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Mathematics is the study ofrepresenting andreasoning about abstractobjects (such asnumbers,points,spaces,sets,structures, andgames). Mathematics is used throughout the world as an essential tool in many fields, includingnatural science,engineering,medicine, and thesocial 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 asstatistics andgame theory. Mathematicians also engage inpure 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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Image 1

1753 portrait

Leonhard Euler (/ˈɔɪlər/OY-lər; 15 April 1707 – 18 September 1783) was a Swisspolymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies ofgraph theory andtopology and made influential discoveries in many other branches of mathematics, such asanalytic number theory,complex analysis, andinfinitesimal calculus. He also introduced much of modern mathematical terminology andnotation, including the notion of amathematical function. He is known for his work inmechanics,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 inSaint Petersburg, Russia, and inBerlin, then the capital ofPrussia.

Euler is credited for popularizing the Greek letter $\pi$ (lowercasepi) to denotethe ratio of a circle's circumference to its diameter, as well as first using the notation $f(x)$ for the value of a function, the letter $i {\displaystyle i}$ to express theimaginary unit ${\sqrt {-1}}$ , the Greek letter $\Sigma$ (capitalsigma) to expresssummations, the Greek letter $\Delta$ (capitaldelta) forfinite 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 thenatural logarithm, now known asEuler's number. Euler made contributions toapplied mathematics andengineering, such as his study of ships, which helped navigation; his three volumes on optics, which contributed to the design ofmicroscopes andtelescopes; and his studies of beam bending and column critical loads. (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 theUnabomber (/ˈjuːnəbɒmər/ ^ⓘYOO-nə-bom-ər), was an American mathematician anddomestic terrorist. A mathematicsprodigy, he abandoned his academic career in 1969 to pursue areclusive primitive lifestyle andlone wolf terrorism campaign.

Kaczynski murdered 3 people and injured 23 others between 1978 and 1995 in a nationwidemail bombing campaign against people he believed to be advancingmodern technology and the destruction of thenatural environment. He authored a roughly 35,000-wordmanifesto andsocial critique calledIndustrial Society and Its Future (1995) which opposes all forms of technology, rejectsleftism andfascism, advocates cultural primitivism, and ultimately suggests violentrevolution. (Full article...)
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Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an Americantheoretical physicist. He studied thepath integral formulation ofquantum mechanics, the theory ofquantum electrodynamics, the physics of thesuperfluidity of supercooledliquid helium, andparticle physics, for which he proposed theparton model. For his contributions to the development of quantum electrodynamics, Feynman received theNobel Prize in Physics in 1965 jointly withJulian Schwinger andShin'ichirō Tomonaga.

Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior ofsubatomic particles, which later became known asFeynman diagrams and is widely used. 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 journalPhysics World, he was ranked the seventh-greatest physicist of all time. (Full article...)
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The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... The graph is situated with positive integers to the right and negative integers to the left.

Inmathematics,1 − 2 + 3 − 4 + ··· is aninfinite series whose terms are the successivepositive integers, givenalternating signs. Usingsigma summation notation the sum of the firstm terms of the series can be expressed as
$\sum _{n=1}^{m}n(-1)^{n-1}.$

The infinite seriesdiverges, meaning that its sequence ofpartial sums,(1, −1, 2, −2, 3, ...), does not tend towards any finitelimit. Nonetheless, in the mid-18th century,Leonhard Euler wrote what he admitted to be aparadoxical equation:
$1-2+3-4+\cdots ={\frac {1}{4}}.$ (Full article...)
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General relativity, also known as thegeneral theory of relativity, and asEinstein's theory of gravity, is thegeometric theory ofgravitation published byAlbert Einstein in 1916 and is the accepted description of gravitation inmodern physics. Generalrelativity generalizesspecial relativity and refinesNewton's law of universal gravitation, providing a unified description of gravity as a geometric property ofspace andtime, or four-dimensionalspacetime. In particular, thecurvature of spacetime is directly related to theenergy,momentum andstress of whatever is present, includingmatter andradiation. The relation is specified by theEinstein field equations, a system of second-orderpartial differential equations.

Newton's law of universal gravitation, which describes gravity in classical mechanics, 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 beyondNewton's law of universal gravitation inclassical physics. These predictions concern the passage of time, thegeometry of space, the motion of bodies infree fall, and the propagation of light, and includegravitational time dilation,gravitational lensing, thegravitational redshift of light, theShapiro time delay andsingularities/black holes. So far, alltests of general relativity have been in agreement with the theory. The time-dependent solutions of general relativity enable us to extrapolate the history of the universe into the past and future, and have provided the modern framework forcosmology, thus leading to the discovery of theBig Bang andcosmic microwave background radiation. Despite the introduction of a number ofalternative theories, general relativity continues to be the simplest theory consistent withexperimental data. (Full article...)
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Portrait,c. 1910, after 1620 original

Johannes Kepler (27 December 1571 – 15 November 1630) was a Germanastronomer,mathematician,astrologer,natural philosopher and writer on music. He is a key figure in the 17th-centuryScientific Revolution, best known for hislaws of planetary motion, and his booksAstronomia nova,Harmonice Mundi, andEpitome Astronomiae Copernicanae. The variety and impact of his work made Kepler one of the founders and fathers of modernastronomy, thescientific method,natural science, andmodern science. He has been described as the "father ofscience fiction" for his novelSomnium.

Kepler was a mathematics teacher at aseminary school inGraz, where he became an associate ofPrince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomerTycho Brahe inPrague, and eventually the imperial mathematician toEmperor Rudolf II and his two successorsMatthias andFerdinand II. He also taught mathematics inLinz, and was an adviser toGeneral Wallenstein. (Full article...)
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The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric groupS̃₃

Theaffine symmetric groups are a family ofmathematical structures that describe the symmetries of thenumber line and the regulartriangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections ofpermutations (rearrangements) of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as agroup with certaingenerators and relations. They are studied incombinatorics andrepresentation theory.

A finitesymmetric group consists of all permutations of a finite set. Each affine symmetric group is an infiniteextension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups.Permutation statistics such asdescents andinversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation. (Full article...)
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The title page of a 1634 version of Hues'sTractatus de globis in the collection of theBiblioteca Nacional de Portugal

Robert Hues (1553 – 24 May 1632) was an Englishmathematician andgeographer. He attendedSt. Mary Hall atOxford, and graduated in 1578. Hues became interested ingeography andmathematics, and studiednavigation at a school set up byWalter Raleigh. During a trip toNewfoundland, he made observations which caused him to doubt the accepted published values forvariations of the compass. Between 1586 and 1588, Hues travelled withThomas Cavendish on acircumnavigation of the globe, performingastronomical observations and taking the latitudes of places they visited. Beginning in August 1591, Hues and Cavendish again set out on anothercircumnavigation 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 variouslatitudes and at theEquator. Cavendish died on the journey in 1592, and Hues returned to England the following year.

In 1594, Hues published his discoveries in theLatin workTractatus 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 byEmery Molyneux in late 1592 or early 1593, and to encourage English sailors to use practical astronomical navigation. Hues's work subsequently went into at least 12 other printings in Dutch, English, French and Latin. (Full article...)
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Bust of Shen at theBeijing Ancient Observatory

Shen Kuo (Chinese:沈括; 1031–1095) orShen Gua,courtesy nameCunzhong (存中) andpseudonymMengqi (now usually given asMengxi)Weng (夢溪翁), was a Chinesepolymath, scientist, and statesman of theSong dynasty (960–1279). Shen was a master in many fields of study includingmathematics,optics, andhorology. In his career as a civil servant, he became afinance minister, governmental state inspector, head official for theBureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as anacademic chancellor. At court his political allegiance was to the Reformist faction known as theNew Policies Group, headed byChancellor Wang Anshi (1021–1085).

In hisDream Pool Essays orDream Torrent Essays (夢溪筆談;Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needlecompass, which would be used for navigation (first described in Europe byAlexander Neckam in 1187). Shen discovered the concept oftrue north in terms ofmagnetic declination towards thenorth pole, with experimentation of suspended magnetic needles and "the improvedmeridian determined by Shen's [astronomical] measurement of the distance between thepole 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 Europefor another four hundred years (evidence of German sundials made circa 1450 show markings similar toChinese geomancers' compasses in regard to declination). (Full article...)
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High-precision test of general relativity by theCassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) aredelayed by the warping ofspacetime (blue lines) due to theSun's mass.

General relativity is atheory ofgravitation developed byAlbert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping ofspacetime.

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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Rejewski,c. 1932

Marian Adam Rejewski (Polish:[ˈmarjanrɛˈjɛfskʲi]^ⓘ; 16 August 1905 – 13 February 1980) was a Polishmathematician andcryptologist who in late 1932 reconstructed the sight-unseen German militaryEnigma cipher machine, aided by limited documents obtained byFrench military intelligence.

Over the next nearly seven years, Rejewski and fellow mathematician-cryptologistsJerzy Różycki andHenryk Zygalski, working at thePolish General Staff'sCipher 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 thecryptologic card catalog and thecryptologic bomb. (Full article...)
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Inclassical mechanics, theLaplace–Runge–Lenz vector (LRL vector) is avector used chiefly to describe the shape and orientation of theorbit of oneastronomical body around another, such as abinary star or a planet revolving around a star. Fortwo bodies interacting byNewtonian gravity, the LRL vector is aconstant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to beconserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by acentral force that varies as theinverse square of the distance between them; such problems are calledKepler problems.

Thus thehydrogen atom is a Kepler problem, since it comprises two charged particles interacting byCoulomb's law ofelectrostatics, another inverse-square central force. The LRL vector was essential in the firstquantum mechanical derivation of thespectrum of the hydrogen atom, before the development of theSchrödinger equation. However, this approach is rarely used today. (Full article...)
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Cantor,c. 1910

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/KAN-tor;German:[ˈɡeːɔʁkˈfɛʁdinantˈluːtvɪçˈfiːlɪpˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation ofset theory, which has become afundamental theory in mathematics. Cantor established the importance ofone-to-one correspondence between the members of two sets, definedinfinite andwell-ordered sets, and proved that thereal numbers are more numerous than thenatural numbers. Cantor's method of proof of this theorem implies the existence of aninfinity of infinities. He defined thecardinal andordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory oftransfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such asLeopold Kronecker andHenri Poincaré and later fromHermann Weyl andL. E. J. Brouwer, whileLudwig Wittgenstein raisedphilosophical objections; seeControversy over Cantor's theory. Cantor, a devoutLutheran Christian, believed the theory had been communicated to him by God. SomeChristian theologians (particularlyneo-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 withpantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopherKonstantin Gutberlet [de;it] was in favor of it and CardinalJohann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). (Full article...)
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Plots of logarithm functions, with three commonly used bases. The special pointslog_b b = 1 are indicated by dotted lines, and all curves intersect inlog_b 1 = 0.

Inmathematics, thelogarithm of a number is theexponent by which another fixed value, thebase, must be raised to produce that number. For example, the logarithm of1000 to base10 is3, because1000 is10 to the3rd power:1000 = 10³ = 10 × 10 × 10. More generally, ifx =b^y, theny is the logarithm ofx to baseb, writtenlog_bx, solog₁₀ 1000 = 3. As a single-variable function, the logarithm to baseb is theinverse ofexponentiation with baseb.

The logarithm base10 is called thedecimal orcommon logarithm and is commonly used in science and engineering. Thenatural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics andphysics because of its very simplederivative. Thebinary logarithm uses base2 and is widely used incomputer science,information theory,music theory, andphotography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is writtenlog x. (Full article...)
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A stamp of Zhang Heng issued byChina Post in 1955

Zhang Heng (Chinese:張衡; AD 78–139),courtesy namePingzi, formerlyromanizedChang Heng, was a Chinesepolymathic scientist and statesman who lived during theEastern Han dynasty. Educated in the capital cities ofLuoyang andChang'an, he achieved success as anastronomer,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 inNanyang. 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 palaceeunuchs during the reign ofEmperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator ofHejian Kingdom in present-dayHebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139. (Full article...)

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graph showing two sets of 4 points, each set perfectly fit by a trend line with positive slope; the set of points on the left is higher and the set on the right lower, so the entire collection of points is best fit by a trend line with negative slope

Simpson's paradox

Credit: Schutz

Simpson's paradox (also known as theYule–Simpson effect) states that an observedassociation between twovariables can reverse when considered at separate levels of a third variable (or, conversely, that the association can reverse when separate groups are combined). Shown here is an illustration of the paradox forquantitative data. In the graph the overall association betweenX andY is negative (asX increases,Y tends to decrease when all of the data is considered, as indicated by the negative slope of the dashed line); but when the blue and red points are considered separately (two levels of a third variable, color), the association betweenX andY appears to be positive in each subgroup (positive slopes on the blue and red lines — note that the effect in real-world data is rarely this extreme). Named after British statisticianEdward H. Simpson, who first described the paradox in 1951 (in the context ofqualitative data), similar effects had been mentioned byKarl Pearson (and coauthors) in 1899, and byUdny Yule in 1903. One famous real-life instance of Simpson's paradox occurred in theUC Berkeley gender-bias case of the 1970s, in which the university was sued forgender discrimination because it had a higher admission rate for male applicants to its graduate schools than for female applicants (and the effect wasstatistically significant). The effect was reversed, however, when the data was split by department: most departments showed a small but significant bias in favor of women. The explanation was that women tended to apply to competitive departments with low rates of admission even among qualified applicants, whereas men tended to apply to less-competitive departments with high rates of admission among qualified applicants. (Note that splitting by department was a more appropriate way of looking at the data since it is individual departments, not the university as a whole, that admit graduate students.)

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Image 1
Graph oflog₂ x as a function of a positive real numberx

Inmathematics, thebinary logarithm (log₂ n) is thepower to which the number2 must beraised to obtain the value n. That is, for any real numberx,
: $x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.$
For example, the binary logarithm of1 is0, the binary logarithm of2 is1, the binary logarithm of4 is 2, and the binary logarithm of32 is 5.

The binary logarithm is thelogarithm to the base2 and is theinverse function of thepower of two function. There are several alternatives to thelog₂ notation for the binary logarithm; see theNotation section below. (Full article...)
Image 2

The Herschel graph.

Ingraph theory, a branch ofmathematics, theHerschel graph is abipartite undirected graph with 11 vertices and 18 edges. It is apolyhedral graph (the graph of aconvex polyhedron), and is the smallest polyhedral graph that does not have aHamiltonian cycle, a cycle passing through all its vertices. The polyhedron whose vertices and edges form this graph is sometimes called theHerschel enneahedron; it is anenneahedron because it has nine faces.

Both the graph and the polyhedron are named after British astronomerAlexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs (but not of this graph). (Full article...)
Image 3
A double bubble. Note that the surface separating the small lower bubble from the large bubble bulges into the large bubble.

In the mathematical theory ofminimal surfaces, thedouble bubble theorem states that the shape that encloses and separates two givenvolumes and has the minimum possiblesurface area is astandard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002.

The proof combines multiple ingredients.Compactness ofrectifiable currents (a generalized definition of surfaces) shows that a solution exists. A symmetry argument proves that the solution must be asurface of revolution, and it can be further restricted to having a bounded number of smooth pieces.Jean Taylor's proof ofPlateau's laws describes how these pieces must be shaped and connected to each other, and a finalcase analysis shows that, among surfaces of revolution connected in this way, only the standard double bubble has locally-minimal area. (Full article...)
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Escher in 1971

Maurits Cornelis Escher (/ˈɛʃər/;Dutch:[ˈmʌurɪtskɔrˈneːlɪsˈɛɕər]; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who madewoodcuts,lithographs, andmezzotints, many of which wereinspired by mathematics.
Despite wide popular interest, for most of his life Escher was neglected in the art world, even in his native Netherlands. He was 70 before a retrospective exhibition was held. In the late twentieth century, he became more widely appreciated, and in the twenty-first century he has been celebrated in exhibitions around the world.

His work features mathematical objects and operations includingimpossible objects, explorations of infinity,reflection,symmetry,perspective,truncated andstellated polyhedra,hyperbolic geometry, andtessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematiciansGeorge Pólya,Roger Penrose, andDonald Coxeter, and thecrystallographer Friedrich Haag, and conducted his own research into tessellation. (Full article...)
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Inmathematics,economics, andcomputer science, theGale–Shapley algorithm (also known as thedeferred acceptance algorithm,propose-and-reject algorithm, orBoston Pool algorithm) is analgorithm for finding a solution to thestable matching problem. It is named forDavid Gale andLloyd Shapley, who published it in 1962, although it had been used for theNational Resident Matching Program since the early 1950s. Shapley andAlvin E. Roth (who pointed out its prior application) won the 2012Nobel 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 matched participants should mutually 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 atruthful 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, andlinear in the size of the input to the algorithm. (Full article...)
Image 6
Theinfinity symbol (∞) is amathematical symbol representing the concept ofinfinity. This symbol is also called alemniscate, after thelemniscate curves of a similar shape studied inalgebraic geometry, or "lazy eight", in the terminology oflivestock branding.

This symbol was first used mathematically byJohn Wallis in the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes (potential infinity) but may also refer to infinite values (actual infinity). It has other related technical meanings, such as the use of long-lasting paper inbookbinding, and has been used for its symbolic value of the infinite in modern mysticism and literature. It is a common element ofgraphic design, for instance in corporate logos as well as in earlier designs such as theMétis flag. (Full article...)
Image 7
As the degree of the Taylor polynomial rises, it approaches the correct function. This image showssinx and its Taylor approximations by polynomials of degree1,3,5,7,9,11, and13 atx = 0.

Inmathematical analysis, theTaylor series orTaylor expansion of afunction is aninfinite sum of terms that are expressed in terms of the function'sderivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named afterBrook Taylor, who introduced them in 1715. A Taylor series is also called aMaclaurin series when 0 is the point where the derivatives are considered, afterColin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

Thepartial sum formed by the firstn + 1 terms of a Taylor series is apolynomial of degreen that is called thenthTaylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate asn increases.Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function isconvergent, its sum is thelimit of theinfinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function isanalytic at a pointx if it is equal to the sum of its Taylor series in someopen interval (oropen disk in thecomplex plane) containingx. This implies that the function is analytic at every point of the interval (or disk). (Full article...)
Image 8
Thehypotenusec of a right triangle with sidesa andb is the Pythagorean sum ofa andb.

Inmathematics,Pythagorean addition is abinary operation on thereal numbers that computes the length of thehypotenuse of aright triangle, given its two sides. Like the more familiar addition and multiplication operations ofarithmetic, it is bothassociative andcommutative.

This operation can be used in the conversion ofCartesian coordinates topolar coordinates, and in the calculation ofEuclidean distance. It also provides a simple notation and terminology for thediameter of acuboid, theenergy-momentum relation inphysics, and the overall noise from independent sources of noise. In its applications tosignal processing andpropagation ofmeasurement uncertainty, the same operation is also calledaddition in quadrature. A scaled version of this operation gives thequadratic mean orroot mean square. (Full article...)
Image 9
The state of avibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinctovertones is given by the projection of the point onto thecoordinate axes in the space.

Inmathematics, aHilbert space is areal orcomplex inner product space that is also acomplete metric space with respect to the metric induced by the inner product. It generalizes the notion ofEuclidean space, toinfinite dimensions. Theinner product, which is the analog of thedot product fromvector calculus, allows lengths and angles to be defined. Furthermore,completeness means that there are enoughlimits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of aBanach space.

Hilbert spaces were studied beginning in the first decade of the 20th century byDavid Hilbert,Erhard Schmidt, andFrigyes Riesz. They are indispensable tools in the theories ofpartial differential equations,quantum mechanics,Fourier analysis (which includes applications tosignal processing andheat transfer), andergodic theory (which forms the mathematical underpinning ofthermodynamics).John von Neumann coined the termHilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era forfunctional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces includespaces of square-integrable functions,spaces of sequences,Sobolev spaces consisting ofgeneralized functions, andHardy spaces ofholomorphic functions. (Full article...)
Image 10

A kite, showing its pairs of equal-length sides and its inscribed circle.

InEuclidean geometry, akite is aquadrilateral withreflection symmetry across adiagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known asdeltoids, but the worddeltoid may also refer to adeltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called adart, particularly if it is not convex.

Every kite is anorthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, atangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential. They include as special cases theright kites, with two opposite right angles; therhombi, with two diagonal axes of symmetry; and thesquares, which are also special cases of both right kites and rhombi. (Full article...)
Image 11
A spiral staircase in theCathedral of St. John the Divine. Severalhelical curves in the staircase project to hyperbolic spirals in its photograph.

Ahyperbolic spiral is a type ofspiral with apitch angle that increases with distance from its center, unlike the constant angles oflogarithmic spirals or decreasing angles ofArchimedean spirals. As this curve widens, it approaches anasymptotic line. It can be found in the view up aspiral staircase and the starting arrangement of certain footraces, and is used to modelspiral galaxies andarchitectural volutes.

As aplane curve, a hyperbolic spiral can be described inpolar coordinates $(r,\varphi )$ by the equation
$r={\frac {a}{\varphi }},$
for an arbitrary choice of thescale factor $a . {\displaystyle a.}$ (Full article...)
Image 12

Walkerc. 1885

Francis Amasa Walker (July 2, 1840 – January 5, 1897) was an Americaneconomist,statistician,journalist,educator,academic administrator, and an officer in theUnion Army. As a prolific author and the third president of theMassachusetts Institute of Technology (MIT) until his death in 1897, Walker was a leading political economist and advocate for a policy ofbimetallism by international agreement. His scholarly contributions are widely recognized as having broadened, liberalized, and modernized economic and statistical theory with contributions towages,wealth distribution,money, andsocial economics, and he is credited with developing the residual theory of wage distribution.

Walker was born into a prominent Boston family, the son of the economist and politicianAmasa Walker, and he graduated fromAmherst College at the age of 20. He received a commission to join the15th Massachusetts Infantry and quickly rose through the ranks as an assistantadjutant general. Walker fought in thePeninsula,Bristoe,Overland, andRichmond-Petersburg Campaigns before being captured by Confederate forces and held at the infamousLibby Prison. In July 1866, he was awarded the honorary grade ofbrevet brigadier general United States Volunteers, to rank from March 13, 1865, when he was 24 years old. (Full article...)

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... that despite a mathematical model deeming theice cream bar flavourGoody Goody Gum Drops impossible, it was still created?
... that the symbol forequality in mathematics was not used for 61 years after its introduction, and was later popularized by Isaac Newton?
... that the BritishNational Hospital Service Reserve trained volunteers to carry out first aid in the aftermath of a nuclear or chemical attack?
... that Olympic historians were unconvinced by speculation that anunknown boy coxswain grew up to bea renowned Georgian mathematician?
... thatFathimath Dheema Ali is the first Olympic qualifier from the Maldives?
... that Leonardo da Vinci invented a device to solveAlhazen's problem, instead of finding a mathematical solution?
... thatpeople in Madagascar performalgebra on tree seeds in order to tell the future?
... thatLivingstone Luboobi claimed that he chose to teach himself double mathematics atA-level because there was no teacher available?

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...that it is impossible totrisect a general angle using only aruler and a compass?
...that in a group of 23 people, there is a more than 50% chance that two peopleshare a birthday?
...that the 1966 publication disprovingEuler's sum of powers conjecture, proposed nearly 200 years earlier, consisted of only two sentences?
...thehyperbolic trigonometric functions of thenatural logarithm can be represented byrational algebraic fractions?
... that economists blamemarket failures onnon-convexity?
... that, according to thepizza theorem, acircular pizza that is sliced off-center into eight equal-angled wedges can still bedivided equally between two people?
... that theclique problem of programming a computer to findcomplete subgraphs in anundirected graph was first studied as a way to find groups of people who all know each other insocial networks?

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Banach–Tarski paradox
Image credit:Benjamin D. Esham

TheBanach–Tarski paradox is atheorem inset-theoretic geometry which states that a solidball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yieldtwo identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball) — solid in the sense of thecontinuum — either one can be reassembled into the other. This is often stated colloquially as "a pea can be chopped up and reassembled into the Sun". (Full article...)

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