Portal:Mathematics

The Mathematics Portal

Carl Friedrich Gauss

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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  Portrait of Kepler in 1627

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

  
  
  In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), 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, they are the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...)
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  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...)
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  In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
  
  The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. (Full article...)
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  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 primitive lifestyle.
  
  Between 1978 and 1995, Kaczynski murdered three individuals and injured 23 others 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 opposing industrialization, rejecting leftism, and advocating for a nature-centered form of anarchism. (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 science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a 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 together with 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" 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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  Stylistic impression of the number, representing how its decimals go on infinitely

  
  
  In mathematics, 0.999... (also written as 0.9, 0..9 or 0.(9)) is a notation for the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal is a numeral that represents the smallest number no less than every number in the sequence $(0.9,0.99,0.999,\ldots )$; that is, the supremum of this sequence. This number is equal to 1. In other words, "0.999..." is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent exactly the same number.
  
  There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined. (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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  General relativity, also known as the general theory of relativity and 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 matter and radiation are present. 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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  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. Considered 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...)
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  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. The name of the corresponding field is actuarial science which covers rigorous mathematical calculations in areas of life expectancy and life insurance. 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.
  
  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 and manage programs that control risk. The actual steps needed to become an actuary are usually country-specific; however, almost all processes share a rigorous schooling or examination structure and take many years to complete. (Full article...)
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  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...)
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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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  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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  The manipulations of the Rubik's Cube form the Rubik's Cube group.

  
  In mathematics, a group is a set with an operation that satisfies the following constraints: the operation is associative and 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...)

Selected image –

animation showing a roughly star-shaped graph being traced out as a smaller circle rolls around inside of a larger circle

Hypotrochoid

Credit: Sam Derbyshire

A hypotrochoid is a curve traced out by a point "attached" to a smaller circle rolling around inside a fixed larger circle. In this example, the hypotrochoid is the red curve that is traced out by the red point 5 units from the center of the black circle of radius 3 as it rolls around inside the blue circle of radius 5. A special case is a hypotrochoid with the inner circle exactly one-half the radius of the outer circle, resulting in an ellipse (see an animation showing this). Mathematical analysis of closely-related curves called hypocycloids lead to special Lie groups. Both hypotrochoids and epitrochoids (where the moving circle rolls around on the outside of the fixed circle) can be created using the Spirograph drawing toy. These curves have applications in the "real world" in epicyclic and hypocycloidal gearing, which were used in World War II in the construction of portable radar gear and may be used today in 3D printing.

Good articles –

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

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  Paterson's worms are a family of cellular automata devised in 1971 by Mike Paterson and John Horton Conway to model the behaviour and feeding patterns of certain prehistoric worms. In the model, a worm moves between points on a triangular grid along line segments, representing food. Its turnings are determined by the configuration of eaten and uneaten line segments adjacent to the point at which the worm currently is. Despite being governed by simple rules the behaviour of the worms can be extremely complex, and the ultimate fate of one variant is still unknown.
  
  The worms were studied in the early 1970s by Paterson, Conway and Michael Beeler, described by Beeler in June 1973, and presented in November 1973 in Martin Gardner's "Mathematical Games" column in Scientific American. (Full article...)
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  In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. For instance, Minkowski's question-mark function produces an isomorphism (a one-to-one order-preserving correspondence) between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals.
  
  The theorem is named after Georg Cantor, who first published it in 1895, using it to characterize the (uncountable) ordering on the real numbers. It can be proved by a back-and-forth method that is also sometimes attributed to Cantor but was actually published later, by Felix Hausdorff. The same back-and-forth method also proves that countable dense unbounded orders are highly symmetric, and can be applied to other kinds of structures. However, Cantor's original proof only used the "going forth" half of this method. In terms of model theory, the isomorphism theorem can be expressed by saying that the first-order theory of unbounded dense linear orders is countably categorical, meaning that it has only one countable model, up to logical equivalence. (Full article...)
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  The regular heptagon cannot be constructed using only a straightedge and compass construction; this can be proven using the field of constructible numbers.

  
  In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
  
  The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. (Full article...)
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  3 + 2 = 5 with apples, a popular choice in textbooks

  
  
  Addition (usually signified by the plus symbol +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (that is, "3 plus 2 is equal to 5").
  
  Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups. (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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  In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with $c$ colors, the same is true for the whole graph. The theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős (1951), after whom it is named.
  
  The De Bruijn–Erdős theorem has several different proofs, all depending in some way on the axiom of choice. Its applications include extending the four-color theorem and Dilworth's theorem from finite graphs and partially ordered sets to infinite ones, and reducing the Hadwiger–Nelson problem on the chromatic number of the plane to a problem about finite graphs. It may be generalized from finite numbers of colors to sets of colors whose cardinality is a strongly compact cardinal. (Full article...)
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  In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.
  
  This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construct three-dimensional visualizations of abstract graphs. Branko Grünbaum has called this theorem "the most important and deepest known result on 3-polytopes." (Full article...)
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  Lighting and reflection calculations, as in the video game OpenArena, use the fast inverse square root code to compute angles of incidence and reflection.

  
  
  Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates ${\textstyle {\frac {1}{\sqrt {x}}}}$, the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number $x$ in IEEE 754 floating-point format. The algorithm is best known for its implementation in 1999 in Quake III Arena, a first-person shooter video game heavily based on 3D graphics. With subsequent hardware advancements, especially the x86 SSE instruction rsqrtss, this algorithm is not generally the best choice for modern computers, though it remains an interesting historical example.
  
  The algorithm accepts a 32-bit floating-point number as the input and stores a halved value for later use. Then, treating the bits representing the floating-point number as a 32-bit integer, a logical shift right by one bit is performed and the result subtracted from the number 0x5F3759DF, which is a floating-point representation of an approximation of ${\textstyle {\sqrt {2^{127}}}}$. This results in the first approximation of the inverse square root of the input. Treating the bits again as a floating-point number, it runs one iteration of Newton's method, yielding a more precise approximation. (Full article...)
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  The Math Myth: And Other STEM Delusions is a 2016 nonfiction book by Queens College political scientist Andrew Hacker analyzing and critiquing the United States educational system's teaching of mathematics as a linear progression towards more advanced fields. Based off a 2012 New York Times op-ed by Hacker titled "Is Algebra Necessary", Hacker argues that the teaching of advanced algebra, trigonometry, and calculus is not useful to the majority of students. He further claims that the requirement of advanced mathematics courses in secondary education contributes to dropout rates and impedes socioeconomically disadvantaged students from pursuing further education. Hacker critiques the Common Core system and American focus on STEM education in lieu of social sciences, arguing that the educational system should prioritize "numeracy" over pure mathematics education.
  
  The Math Myth received broadly critical coverage from critics and mathematicians, some citing Hacker's arguments as "disingenuous" and contributing to an elitist attitude towards mathematics, with many citing a lack of exploration on mathematics in early childhood and primary education. Others praised Hacker's work, describing the book as offering a convincing critique of STEM education in the United States and empowering to students struggling in mathematics. (Full article...)
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  Convergence of a convex curve to a circle under the curve-shortening flow. Inner curves (lighter color) are flowed versions of the outer curves. Time steps between curves are not uniform.

  
  In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, and arc length evolution.
  
  As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth. It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of the curve decreases as the curve converges to a circular shape, before collapsing to a single point of singularity. If two disjoint simple smooth closed curves evolve, they remain disjoint until one of them collapses to a point.
  The circle is the only simple closed curve that maintains its shape under the curve-shortening flow, but some curves that cross themselves or have infinite length keep their shape, including the grim reaper curve, an infinite curve that translates upwards, and spirals that rotate while remaining the same size and shape. (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 polyhedron and its midsphere. The red circles are the boundaries of spherical caps within which the surface of the sphere can be seen from each vertex.

  
  In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals (Catalan solids) all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.
  
  When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their corresponding circles in this circle packing. (Full article...)

Did you know (auto-generated) –

• ... that after Florida schools banned 54 mathematics books, Chaz Stevens petitioned that they also ban the Bible?
• ... that Arithmetic was the first mathematics text book written in the Russian language?
• ... that Ukrainian baritone Danylo Matviienko, who holds a master's degree in mathematics, appeared as Demetrius in Britten's opera A Midsummer Night's Dream at the Oper Frankfurt?
• ... that Latvian-Soviet artist Karlis Johansons exhibited a skeletal tensegrity form of the Schönhardt polyhedron seven years before Erich Schönhardt's 1928 paper on its mathematics?
• ... that multiple mathematics competitions have made use of Sophie Germain's identity?
• ... 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 the prologue to The Polymath was written by Martin Kemp, a leading expert on Leonardo da Vinci?
• ... that the discovery of Descartes' theorem in geometry came from a too-difficult mathematics problem posed to a princess?

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?

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Example of a four color map Image credit: User:Inductiveload

The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. "Color by Number" worksheets and exercises, which combine learning art and math for people of young ages, are a good example of the four color theorem.

It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.

The four color theorem was the first major theorem to be proven using a computer, and the proof is disputed by some mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.

The lack of mathematical elegance was another factor, and to paraphrase comments of the time, "a good mathematical proof is like a poem — this is a telephone directory!" (Full article...)

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