... that according to Kawasaki's theorem, an origami crease pattern with one vertex may be folded flat if and only if the sum of every other angle between consecutive creases is 180º?
• ... that, in the Rule 90 cellular automaton, any finite pattern eventually fills the whole array of cells with copies of itself?

Selected article –

The continuum hypothesis is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following:

There is no set whose size is strictly between that of the integers and that of the real numbers.

Or mathematically speaking, noting that the cardinality for the integers $|\mathbb {Z} |$ is $\aleph _{0}$ ("aleph-null") and the cardinality of the real numbers $|\mathbb {R} |$ is $2^{\aleph _{0}}$, the continuum hypothesis says

$\nexists \mathbb {A} :\aleph _{0}<|\mathbb {A} |<2^{\aleph _{0}}.$

This is equivalent to:

$2^{\aleph _{0}}=\aleph _{1}$

The real numbers have also been called the continuum, hence the name. (Full article...)

View all selected articles

Subcategories

Algebra | Arithmetic | Analysis | Complex analysis | Applied mathematics | Calculus | Category theory | Chaos theory | Combinatorics | Dynamical systems | Fractals | Game theory | Geometry | Algebraic geometry | Graph theory | Group theory | Linear algebra | Mathematical logic | Model theory | Multi-dimensional geometry | Number theory | Numerical analysis | Optimization | Order theory | Probability and statistics | Set theory | Statistics | Topology | Algebraic topology | Trigonometry | Linear programming

---

Mathematics | History of mathematics | Mathematicians | Awards | Education | Literature | Notation | Organizations | Theorems | Proofs | Unsolved problems

Full category tree. Select [►] to view subcategories.

Mathematics

Mathematics by country

Fields of mathematics

Mathematics-related lists

Mathematicians

Women in mathematics

Works about mathematics

Mathematics and art

Mathematical classification systems

Mathematical concepts

Mathematical constants

Mathematics and culture

Eponyms in mathematics

Mathematical examples

FL-Class Polyhedra articles

History of mathematics

Mathematical notation

Mathematical science occupations

Outlines of mathematics and logic

Philosophy of mathematics

Mathematical projects

Mathematical proofs

Pseudomathematics

Set index articles on mathematics

Mathematical terminology

Mathematical theorems

Mathematical tools

Mathematics stubs

Topics in mathematics

General	Foundations	Number theory	Discrete mathematics
Mathematicians History of mathematics Philosophy of mathematics Mathematical notation Mathematical beauty Mathematics education Glossary of mathematical symbols	Foundations of mathematics Mathematical logic Proof theory Gödel's incompleteness theorems Model theory Recursion theory Set theory Naive set theory Axiomatic set theory Category theory Topos theory	Number theory Algebraic number theory Analytic number theory Arithmetic Fundamental theorem of arithmetic Numbers Natural numbers Prime numbers Rational numbers Algebraic numbers	Discrete mathematics Combinatorics Combinatorial geometry Coding theory Combinatorial design Enumerative combinatorics Combinatorial optimization Graph theory Order theory Lattice theory Digital Signal Processing Theoretical computer science Computability theory Complexity theory
Algebra	Analysis	Geometry and topology	Applied mathematics
Algebra Elementary algebra Abstract algebra Group theory Ring theory Field theory Commutative algebra Geometric algebra Linear algebra Matrix theory Multilinear algebra Universal algebra Fundamental theorem of algebra	Analysis Calculus Fundamental theorem of calculus Vector calculus Geometric calculus Measure theory Real analysis Complex analysis Differential equations Ordinary differential equations Partial differential equations Integral equations Approximation theory Special functions Potential theory Harmonic analysis Fourier analysis Functional analysis Operator theory	Geometry Euclidean geometry Trigonometry Analytic geometry Non-Euclidean geometry Affine geometry Projective geometry Convex geometry Discrete geometry Algebraic geometry Differential geometry K theory Gauge theory Riemannian geometry Lie groups Topology General topology Algebraic topology Geometric topology Differential topology	Applied mathematics Mathematical modeling Mathematical physics Dynamical systems Control theory Calculus of variations Optimization Mathematical economics Game theory Mathematical finance Statistics Probability theory Stochastic processes Numerical analysis Cryptography Information theory

Index of mathematics 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

Related portals

WikiProjects

The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.

Project pages

• Participants
• Current activity
• Manual of style
• Conventions

Essays

• Advice on using Wikipedia for mathematics self-study
• Proofs

Subprojects

• PlanetMath Exchange (Talk)
• Graphics (Talk)
• Article assessment (Talk)

Related projects

Computer science | Cryptography | Game theory | Numbers | Physics | Science | Statistics

Things you can do

• Create a new article: List of requested articles
• Add a new image: Image requests
• Expand an article: Mathematics stubs
• Lend your expertise: Expert attention needed

In other Wikimedia projects

The following Wikimedia Foundation sister projects provide more on this subject:

• 

  Commons
  Free media repository

• 

  Wikibooks
  Free textbooks and manuals

• 

  Wikidata
  Free knowledge base

• 

  Wikinews
  Free-content news

• 

  Wikiquote
  Collection of quotations

• 

  Wikisource
  Free-content library

• 

  Wikiversity
  Free learning tools

• 

  Wiktionary
  Dictionary and thesaurus

More portals

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