Portal:Mathematics

The Mathematics Portal

Euclid

Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

Featured articles –

  Featured articles are displayed here, which represent some of the best content on English Wikipedia.

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  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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  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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  Plots of logarithm functions, with three commonly used bases. The special points log_b b = 1 are indicated by dotted lines, and all curves intersect in log_b 1 = 0.

  
  
  In mathematics, the logarithm to base b is the inverse function of exponentiation with base b. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 10³, the logarithm base $10$ of 1000 is 3, or log₁₀ (1000) = 3. The logarithm of x to base b is denoted as log_b (x), or without parentheses, log_b x. When the base is clear from the context or is irrelevant it is sometimes written log x.
  
  The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base 2 and is frequently used in computer science. (Full article...)
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  Archimedes Thoughtful
  by Domenico Fetti (1620)

  
  Archimedes of Syracuse (/ˌɑːrkɪˈmiːdiːz/ AR-kim-EE-deez; c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.
  
  Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy known as Archimedes' principle. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. (Full article...)
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  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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  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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  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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  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 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 which opposes all forms of technology, rejecting leftism, and advocating a nature-centered form of anarchism. (Full article...)
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  Portrait by Jakob Emanuel Handmann, 1753

  
  Leonhard Euler (/ˈɔɪlər/ OY-lər; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleːɔnhart ˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.
  
  Euler is regarded as one of the greatest, most prolific mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. (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. (Full article...)
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  In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
  
  The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. (Full article...)
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  Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is known for the work he did 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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  General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations.
  
  Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data. (Full article...)

Selected image –

animation showing what looks like a smaller inner cube with corners connected to those of a larger outer cube; the smaller cube passes through one face of the larger cube and becomes larger as the larger cube becomes smaller; eventually the smaller and larger cubes have switched positions and the animation repeats

Tesseract

Credit: User:JasonHise

A three-dimensional projection of a tesseract performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom. Also called an 8-cell or octachoron, a tesseract is the four-dimensional analog of the cube (i.e., a 4-D hypercube, or 4-cube), where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. The tesseract is to the cube as the cube is to the square. Tesseracts and other polytopes can be used as the basis for the network topology when linking multiple processors in parallel computing.

Good articles –

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

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  Title page of the second (1523) edition

  
  Summa de arithmetica, geometria, proportioni et proportionalita (Summary of arithmetic, geometry, proportions and proportionality) is a book on mathematics written by Luca Pacioli and first published in 1494. It contains a comprehensive summary of Renaissance mathematics, including practical arithmetic, basic algebra, basic geometry and accounting, written for use as a textbook and reference work.
  
  Written in vernacular Italian, the Summa is the first printed work on algebra, and it contains the first published description of the double-entry bookkeeping system. It set a new standard for writing and argumentation about algebra, and its impact upon the subsequent development and standardization of professional accounting methods was so great that Pacioli is sometimes referred to as the "father of accounting". (Full article...)
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  In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares.

  
  In geometry, Keller's conjecture is the conjecture that in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other. For instance, in any tiling of the plane by identical squares, some two squares must share an entire edge, as they do in the illustration.
  
  This conjecture was introduced by Ott-Heinrich Keller (1930), after whom it is named. A breakthrough by Lagarias and Shor (1992) showed that it is false in ten or more dimensions, and after subsequent refinements, it is now known to be true in spaces of dimension at most seven and false in all higher dimensions. The proofs of these results use a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. (Full article...)
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  Two right triangles with the two legs of the top one equal to the leg and hypotenuse of the bottom one. For these lengths, $a^{2}$, $b^{2}$, and $c^{2}$ form an arithmetic progression separated by a gap of $d^{2}$. It is not possible for all four lengths $a$, $b$, $c$, and $d$ to be integers.

  
  Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has many equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci. In its geometric forms, it states:

  • A right triangle in the Euclidean plane for which all three side lengths are rational numbers cannot have an area that is the square of a rational number. The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square.
  • A right triangle and a square with equal areas cannot have all sides commensurate with each other.
  • There do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle.

  
  More abstractly, as a result about Diophantine equations (integer or rational-number solutions to polynomial equations), it is equivalent to the statements that:

  • If three square numbers form an arithmetic progression, then the gap between consecutive numbers in the progression (called a congruum) cannot itself be square.
  • The only rational points on the elliptic curve $y^{2}=x(x-1)(x+1)$ are the three trivial points with $x\in \{-1,0,1\}$ and $y=0$.
  • The quartic equation $x^{4}-y^{4}=z^{2}$ has no nonzero integer solution.

  
  An immediate consequence of the last of these formulations is that Fermat's Last Theorem is true in the special case that its exponent is 4. (Full article...)
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  16 polygonalizations of a set of six points

  
  In computational geometry, a polygonalization of a finite set of points in the Euclidean plane is a simple polygon with the given points as its vertices. A polygonalization may also be called a polygonization, simple polygonalization, Hamiltonian polygon, non-crossing Hamiltonian cycle, or crossing-free straight-edge spanning cycle.
  
  Every point set that does not lie on a single line has at least one polygonalization, which can be found in polynomial time. For points in convex position, there is only one, but for some other point sets there can be exponentially many. Finding an optimal polygonalization under several natural optimization criteria is a hard problem, including as a special case the travelling salesman problem. The complexity of counting all polygonalizations remains unknown. (Full article...)
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  Huygens by Caspar Netscher (1671), Museum Boerhaave, Leiden

  
  Christiaan Huygens, Lord of Zeelhem, FRS (/ˈhaɪɡənz/ HY-gənz, US also /ˈhɔɪɡənz/ HOY-gənz; Dutch: [ˈkrɪstijaːn ˈɦœyɣə(n)s] ; also spelled Huyghens; Latin: Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution. In physics, Huygens made seminal contributions to optics and mechanics, while as an astronomer he studied the rings of Saturn and discovered its largest moon, Titan. As an engineer and inventor, he improved the design of telescopes and invented the pendulum clock, the most accurate timekeeper for almost 300 years. A talented mathematician and physicist, his works contain the first idealization of a physical problem by a set of mathematical parameters, and the first mathematical and mechanistic explanation of an unobservable physical phenomenon.
  
  Huygens first identified the correct laws of elastic collision in his work De Motu Corporum ex Percussione, completed in 1656 but published posthumously in 1703. In 1659, Huygens derived geometrically the formula in classical mechanics for the centrifugal force in his work De vi Centrifuga, a decade before Newton. In optics, he is best known for his wave theory of light, which he described in his Traité de la Lumière (1690). His theory of light was initially rejected in favour of Newton's corpuscular theory of light, until Augustin-Jean Fresnel adapted Huygens's principle to give a complete explanation of the rectilinear propagation and diffraction effects of light in 1821. Today this principle is known as the Huygens–Fresnel principle. (Full article...)
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  Three of the ordinary lines in a 4 × 4 grid of points

  
  The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.
  
  A line that contains exactly two of a set of points is known as an ordinary line. Another way of stating the theorem is that every finite set of points that is not collinear has an ordinary line. According to a strengthening of the theorem, every finite point set (not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of $n$ points in time $O(n\log n)$. (Full article...)
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  An anti­parallelogram

  
  
  In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, each pair of sides is antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also called contraparallelograms or crossed parallelograms.
  
  Antiparallelograms occur as the vertex figures of certain nonconvex uniform polyhedra. In the theory of four-bar linkages, the linkages with the form of an antiparallelogram are also called butterfly linkages or bow-tie linkages, and are used in the design of non-circular gears. In celestial mechanics, they occur in certain families of solutions to the 4-body problem. (Full article...)
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  The Petersen graph is the smallest snark.

  
  In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three colors. In order to avoid trivial cases, snarks are often restricted to have additional requirements on their connectivity and on the length of their cycles. Infinitely many snarks exist.
  
  One of the equivalent forms of the four color theorem is that every snark is a non-planar graph. Research on snarks originated in Peter G. Tait's work on the four color theorem in 1880, but their name is much newer, given to them by Martin Gardner in 1976. Beyond coloring, snarks also have connections to other hard problems in graph theory: writing in the Electronic Journal of Combinatorics, Miroslav Chladný and Martin Škoviera state thatAs well as the problems they mention, W. T. Tutte's snark conjecture concerns the existence of Petersen graphs as graph minors of snarks; its proof has been long announced but remains unpublished, and would settle a special case of the existence of nowhere zero 4-flows. (Full article...)
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  In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences of mathematical logic. There are several variations in the types of logical operation that can be used in these sentences. The first-order logic of graphs concerns sentences in which the variables and predicates concern individual vertices and edges of a graph, while monadic second-order graph logic allows quantification over sets of vertices or edges. Logics based on least fixed point operators allow more general predicates over tuples of vertices, but these predicates can only be constructed through fixed-point operators, restricting their power.
  
  A sentence $S$ may be true for some graphs, and false for others; a graph $G$ is said to model $S$, written $G\models S$, if $S$ is true of the vertices and adjacency relation of $G$. The algorithmic problem of model checking concerns testing whether a given graph models a given sentence. The algorithmic problem of satisfiability concerns testing whether there exists a graph that models a given sentence.
  Although both model checking and satisfiability are hard in general, several major algorithmic meta-theorems show that properties expressed in this way can be tested efficiently for important classes of graphs. (Full article...)
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  The logo of the International Mathematical Olympiad

  
  
  The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. It is "the most prestigious" mathematical competition in the world. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. More than 100 countries participate. Each country sends a team of up to six students, plus one team leader, one deputy leader, and observers.
  
  The content ranges from extremely difficult algebra and pre-calculus problems to problems in branches of mathematics not conventionally covered in secondary or high school and often not at university level either, such as projective and complex geometry, functional equations, combinatorics, and well-grounded number theory, of which extensive knowledge of theorems is required. Calculus, though allowed in solutions, is never required, as there is a principle that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge. Supporters of this principle claim that this allows more universality and creates an incentive to find elegant, deceptively simple-looking problems which nevertheless require a certain level of ingenuity, often times a great deal of ingenuity to net all points for a given IMO problem. (Full article...)
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  As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.

  
  In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives 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 after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
  
  The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite 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 is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk). (Full article...)
• 
  In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with $c$ colors, the same is true for the whole graph. The theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős (1951), after whom it is named.
  
  The De Bruijn–Erdős theorem has several different proofs, all depending in some way on the axiom of choice. Its applications include extending the four-color theorem and Dilworth's theorem from finite graphs and partially ordered sets to infinite ones, and reducing the Hadwiger–Nelson problem on the chromatic number of the plane to a problem about finite graphs. It may be generalized from finite numbers of colors to sets of colors whose cardinality is a strongly compact cardinal. (Full article...)

Did you know (auto-generated) –

• ... that mathematician Mathias Metternich was one of the founders of the Jacobin club of the Republic of Mainz?
• ... that Catechumen, a Christian first-person shooter, was funded only in the aftermath of the Columbine High School massacre?
• ... that in the aftermath of the American Civil War, the only Black-led organization providing teachers to formerly enslaved people was the African Civilization Society?
• ... that the music of math rock band Jyocho has been alternatively described as akin to "madness" or "contemplative and melancholy"?
• ... that Green Day's "Wake Me Up When September Ends" became closely associated with the aftermath of Hurricane Katrina?
• ... that circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?
• ... that despite published scholarship to the contrary, Andrew Planta neither received a doctorate nor taught mathematics at Erlangen?
• ... that people in Madagascar perform algebra on tree seeds in order to tell the future?

More did you know –

Did you know...

• ...that it is impossible to trisect a general angle using only a ruler and a compass?
• ...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?
• ...that the 1966 publication disproving Euler's sum of powers conjecture, proposed nearly 200 years earlier, consisted of only two sentences?
• ...the hyperbolic trigonometric functions of the natural logarithm can be represented by rational algebraic fractions?
• ... that economists blame market failures on non-convexity?
• ... that, according to the pizza theorem, a circular pizza that is sliced off-center into eight equal-angled wedges can still be divided equally between two people?
• ... that the clique problem of programming a computer to find complete subgraphs in an undirected graph was first studied as a way to find groups of people who all know each other in social networks?

Selected article –

Fractals arise in surprising places, in this case, the famous Collatz conjecture in number theory. Image credit: Pokipsy76

A fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole". The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured".

A fractal as a geometric object generally has the following features:

• It has a fine structure at arbitrarily small scales.
• It is too irregular to be easily described in traditional Euclidean geometric language.
• It is self-similar (at least approximately or stochastically).
• It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
• It has a simple and recursive definition.

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics. Fractals, when zoomed in, will keep showing more and more of itself, and it keeps going for infinity. (Full article...)

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