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Las matemáticas son el estudio de la representación y el razonamiento sobre objetos abstractos(como números , puntos , espacios , conjuntos , estructuras y juegos ). Las matemáticas se utilizan en todo el mundo como una herramienta esencial en muchos campos, incluidas las ciencias naturales , la ingeniería , la medicina y las ciencias sociales . Las matemáticas aplicadas , la rama de las matemáticas que se ocupa de la aplicación del conocimiento matemático a otros campos, inspiran y hacen uso de nuevos descubrimientos matemáticos y, a veces, conducen al desarrollo de disciplinas matemáticas completamente nuevas, como la estadística y la teoría de juegos . Los matemáticos también se dedican a las matemáticas puras , o las matemáticas por sí mismas, sin tener ninguna aplicación en mente. No existe una línea clara que separe las matemáticas puras de las aplicadas, y a menudo se descubren aplicaciones prácticas para lo que comenzó como matemáticas puras. ( Artículo completo... )

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Richard Phillips Feynman ( 11 de mayo de 1918 - 15 de febrero de 1988) fue un físico teórico estadounidense . Es más conocido por su trabajo en la formulación de la integral de trayectorias de la mecánica cuántica , la teoría de la electrodinámica cuántica , la física de la superfluidez del helio líquido superenfriadoy en la física de partículas , para la que propuso el modelo partónico . Por sus contribuciones al desarrollo de la electrodinámica cuántica, Feynman recibió el Premio Nobel de Física en 1965 junto con Julian Schwinger y Shin'ichirō Tomonaga .

Feynman desarrolló un esquema de representación pictórica ampliamente utilizado para las expresiones matemáticas que describen el comportamiento de las partículas subatómicas , que más tarde se conocerían como diagramas de Feynman . Durante su vida, Feynman se convirtió en uno de los científicos más conocidos del mundo. En una encuesta realizada en 1999 entre 130 físicos destacados de todo el mundo por la revista británica Physics World , fue clasificado como el séptimo físico más grande de todos los tiempos. ( Artículo completo... )
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Retrato de August Köhler, c. 1910 , según el original de 1627

Johannes Kepler ( / ˈ k ɛ p l ər / ; alemán: [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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Logic studies valid forms of inference like modus ponens.

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work". Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like $\land$ (and) or $\to$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. (Full article...)
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High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of spacetime (blue lines) due to the Sun's mass.

General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.

By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion. (Full article...)
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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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Portrait by Jakob Emanuel Handmann, 1753

Leonhard Euler (/ˈɔɪlər/ OY-lər; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] ^ⓘ, Swiss Standard German: [ˈleɔnhard ˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swiss 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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The number $π$ (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number $π$ appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ${\tfrac {22}{7}}$ are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers. The transcendence of $π$ implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of $π$ appear to be randomly distributed, but no proof of this conjecture has been found.

For thousands of years, mathematicians have attempted to extend their understanding of $π$ , sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of $π$ for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate $π$ with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated $π$ to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for $π$ , based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. (Full article...)
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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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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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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...)
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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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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, that number is 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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In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously. (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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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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Animación del acto de "desenrollar" la circunferencia de un círculo, que ilustra la relación pi (π)

Crédito: John Reid

Pi , representada por la letra griega π , es una constante matemática cuyo valor es la relación entrela circunferencia de un círculo y su diámetro en el espacio euclidiano (es decir, en un plano ); también es la relación entre el área de un círculo y el cuadrado de su radio. (Estos hechos se reflejan en las conocidas fórmulas de la geometría ,

C = π d

A = π r 2

). En esta animación, el círculo tiene un diámetro de 1 unidad, lo que le da una circunferencia de π. El rodamiento muestra que la distancia que un punto del círculo se mueve linealmente en una revolución completa es igual a π. Pi es un número irracional y, por lo tanto, no se puede expresar como la relación de dos números enteros ; como resultado, la expansión decimal de π no es terminal ni periódica. Con 50 decimales, π ≈ 3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510, un valor de precisión suficiente para permitir el cálculo del volumen de una esfera del tamaño de la órbita de Neptuno alrededor del Sol (asumiendo un valor exacto para este radio) con una precisión de 1 angstrom cúbico . Según el teorema de Lindemann-Weierstrass , demostrado por primera vez en 1882, π también es un número trascendental (o no algebraico ), lo que significa que no es la raíz de ningún polinomio distinto de cerocon coeficientes racionales . (Esto implica que no se puede expresar utilizando ninguna expresión algebraica de forma cerrada —y también que es imposible resolver el antiguo problema de la cuadratura del círculo utilizando una construcción con regla y compás ). Tal vez la expresión cerrada no algebraica más simple para π sea

4 arctan 1

, basada en la función tangente inversa (una función trascendental ). También hay muchas series infinitas y algunos productos infinitos que convergen a π o a una función simple de este, como 2/π; uno de ellos es la representación en serie infinita de la expresión tangente inversa que acabamos de mencionar. Tales enfoques iterativos para aproximar π aparecieron por primera vez en la India del siglo XV y luego fueron redescubiertos (quizás no de forma independiente) en la Europa de los siglos XVII y XVIII (junto con varias fracciones continuas ).

representaciones). Aunque estos métodos a menudo sufren de una tasa de convergencia impracticablemente lenta, una serie infinita moderna que converge a 1/π muy rápidamente es dada por el algoritmo de Chudnovsky , publicado por primera vez en 1989; cada término de esta serie da una asombrosa precisión de 14 decimales adicionales. Además de la geometría y la trigonometría , π aparece en muchas otras áreas de las matemáticas, incluyendo la teoría de números , el cálculo y la probabilidad .

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El teorema de Euclides-Euler es un teorema de la teoría de números que relaciona los números perfectos con los primos de Mersenne . Afirma que un número par es perfecto si y solo si tiene la forma $2 p -1 (2 p - 1)$ , donde $2 p - 1$ es un número primo . El teorema recibe su nombre de los matemáticos Euclides y Leonhard Euler , quienes demostraron respectivamente los aspectos "si" y "solo si" del teorema.

Se ha conjeturado que existen infinitos primos de Mersenne. Aunque la verdad de esta conjetura sigue siendo desconocida, es equivalente, por el teorema de Euclides-Euler, a la conjetura de que existen infinitos números perfectos pares. Sin embargo, también se desconoce si existe incluso un solo número perfecto impar. ( Artículo completo... )
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Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Each row of $k$ squares of side length 1/ $k$ has total area 1/ $k$ , and all the squares together exactly cover a larger square with area 1. The bottom row of 47058 squares with side length 1/47058 is too small to see in the figure and is not shown.

In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time.

The initial terms of Sylvester's sequence almost solve this problem, except that the last chosen term equals one plus the product of the others, rather than being a proper divisor. Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each $k\geq 5$ . Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values. (Full article...)
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Einstein in 1921

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who is widely held as one of the most influential scientists. Best known for developing the theory of relativity, Einstein also made important contributions to quantum mechanics. His mass–energy equivalence formula $E = mc 2$ , which arises from special relativity, has been called "the world's most famous equation". He received the 1921 Nobel Prize in Physics.

Born in the German Empire, Einstein moved to Switzerland in 1895, and at the age of seventeen he enrolled in the mathematics and physics teaching diploma program at the Swiss federal polytechnic school. In 1903, he secured a permanent position at the Swiss Patent Office. In 1905, he submitted a successful PhD dissertation to the University of Zurich. In 1914, he moved to Berlin to join the Prussian Academy of Sciences and the Humboldt University of Berlin, becoming director of the Kaiser Wilhelm Institute for Physics in 1917. In 1933, while Einstein was visiting the United States, Adolf Hitler came to power in Germany. Horrified by the Nazi persecution of his fellow Jews, Einstein decided to remain in the US. On the eve of World War II, he endorsed a letter to President Franklin D. Roosevelt alerting him to the potential German nuclear weapons program and recommended that the US begin similar research, though he generally viewed the idea of nuclear weapons with great dismay. (Full article...)
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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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Fredrik Carl Mülertz Størmer (3 September 1874 – 13 August 1957) was a Norwegian mathematician and astrophysicist. In mathematics, he is known for his work in number theory, including the calculation of π and Størmer's theorem on consecutive smooth numbers. In physics, he is known for studying the movement of charged particles in the magnetosphere and the formation of aurorae, and for his book on these subjects, From the Depths of Space to the Heart of the Atom. He worked for many years as a professor of mathematics at the University of Oslo in Norway. A crater on the far side of the Moon is named after him. (Full article...)
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Graham in 1998

Ronald Lewis Graham (October 31, 1935 – July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences.

After graduate study at the University of California, Berkeley, Graham worked for many years at Bell Labs and later at the University of California, San Diego. He did important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness, and many topics in mathematics are named after him. He published six books and about 400 papers, and had nearly 200 co-authors, including many collaborative works with his wife Fan Chung and with Paul Erdős. (Full article...)
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Squaring the circle: the areas of this square and this circle are both equal to $π$ . In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.

Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( $\pi$ ) is a transcendental number.
That is, $\pi$ is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if $\pi$ were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. (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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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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Advanced Placement (AP) Statistics (also known as AP Stats) is a college-level high school statistics course offered in the United States through the College Board's Advanced Placement program. This course is equivalent to a one semester, non-calculus-based introductory college statistics course and is normally offered to sophomores, juniors and seniors in high school.

One of the College Board's more recent additions, the AP Statistics exam was first administered in May 1996 to supplement the AP program's math offerings, which had previously consisted of only AP Calculus AB and BC. In the United States, enrollment in AP Statistics classes has increased at a higher rate than in any other AP class. (Full article...)
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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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Natural patterns form as wind blows sand in the dunes of the Namib Desert. The crescent shaped dunes and the ripples on their surfaces repeat wherever there are suitable conditions.

Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. The Hungarian biologist Aristid Lindenmayer and the French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns. (Full article...)

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... que " Wake Me Up When September Ends " de Green Day quedó estrechamente asociado con las consecuencias del huracán Katrina ?
... que la gente de Madagascar practica álgebra con semillas de árboles para predecir el futuro ?
... que después de que Arquímedes definiera por primera vez las curvas convexas , los matemáticos perdieron el interés en su análisis hasta el siglo XIX, más de dos milenios después?
... que Fathimath Dheema Ali es la primera deportista de Maldivas que se clasifica para los Juegos Olímpicos?
... que después de que las escuelas de Florida prohibieran 54 libros de matemáticas, Chaz Stevens solicitó que también prohibieran la Biblia ?
... ¿que el propietario Matthew Benham influyó tanto en el Brentford FC en el Reino Unido como en el FC Midtjylland en Dinamarca para que utilizaran modelos matemáticos para reclutar jugadores de fútbol infravalorados?
... que la Reserva del Servicio Hospitalario Nacional Británico entrenó a voluntarios para brindar primeros auxilios después de un ataque nuclear o químico?
... que la música de la banda de math rock Jyocho ha sido descrita alternativamente como similar a la "locura" o "contemplativa y melancólica"?

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... que como la dimensión de una hiperesfera tiende a infinito, su " volumen " (contenido) tiende a 0?
...que la primalidad de un número se puede determinar utilizando sólo una única división utilizando el Teorema de Wilson ?
...que la línea que separa el numerador y el denominador de una fracción se llama solidus si se escribe como una línea diagonal o vinculum si se escribe como una línea horizontal?
...que un mono que presiona teclas al azar en un teclado de máquina de escribir durante un tiempo infinito casi con seguridad escribirá las obras completas de William Shakespeare?
... que existen 115.200 soluciones al problema del ménage à permutar seis parejas de mujeres y hombres en una mesa de doce personas, de modo que los hombres y las mujeres se alternen y se sienten lejos de sus parejas?
... que el matemático Paul Erdős llamó a la conjetura de Hadwiger , una generalización aún abierta del problema de los cuatro colores , "uno de los problemas más profundos sin resolver en la teoría de grafos "?
...que las seis permutaciones del vector (1,2,3) forman un hexágono regular en el espacio 3d , las 24 permutaciones de (1,2,3,4) forman un octaedro truncado en cuatro dimensiones , y ambas son ejemplos de permutoedros ?

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Georg Ferdinand Ludwig Philipp Cantor ( San Petersburgo , Rusia, 3 de diciembre de 1845 - Halle , Alemania , 6 de enero de 1918) fue un matemático alemán conocido por ser el creador de la teoría de conjuntos . Cantor estableció la importancia de la correspondencia biunívoca entre conjuntos, definió los conjuntos infinitos y bien ordenados y demostró que los números reales son «más numerosos» que los números naturales . De hecho, el teorema de Cantor implica la existencia de una «infinidad de infinitos». Definió los números cardinales y ordinales , y su aritmética. La obra de Cantor es de gran interés filosófico, hecho del que era muy consciente.

El trabajo de Cantor encontró resistencia por parte de matemáticos contemporáneos como Leopold Kronecker y Henri Poincaré , y más tarde de Hermann Weyl y LEJ Brouwer . Ludwig Wittgenstein planteó objeciones filosóficas . Hoy en día, la gran mayoría de matemáticos que no son ni constructivistas ni finitistas aceptan el trabajo de Cantor sobre los conjuntos transfinitos y la aritmética, reconociéndolo como un importante cambio de paradigma . ( Artículo completo... )