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

• 

  

  
  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...)
• 
  
  

  

  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 Eastern 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...)
• 

  

  
  Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and in particle physics, for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.
  
  Feynman developed a widely used pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World, he was ranked the seventh-greatest physicist of all time. (Full article...)
• 
  
  

  

  Portrait by August Köhler, c. 1910, after 1627 original

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

  

  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...)
• 
  The number π (/paɪ/ ^ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
  
  The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ${\tfrac {22}{7}}$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found. (Full article...)
• 
  
  

  

  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...)
• 

  

  Plots of logarithm functions, with three commonly used bases. The special points log_b b = 1 are indicated by dotted lines, and all curves intersect in log_b 1 = 0.

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

  

  Damage from Hurricane Katrina in 2005. Actuaries need to estimate long-term levels of such damage in order to accurately price property insurance, set appropriate reserves, and design appropriate reinsurance and capital management strategies.

  
  An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require asset management, liability management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms. The name of the corresponding academic discipline is actuarial science.
  
  While the concept of insurance dates to antiquity, the concepts needed to scientifically measure and mitigate risks have their origins in the 17th century studies of probability and annuities. Actuaries of the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems to design programs that manage risk, by determining if the implementation of strategies proposed for mitigating potential risks, does not exceed the expected cost of those risks actualized. The steps needed to become an actuary, including education and licensing, are specific to a given country, with various additional requirements applied by regional administrative units; however, almost all processes impart universal principles of risk assessment, statistical analysis, and risk mitigation, involving rigorously structured training and examination schedules, taking many years to complete. (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 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...)
• 

  

  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...)
• 
  
  

  

  Bust of Shen at the Beijing Ancient Observatory

  
  Shen Kuo (Chinese: 沈括; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁), was a Chinese polymath, scientist, and statesman of the Song dynasty (960–1279). Shen was a master in many fields of study including mathematics, optics, and horology. In his career as a civil servant, he became a finance minister, governmental state inspector, head official for the Bureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as an academic chancellor. At court his political allegiance was to the Reformist faction known as the New Policies Group, headed by Chancellor Wang Anshi (1021–1085).
  
  In his Dream Pool Essays or Dream Torrent Essays (夢溪筆談; Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needle compass, which would be used for navigation (first described in Europe by Alexander Neckam in 1187). Shen discovered the concept of true north in terms of magnetic declination towards the north pole, with experimentation of suspended magnetic needles and "the improved meridian determined by Shen's [astronomical] measurement of the distance between the pole star and true north". This was the decisive step in human history to make compasses more useful for navigation, and may have been a concept unknown in Europe for another four hundred years (evidence of German sundials made circa 1450 show markings similar to Chinese geomancers' compasses in regard to declination). (Full article...)
• 
  
  

  

  Elementary algebra studies which values solve equations formed using arithmetical operations.

  
  Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
  
  Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. (Full article...)
• 

  

  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...)

Good articles

• 

  

  The Laves graph

  
  In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance ${\sqrt {2}}$. It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices.
  H. S. M. Coxeter (1955) named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932. It has also been called the K₄ crystal, (10,3)-a network, diamond twin, triamond, and the srs net. The regions of space nearest each vertex of the graph are congruent 17-sided polyhedra that tile space. Its edges lie on diagonals of the regular skew polyhedron, a surface with six squares meeting at each integer point of space. (Full article...)
• 

  

  Graph of the equation y = 1/x. Here, e is the unique number larger than 1 that makes the shaded area under the curve equal to 1.

  
  The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted $\gamma$. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.
  
  The number e is of great importance in mathematics, alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity $e^{i\pi }+1=0$ and play important and recurring roles across mathematics. Like the constant π, e is irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients. To 30 decimal places, the value of e is: (Full article...)
• 
  Klaus Friedrich Roth FRS (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society.
  
  Roth moved to England as a child in 1933 to escape the Nazis, and was educated at the University of Cambridge and University College London, finishing his doctorate in 1950. He taught at University College London until 1966, when he took a chair at Imperial College London. He retired in 1988. (Full article...)
• 

  

  Rule 184, run for 128 steps from random configurations with each of three different starting densities: top 25%, middle 50%, bottom 75%. The view shown is a 300-pixel crop from a wider simulation.

  
  Rule 184 is a one-dimensional binary cellular automaton rule, notable for solving the majority problem as well as for its ability to simultaneously describe several, seemingly quite different, particle systems:
  

  • Rule 184 can be used as a simple model for traffic flow in a single lane of a highway, and forms the basis for many cellular automaton models of traffic flow with greater sophistication. In this model, particles (representing vehicles) move in a single direction, stopping and starting depending on the cars in front of them. The number of particles remains unchanged throughout the simulation. Because of this application, Rule 184 is sometimes called the "traffic rule".
  • Rule 184 also models a form of deposition of particles onto an irregular surface, in which each local minimum of the surface is filled with a particle in each step. At each step of the simulation, the number of particles increases. Once placed, a particle never moves.
  • Rule 184 can be understood in terms of ballistic annihilation, a system of particles moving both leftwards and rightwards through a one-dimensional medium. When two such particles collide, they annihilate each other, so that at each step the number of particles remains unchanged or decreases.

  
  The apparent contradiction between these descriptions is resolved by different ways of associating features of the automaton's state with particles. (Full article...)
• 

  

  16 polygonalizations of a set of six points

  
  In computational geometry, a polygonalization of a finite set of points in the Euclidean plane is a simple polygon with the given points as its vertices. A polygonalization may also be called a polygonization, simple polygonalization, Hamiltonian polygon, non-crossing Hamiltonian cycle, or crossing-free straight-edge spanning cycle.
  
  Every point set that does not lie on a single line has at least one polygonalization, which can be found in polynomial time. For points in convex position, there is only one, but for some other point sets there can be exponentially many. Finding an optimal polygonalization under several natural optimization criteria is a hard problem, including as a special case the travelling salesman problem. The complexity of counting all polygonalizations remains unknown. (Full article...)
• 
  In geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. The theorem is named after Frank S. Beckman and Donald A. Quarles Jr., who published this result in 1953; it was later rediscovered by other authors and re-proved in multiple ways. Analogous theorems for rational subsets of Euclidean spaces, or for non-Euclidean geometry, are also known. (Full article...)
• 
  
  

  

  Maxwell, c. 1870s

  
  James Clerk Maxwell FRS FRSE (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism achieved the second great unification in physics, where the first one had been realised by Isaac Newton. Maxwell was also key in the creation of statistical mechanics.
  
  With the publication of "A Dynamical Theory of the Electromagnetic Field" in 1865, Maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light. He proposed that light is an undulation in the same medium that is the cause of electric and magnetic phenomena. The unification of light and electrical phenomena led to his prediction of the existence of radio waves, and the paper contained his final version of his equations, which he had been working on since 1856. As a result of his equations, and other contributions such as introducing an effective method to deal with network problems and linear conductors, he is regarded as a founder of the modern field of electrical engineering. In 1871, Maxwell became the first Cavendish Professor of Physics, serving until his death in 1879. (Full article...)
• 

  

  Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + ⁠96/2⁠ − 1 = 48

  
  In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical Snapshots. It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. (Full article...)
• 
  Fleiss' kappa (named after Joseph L. Fleiss) is a statistical measure for assessing the reliability of agreement between a fixed number of raters when assigning categorical ratings to a number of items or classifying items. This contrasts with other kappas such as Cohen's kappa, which only work when assessing the agreement between not more than two raters or the intra-rater reliability (for one appraiser versus themself). The measure calculates the degree of agreement in classification over that which would be expected by chance.
  
  Fleiss' kappa can be used with binary or nominal-scale. It can also be applied to ordinal data (ranked data): the MiniTab online documentation gives an example. However, this document notes: "When you have ordinal ratings, such as defect severity ratings on a scale of 1–5, Kendall's coefficients, which account for ordering, are usually more appropriate statistics to determine association than kappa alone." Keep in mind however, that Kendall rank coefficients are only appropriate for rank data. (Full article...)
• 

  

  Mathematics in art: Albrecht Dürer's copper plate engraving Melencolia I, 1514. Mathematical references include a compass for geometry, a magic square and a truncated rhombohedron, while measurement is indicated by the scales and hourglass.

  
  Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.
  
  Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions conjectured to have been based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De divina proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas vaulting. (Full article...)
• 

  

  The maximum spacing method tries to find a distribution function such that the spacings, D_(i), are all approximately of the same length. This is done by maximizing their geometric mean.

  
  
  In statistics, maximum spacing estimation (MSE or MSP), or maximum product of spacing estimation (MPS), is a method for estimating the parameters of a univariate statistical model. The method requires maximization of the geometric mean of spacings in the data, which are the differences between the values of the cumulative distribution function at neighbouring data points.
  
  The concept underlying the method is based on the probability integral transform, in that a set of independent random samples derived from any random variable should on average be uniformly distributed with respect to the cumulative distribution function of the random variable. The MPS method chooses the parameter values that make the observed data as uniform as possible, according to a specific quantitative measure of uniformity. (Full article...)
• 
  Stanisław Marcin Ulam (Polish: [sta'ɲiswaf 'mart͡ɕin 'ulam]; 13 April 1909 – 13 May 1984) was a Polish mathematician, nuclear physicist and computer scientist. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved a number of theorems and proposed several conjectures.
  
  Born into a wealthy Polish Jewish family in Lemberg, Austria-Hungary; Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski and Włodzimierz Stożek. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, where he worked to establish important results regarding ergodic theory. On 20 August 1939, he sailed for the United States for the last time with his 17-year-old brother Adam Ulam. He became an assistant professor at the University of Wisconsin–Madison in 1940, and a United States citizen in 1941. (Full article...)

Did you know

• ... 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 multiple mathematics competitions have made use of Sophie Germain's identity?
• ... that ten-sided gaming dice have kite-shaped faces?
• ... that the word algebra is derived from an Arabic term for the surgical treatment of bonesetting?
• ... that Catechumen, a Christian first-person shooter, was funded only in the aftermath of the Columbine High School massacre?
• ... that subgroup distortion theory, introduced by Misha Gromov in 1993, can help encode text?
• ... that Green Day's "Wake Me Up When September Ends" became closely associated with the aftermath of Hurricane Katrina?
• ... 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?

Did you know...

• ... that, while the criss-cross algorithm visits all eight corners of the Klee–Minty cube when started at a worst corner, it visits only three more corners on average when started at a random corner?
• ...that in senary, all prime numbers other than 2 and 3 end in 1 or a 5?
• ...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?

Featured pictures

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