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.

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  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...)
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  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...)
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  Hilary Putnam

  
  The Quine–Putnam indispensability argument is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. It was named after the philosophers Willard 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...)
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  Logic studies valid forms of inference like modus ponens.

  
  
  Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.
  
  Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work". Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like $\land$ (and) or $\to$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. (Full article...)
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  The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... The graph is situated with positive integers to the right and negative integers to the left.

  
  
  In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
  $$\sum _{n=1}^{m}n(-1)^{n-1}.$$
  
  The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:
  $$1-2+3-4+\cdots ={\frac {1}{4}}.$$ (Full article...)
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  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 to base b is the inverse function of exponentiation with base b. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 10³, the logarithm base $10$ of 1000 is 3, or log₁₀ (1000) = 3. The logarithm of x to base b is denoted as log_b (x), or without parentheses, log_b x. When the base is clear from the context or is irrelevant it is sometimes written log x.
  
  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 frequently used in computer science. (Full article...)
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  In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.
  
  Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously. (Full article...)
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  Bust of Shen at the Beijing Ancient Observatory

  
  Shen Kuo (Chinese: 沈括; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁), was a Chinese polymath, scientist, and statesman of the Song dynasty (960–1279). Shen was a master in many fields of study including mathematics, optics, and horology. In his career as a civil servant, he became a finance minister, governmental state inspector, head official for the Bureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as an academic chancellor. At court his political allegiance was to the Reformist faction known as the New Policies Group, headed by Chancellor Wang Anshi (1021–1085).
  
  In his Dream Pool Essays or Dream Torrent Essays (夢溪筆談; Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needle compass, which would be used for navigation (first described in Europe by Alexander Neckam in 1187). Shen discovered the concept of true north in terms of magnetic declination towards the north pole, with experimentation of suspended magnetic needles and "the improved meridian determined by Shen's [astronomical] measurement of the distance between the pole star and true north". This was the decisive step in human history to make compasses more useful for navigation, and may have been a concept unknown in Europe for another four hundred years (evidence of German sundials made circa 1450 show markings similar to Chinese geomancers' compasses in regard to declination). (Full article...)
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  One of Molyneux's celestial globes, which is displayed in Middle Temple Library – from the frontispiece of the Hakluyt Society's 1889 reprint of A Learned Treatise of Globes, both Cœlestiall and Terrestriall, one of the English editions of Robert Hues' Latin work Tractatus de Globis (1594)

  
  Emery Molyneux (/ˈɛməri ˈmɒlɪnoʊ/ EM-ər-ee MOL-in-oh; died June 1598) was an English Elizabethan maker of globes, mathematical instruments and ordnance. His terrestrial and celestial globes, first published in 1592, were the first to be made in England and the first to be made by an Englishman.
  
  Molyneux was known as a mathematician and maker of mathematical instruments such as compasses and hourglasses. He became acquainted with many prominent men of the day, including the writer Richard Hakluyt and the mathematicians Robert Hues and Edward Wright. He also knew the explorers Thomas Cavendish, Francis Drake, Walter Raleigh and John Davis. Davis probably introduced Molyneux to his own patron, the London merchant William Sanderson, who largely financed the construction of the globes. When completed, the globes were presented to Elizabeth I. Larger globes were acquired by royalty, noblemen and academic institutions, while smaller ones were purchased as practical navigation aids for sailors and students. The globes were the first to be made in such a way that they were unaffected by the humidity at sea, and they came into general use on ships. (Full article...)
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  Émile Michel Hyacinthe Lemoine (French: [emil ləmwan]; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most notably, the École Polytechnique. Lemoine taught as a private tutor for a short period after his graduation from the latter school.
  
  Lemoine is best known for his proof of the existence of the Lemoine point (or the symmedian point) of a triangle. Other mathematical work includes a system he called Géométrographie and a method which related algebraic expressions to geometric objects. He has been called a co-founder of modern triangle geometry, as many of its characteristics are present in his work. (Full article...)
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  Title page of the first edition of Wright's Certaine Errors in Navigation (1599)

  
  Edward Wright (baptised 8 October 1561; died November 1615) was an English mathematician and cartographer noted for his book Certaine Errors in Navigation (1599; 2nd ed., 1610), which for the first time explained the mathematical basis of the Mercator projection by building on the works of Pedro Nunes, and set out a reference table giving the linear scale multiplication factor as a function of latitude, calculated for each minute of arc up to a latitude of 75°. This was in fact a table of values of the integral of the secant function, and was the essential step needed to make practical both the making and the navigational use of Mercator charts.
  
  Wright was born at Garveston in Norfolk and educated at Gonville and Caius College, Cambridge, where he became a fellow from 1587 to 1596. In 1589 the college granted him leave after Elizabeth I requested that he carry out navigational studies with a raiding expedition organised by the Earl of Cumberland to the Azores to capture Spanish galleons. The expedition's route was the subject of the first map to be prepared according to Wright's projection, which was published in Certaine Errors in 1599. The same year, Wright created and published the first world map produced in England and the first to use the Mercator projection since Gerardus Mercator's original 1569 map. (Full article...)
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  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...)
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  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...)
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  Noether c. 1900–1910

  
  Amalie Emmy Noether (US: /ˈnʌtər/, UK: /ˈnɜːtə/; German: [ˈnøːtɐ]; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
  
  Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations but instead studied mathematics at the University of Erlangen, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent. (Full article...)
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  The title page of a 1634 version of Hues' Tractatus de globis in the collection of the Biblioteca Nacional de Portugal

  
  Robert Hues (1553 – 24 May 1632) was an English mathematician and geographer. He attended St. Mary Hall at Oxford, and graduated in 1578. Hues became interested in geography and mathematics, and studied navigation at a school set up by Walter Raleigh. During a trip to Newfoundland, he made observations which caused him to doubt the accepted published values for variations of the compass. Between 1586 and 1588, Hues travelled with Thomas Cavendish on a circumnavigation of the globe, performing astronomical observations and taking the latitudes of places they visited. Beginning in August 1591, Hues and Cavendish again set out on another circumnavigation of the globe. During the voyage, Hues made astronomical observations in the South Atlantic, and continued his observations of the variation of the compass at various latitudes and at the Equator. Cavendish died on the journey in 1592, and Hues returned to England the following year.
  
  In 1594, Hues published his discoveries in the Latin work Tractatus de globis et eorum usu (Treatise on Globes and Their Use) which was written to explain the use of the terrestrial and celestial globes that had been made and published by Emery Molyneux in late 1592 or early 1593, and to encourage English sailors to use practical astronomical navigation. Hues' work subsequently went into at least 12 other printings in Dutch, English, French and Latin. (Full article...)

Selected image –

three-dimensional rendering of a pink, translucent Klein bottle

Klein bottle

Credit: Wridgers

A Klein bottle is an example of a closed surface (a two-dimensional manifold) that is non-orientable (no distinction between the "inside" and "outside"). This image is a representation of the object in everyday three-dimensional space, but a true Klein bottle is an object in four-dimensional space. When it is constructed in three-dimensions, the "inner neck" of the bottle curves outward and intersects the side; in four dimensions, there is no such self-intersection (the effect is similar to a two-dimensional representation of a cube, in which the edges seem to intersect each other between the corners, whereas no such intersection occurs in a true three-dimensional cube). Also, while any real, physical object would have a thickness to it, the surface of a true Klein bottle has no thickness. Thus in three dimensions there is an inside and outside in a colloquial sense: liquid forced through the opening on the right side of the object would collect at the bottom and be contained on the inside of the object. However, on the four-dimensional object there is no inside and outside in the way that a sphere has an inside and outside: an unbroken curve can be drawn from a point on the "outer" surface (say, the object's lowest point) to the right, past the "lip" to the "inside" of the narrow "neck", around to the "inner" surface of the "body" of the bottle, then around on the "outer" surface of the narrow "neck", up past the "seam" separating the inside and outside (which, as mentioned before, does not exist on the true 4-D object), then around on the "outer" surface of the body back to the starting point (see the light gray curve on this simplified diagram). In this regard, the Klein bottle is a higher-dimensional analog of the Möbius strip, a two-dimensional manifold that is non-orientable in ordinary 3-dimensional space. In fact, a Klein bottle can be constructed (conceptually) by "gluing" the edges of two Möbius strips together.

Good articles –

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

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  Walton in Pomona College's Walker Lounge c. 1957

  
  Jean Brosius Walton (March 6, 1914 – July 5, 2006) was an American academic administrator and women's studies scholar. She spent the bulk of her career at Pomona College in Claremont, California.
  
  Born to a Pennsylvania Quaker family, Walton grew up at George School and studied mathematics at Swarthmore College, Brown University and the University of Pennsylvania. She joined Pomona College in 1949 as the Dean of Women, and was promoted to dean of students in 1969 and vice president for student affairs in 1976, three years before her formal retirement. During her tenure, she advocated for women's education, engaged with student protests against the Vietnam War, oversaw reform of residential life policies to eliminate parietal rules, and co-founded the Claremont Colleges' Intercollegiate Women's Studies Program. She earned widespread recognition for her work and was praised by colleagues for her independent and dignified personality. (Full article...)
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  Composite numbers can be arranged into rectangles but prime numbers cannot.

  
  
  A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
  
  The property of being prime is called primality. A simple but slow method of checking the primality of a given number $n$, called trial division, tests whether $n$ is a multiple of any integer between 2 and ${\sqrt {n}}$. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of October 2024 the largest known prime number is a Mersenne prime with 41,024,320 decimal digits. (Full article...)
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  In mathematics, the factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. The factorial of $n$ also equals the product of $n$ with the next smaller factorial:
  $$<br>{\begin{aligned}n!&=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\\<br>&=n\times (n-1)!\\<br>\end{aligned}}$$
  For example,
  $$5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.$$
  The value of 0! is 1, according to the convention for an empty product.
  
  Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of $n$ distinct objects: there are $n!$. In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science. (Full article...)
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  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...)
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  In graph theory, a cop-win graph is an undirected graph on which the pursuer (cop) can always win a pursuit–evasion game against a robber, with the players taking alternating turns in which they can choose to move along an edge of a graph or stay put, until the cop lands on the robber's vertex. Finite cop-win graphs are also called dismantlable graphs or constructible graphs, because they can be dismantled by repeatedly removing a dominated vertex (one whose closed neighborhood is a subset of another vertex's neighborhood) or constructed by repeatedly adding such a vertex. The cop-win graphs can be recognized in polynomial time by a greedy algorithm that constructs a dismantling order. They include the chordal graphs, and the graphs that contain a universal vertex. (Full article...)
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  A binary tiling in the Poincaré disk model of the hyperbolic plane. Each tile edge lies on a horocycle (shown as circles interior to the disk) or a hyperbolic line (arcs perpendicular to the disk boundary). The horocycles and lines are asymptotic to an ideal point located at the right side of the Poincaré disk.

  
  
  In geometry, a binary tiling (sometimes called a Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. The tiles are congruent, each adjoining five others. They may be convex pentagons, or non-convex shapes with four sides, alternatingly line segments and horocyclic arcs, meeting at four right angles.
  
  There are uncountably many distinct binary tilings for a given shape of tile. They are all weakly aperiodic, which means that they can have a one-dimensional symmetry group but not a two-dimensional family of symmetries. There exist binary tilings with tiles of arbitrarily small area. (Full article...)
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  In mathematics and computer science, the BIT predicate, sometimes written ${\text{BIT}}(i,j)$, is a predicate that tests whether the $j$th bit of the number $i$ (starting from the least significant digit) is 1, when $i$ is written as a binary number. Its mathematical applications include modeling the membership relation of hereditarily finite sets, and defining the adjacency relation of the Rado graph. In computer science, it is used for efficient representations of set data structures using bit vectors, in defining the private information retrieval problem from communication complexity, and in descriptive complexity theory to formulate logical descriptions of complexity classes. (Full article...)
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  In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that certain polyhedra with equal volume cannot be dissected into each other.
  
  Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes and Dehn invariants are equal. Having Dehn invariant zero is a necessary (but not sufficient) condition for being a space-filling polyhedron, and a polyhedron can be cut up and reassembled into a space-filling polyhedron if and only if its Dehn invariant is zero. The Dehn invariant of a self-intersection-free flexible polyhedron is invariant as it flexes. Dehn invariants are also an invariant for dissection in higher dimensions, and (with volume) a complete invariant in four dimensions. (Full article...)
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  Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it in 1967. In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by Adrien Douady under the name six-beaked shaddock; later authors have applied variants of this name more specifically to Jessen's icosahedron.
  
  The faces of Jessen's icosahedron meet only in right angles, even though it has no orientation where they are all parallel to the coordinate planes. It is a "shaky polyhedron", meaning that (like a flexible polyhedron) it is not infinitesimally rigid. Outlining the edges of this polyhedron with struts and cables produces a widely-used tensegrity structure, also called the six-bar tensegrity, tensegrity icosahedron, or expanded octahedron. (Full article...)
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  Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
  
  Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications. (Full article...)
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  Ronald Paul "Ron" Fedkiw (born February 27, 1968) is a full professor in the Stanford University department of computer science and a leading researcher in the field of computer graphics, focusing on topics relating to physically based simulation of natural phenomena and machine learning. His techniques have been employed in many motion pictures. He has earned recognition at the 80th Academy Awards and the 87th Academy Awards as well as from the National Academy of Sciences.
  
  His first Academy Award was awarded for developing techniques that enabled many technically sophisticated adaptations including the visual effects in 21st century movies in the Star Wars, Harry Potter, Terminator, and Pirates of the Caribbean franchises. Fedkiw has designed a platform that has been used to create many of the movie world's most advanced special effects since it was first used on the T-X character in Terminator 3: Rise of the Machines. His second Academy Award was awarded for computer graphics techniques for special effects for large scale destruction. Although he has won an Oscar for his work, he does not design the visual effects that use his technique. Instead, he has developed a system that other award-winning technicians and engineers have used to create visual effects for some of the world's most expensive and highest-grossing movies. (Full article...)
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  An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tiles

  
  A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
  
  A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A tessellation of space, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. (Full article...)

Did you know (auto-generated) –

• ... that multiple mathematics competitions have made use of Sophie Germain's identity?
• ... 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 circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?
• ... that the discovery of Descartes' theorem in geometry came from a too-difficult mathematics problem posed to a princess?
• ... that a folded paper lantern shows that certain mathematical definitions of surface area are incorrect?
• ... that ten-sided gaming dice have kite-shaped faces?
• ... that two members of the French parliament were killed when a delayed-action German bomb exploded in the town hall at Bapaume on 25 March 1917?
• ... that mathematics professor Ari Nagel has fathered more than a hundred children?

More did you know –

Did you know...

• ...an infinite, nonrepeating decimal can be represented using only the number 1 using continued fractions?
• ...that 253931039382791 and the following 18 prime numbers all end in the digit 1?
• ...that the Electronic Frontier Foundation funds awards for the discovery of prime numbers beyond certain sizes?
• ...that pi can be computed using only the number 2 by the work of Viète?
• … that the Riemann Hypothesis, one of the Millennium Problems, depends on the asymptotic growth of the Mertens Function?
• … that every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater?
• … that the best known lower bound for the length of the smallest superpermutation was first posted anonymously to the internet imageboard 4chan?

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In this shear transformation of the Mona Lisa, the central vertical axis (red vector) is unchanged, but the diagonal vector (blue) has changed direction. Hence the red vector is said to be an eigenvector of this particular transformation and the blue vector is not. Image credit: User:Voyajer

In mathematics, an eigenvector of a transformation is a vector, different from the zero vector, which that transformation simply multiplies by a constant factor, called the eigenvalue of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. The eigenspace for a factor is the set of eigenvectors with that factor as eigenvalue, together with the zero vector.

In the specific case of linear algebra, the eigenvalue problem is this: given an n by n matrix A, what nonzero vectors x in $R^{n}$ exist, such that Ax is a scalar multiple of x?

The scalar multiple is denoted by the Greek letter λ and is called an eigenvalue of the matrix A, while x is called the eigenvector of A corresponding to λ. These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear situations.

It is common to prefix any natural name for the vector with eigen instead of saying eigenvector. For example, eigenfunction if the eigenvector is a function, eigenmode if the eigenvector is a harmonic mode, eigenstate if the eigenvector is a quantum state, and so on. Similarly for the eigenvalue, e.g. eigenfrequency if the eigenvalue is (or determines) a frequency. (Full article...)

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