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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...)

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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...)
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Cantor, c. 1910

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor, German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). (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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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...)
$Image 5 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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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 $252 = 21 \times 12$ and $105 = 21 \times 5)$ , 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, $21 = 5 \times 105 + (-2) \times 252$ ). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...)
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Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist, 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, as well as his work 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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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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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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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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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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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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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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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 is the inverse function to exponentiation. 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 $1000 = 10 3$ , the logarithm base $10$ of $1000$ is $3$ , or $log 10 (1000) = 3$ . 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, such as in big O notation, 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 $e \approx 2.718$ 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 frequently used in computer science. (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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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 all forms of technology, rejecting leftism, and advocating for the abolition of industrial society. (Full article...)

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Knot theory

Credit: Jkasd

This is a chart of all prime knots having seven or fewer crossings (not including mirror images) along with the unknot (or "trivial knot"), a closed loop that is not a prime knot. The knots are labeled with Alexander-Briggs notation. Many of these knots have special names, including the trefoil knot (3₁) and figure-eight knot (4₁). Knot theory is the study of knots viewed as different possible embeddings of a 1-sphere (a circle) in three-dimensional Euclidean space (

R 3

). These mathematical objects are inspired by real-world knots, such as knotted ropes or shoelaces, but don't have any free ends and so cannot be untied. (Two other closely related mathematical objects are braids, which can have loose ends, and links, in which two or more knots may be intertwined.) One way of distinguishing one knot from another is by the number of times its two-dimensional depiction crosses itself, leading to the numbering shown in the diagram above. The prime knots play a role very similar to prime numbers in number theory; in particular, any given (non-trivial) knot can be uniquely expressed as a "sum" of prime knots (a series of prime knots spliced together) or is itself prime. Early knot theory enjoyed a brief period of popularity among physicists in the late 19th century after William Thomson suggested that atoms are knots in the luminiferous aether. This led to the first serious attempts to catalog all possible knots (which, along with links, now number in the billions). In the early 20th century, knot theory was recognized as a subdiscipline within geometric topology. Scientific interest was resurrected in the latter half of the 20th century by the need to understand knotting problems in organic chemistry, including the behavior of DNA, and the recognition of connections between knot theory and quantum field theory.

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In geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. The theorem is named after Frank S. Beckman and Donald A. Quarles Jr., who published this result in 1953; it was later rediscovered by other authors and re-proved in multiple ways. Analogous theorems for rational subsets of Euclidean spaces, or for non-Euclidean geometry, are also known. (Full article...)
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The infinity symbol ( $\infty$ ) is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate, after the lemniscate curves of a similar shape studied in algebraic geometry, or "lazy eight", in the terminology of livestock branding.

This symbol was first used mathematically by John Wallis in the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes (potential infinity) rather than infinite values (actual infinity). It has other related technical meanings, such as the use of long-lasting paper in bookbinding, and has been used for its symbolic value of the infinite in modern mysticism and literature. It is a common element of graphic design, for instance in corporate logos as well as in older designs such as the Métis flag. (Full article...)
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Using the Pythagorean theorem to compute two-dimensional Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance.

These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's Elements, distances were not represented as numbers but line segments of the same length, which were considered "equal". The notion of distance is inherent in the compass tool used to draw a circle, whose points all have the same distance from a common center point. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century. (Full article...)
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In mathematics, economics, and computer science, the Gale–Shapley algorithm (also known as the deferred acceptance algorithm, propose-and-reject algorithm, or Boston Pool algorithm) is an algorithm for finding a solution to the stable matching problem. It is named for David Gale and Lloyd Shapley, who published it in 1962, although it had been used for the National Resident Matching Program since the early 1950s. Shapley and Alvin E. Roth (who pointed out its prior application) won the 2012 Nobel Prize in Economics for work including this algorithm.

The stable matching problem seeks to pair up equal numbers of participants of two types, using preferences from each participant. The pairing must be stable: no pair of participants should prefer each other to their assigned match. In each round of the Gale–Shapley algorithm, unmatched participants of one type propose a match to the next participant on their preference list. Each proposal is accepted if its recipient prefers it to their current match. The resulting procedure is a truthful mechanism from the point of view of the proposing participants, who receive their most-preferred pairing consistent with stability. In contrast, the recipients of proposals receive their least-preferred pairing. The algorithm can be implemented to run in time quadratic in the number of participants, and linear in the size of the input to the algorithm. (Full article...)
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Stars is a wood engraving print created by the Dutch artist M. C. Escher in 1948, depicting two chameleons in a polyhedral cage floating through space.

The compound of three octahedra used for the central cage in Stars had been studied before in mathematics, and Escher likely learned of it from the book Vielecke und Vielflache by Max Brückner. Escher used similar compound polyhedral forms in several other works, including Crystal (1947), Study for Stars (1948), Double Planetoid (1949), and Waterfall (1961). (Full article...)
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Translation of an English sentence to first-order logic

Logic translation is the process of representing a text in the formal language of a logical system. If the original text is formulated in ordinary language then the term natural language formalization is often used. An example is the translation of the English sentence "some men are bald" into first-order logic as $\exists x(M(x)\land B(x))$ . The purpose is to reveal the logical structure of arguments. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guide reasoning by arriving at new conclusions.

Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the sciences and humanities. It may play a key role for logic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of informal logic is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another, for example, from modal logic to first-order logic. This form of logic translation is specifically relevant for logic programming and metalogic. (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 convex hull of the red set is the blue and red convex set.

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points.
The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. They can be solved in time $O(n\log n)$ for two or three dimensional point sets, and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions. (Full article...)
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Mathematics in art: Albrecht Dürer's copper plate engraving Melencolia I, 1514. Mathematical references include a compass for geometry, a magic square and a truncated rhombohedron, while measurement is indicated by the scales and hourglass.

Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.

Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions conjectured to have been based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De divina proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas vaulting. (Full article...)
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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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Wigner in 1963

Eugene Paul Wigner (Hungarian: Wigner Jenő Pál, pronounced [ˈviɡnɛr ˈjɛnøː ˈpaːl]; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".

A graduate of the Technical University of Berlin, Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen. Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus. In 1930, Princeton University recruited Wigner, along with John von Neumann, and he moved to the United States, where he obtained citizenship in 1937. (Full article...)
Image 12
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...)

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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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