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

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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. 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...)
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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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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...)
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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 reclusive primitive lifestyle.

Kaczynski murdered three people and injured 23 others between 1978 and 1995 in a nationwide mail bombing campaign against people he believed to be advancing modern technology and the destruction of the natural environment. He authored Industrial Society and Its Future, a 35,000-word manifesto and social critique opposing all forms of technology, rejecting leftism, and advocating a nature-centered form of anarchism. (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 is
present matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations.

Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data. (Full article...)
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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...)
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Portrait by Jakob Emanuel Handmann, 1753

Leonhard Euler (/ˈɔɪlər/ OY-lər; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleɔnhard ˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.

Euler is credited for popularizing the Greek letter π {\displaystyle \pi } (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle \Sigma } (capital sigma) to express summations, the Greek letter Δ {\displaystyle \Delta } (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant e {\displaystyle e} , the base of the natural logarithm, now known as Euler's number. Euler made contributions to applied mathematics and engineering, such as his study of ships which helped navigation, his three volumes on optics contributed to the design of microscopes and telescopes, and he studied the bending of beams and the critical load of columns. (Full article...)
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In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.

The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. (Full article...)
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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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Figure 1: A solution (in purple) to Apollonius's problem. The given circles are shown in black.

In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 BC – c. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts).

In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN. (Full article...)
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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 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 $3$ rd power: $1000 = 10 3 = 10 \times 10 \times 10$ . More generally, if $x = b y$ , then $y$ is the logarithm of $x$ to base $b$ , written $log b x$ , so $log 10 1000 = 3$ . 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 $e \approx 2.718$ 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...)
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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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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...)
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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...)
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Archimedes Thoughtful by Fetti (1620)

Archimedes of Syracuse (/ˌɑːrkɪˈmiːdiːz/ AR-kim-EE-deez; c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is considered one of the leading scientists in classical antiquity. Regarded as the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.

Archimedes' other mathematical achievements include deriving an approximation of pi ( $π$ ), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy known as Archimedes' principle. In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. (Full article...)

Good articles

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Georg Cantor, c. 1870

Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the topological notion of a set being dense in an interval.

Cantor's article also contains a proof of the existence of transcendental numbers. Both constructive and non-constructive proofs have been presented as "Cantor's proof." The popularity of presenting a non-constructive proof has led to a misconception that Cantor's arguments are non-constructive. Since the proof that Cantor published either constructs transcendental numbers or does not, an analysis of his article can determine whether or not this proof is constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs: a non-constructive proof that uses the uncountability of the real numbers and a constructive proof that does not use uncountability. (Full article...)
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Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30.

In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.

As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first $n$ positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number. (Full article...)
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A graph with three components

In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components.

The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not. (Full article...)
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1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral.

In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions. (Full article...)
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In number theory, a Wieferich prime is a prime number p such that p² divides $2 p - 1 - 1$ , therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides $2 p - 1 - 1$ . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture. (Full article...)
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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...)
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The Shapley–Folkman lemma is illustrated by the Minkowski addition of four sets. The point (+) in the convex hull of the Minkowski sum of the four non-convex sets (right) is the sum of four points (+) from the (left-hand) sets—two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown as red dots).

The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ross M. Starr.

The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex. (Full article...)
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Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.

In mathematical analysis, the Dirac delta function (or $δ$ distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as

$\delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}$ (Full article...)
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Fredrik Carl Mülertz Størmer (3 September 1874 – 13 August 1957) was a Norwegian mathematician and astrophysicist. In mathematics, he is known for his work in number theory, including the calculation of $π$ and Størmer's theorem on consecutive smooth numbers. In physics, he is known for studying the movement of charged particles in the magnetosphere and the formation of aurorae, and for his book on these subjects, From the Depths of Space to the Heart of the Atom. He worked for many years as a professor of mathematics at the University of Oslo in Norway. A crater on the far side of the Moon is named after him. (Full article...)
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A unit distance graph with 16 vertices and 40 edges

In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs.

An unsolved problem of Paul Erdős asks how many edges a unit distance graph on $n$ vertices can have. The best known lower bound is slightly above linear in $n$ —far from the upper bound, proportional to $n^{4/3}$ . The number of colors required to color unit distance graphs is also unknown (the Hadwiger–Nelson problem): some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number there is a unit distance graph with two vertices that must be that distance apart. According to the Beckman–Quarles theorem, the only plane transformations that preserve all unit distance graphs are the isometries. (Full article...)
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Rosa Margaret Morris (16 July 1914 – 15 October 2011) was a Welsh applied mathematician, working in potential theory and aerodynamics. When she was 23, her research and examination results made national and international news. She received several fellowships and awards before graduating with a PhD in mathematics from the University of Cambridge in 1940. In her later career, she taught at the University College of South Wales and Monmouthshire (now Cardiff University), where she co-authored a successful textbook on Mathematical Methods of Physics with Roy Chisholm and became one of the first female Heads of School of Mathematics in the United Kingdom. (Full article...)
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The complete graph $K 4$ has ten matchings, corresponding to the value $T (4) = 10$ of the fourth telephone number.

In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways $n$ people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on $n$ vertices, the number of permutations on $n$ elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with $n$ cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values (starting from $n = 0$ ) (Full article...)

Did you know

... that despite a mathematical model deeming the ice cream bar flavour Goody Goody Gum Drops impossible, it was still created?
... that the British National Hospital Service Reserve trained volunteers to carry out first aid in the aftermath of a nuclear or chemical attack?
... that Green Day's "Wake Me Up When September Ends" became closely associated with the aftermath of Hurricane Katrina?
... that more than 60 scientific papers authored by mathematician Paul Erdős were published posthumously?
... that subgroup distortion theory, introduced by Misha Gromov in 1993, can help encode text?
... that The Math Myth advocates for American high schools to stop requiring advanced algebra?
... that Fairleigh Dickinson's upset victory over Purdue was the biggest upset in terms of point spread in NCAA tournament history, with Purdue being a 23+1⁄2-point favorite?
... that in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?

Did you know...

...that people are significantly slower to identify the parity of zero than other whole numbers, regardless of age, language spoken, or whether the symbol or word for zero is used?
...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?

More did you know facts

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Featured pictures

Image 1Tetrahedral group at Symmetry group, by Debivort (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 2Helicatenoid, by Wickerprints (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 3Mandelbrot set, step 8, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 4Mandelbrot set, step 13, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 5Radian, by Lucas V. Barbosa (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 6Mandelbrot set, step 11, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 7Topological knots, by Jkasd (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 8Mandelbrot set, start, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 9Fields Medal, back, by Stefan Zachow (edited by King of Hearts) (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 10Cellular automata at Reflector (cellular automaton), by Simpsons contributor (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 11Breeder, by Protious/Hyperdeath (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 12Mandelbrot set, step 1, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 13Mandelbulb, by Ondřej Karlík (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 14Mandelbrot set, step 3, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 15Mandelbrot set, step 6, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 16Proof of the Pythagorean theorem, by Joaquim Alves Gaspar (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 17Mandelbrot set, step 7, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 18Anscombe's quartet, by Schutz (edited by Avenue) (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 19Hypotrochoid, by Sam Derbyshire (edited by Anevrisme and Perhelion) (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 20Missing square puzzle, by Fibonacci (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 21Mandelbrot set, step 4, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 22Mandelbrot set, by Simpsons contributor (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 23Mandelbrot set, step 9, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 24Line integral of scalar field, by Lucas V. Barbosa (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 25Mandelbrot set, step 5, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 26Lorenz attractor at Chaos theory, by Wikimol (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 27Mandelbrot set, step 12, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 28Fields Medal, front, by Stefan Zachow (edited by King of Hearts) (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 29Desargues' theorem, by Dynablast (edited by Jujutacular and Julia W) (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 30Mandelbrot set, step 10, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 31Mandelbrot set, step 2, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)
$Image 32Lyapunov fractal, by BernardH (from Wikipedia:Featured pictures/Sciences/Mathematics)$

Image 32Lyapunov fractal, by BernardH (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 33Non-uniform rational B-spline, by Greg L (from Wikipedia:Featured pictures/Sciences/Mathematics)
Image 34Mandelbrot set, step 14, by Wolfgangbeyer (from Wikipedia:Featured pictures/Sciences/Mathematics)

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