Portal:Mathematics
Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)
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General relativity, also known as the general theory of relativity and 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 matter and radiation are present. 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...) -
Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because the remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300).
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).
It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules,
and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...) -
Stylistic impression of the number, representing how its decimals go on infinitely
In mathematics, 0.999... (also written as 0.9, 0. or 0.(9)) is a notation for the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal is a numeral that represents the smallest number no less than every number in the sequence ; that is, the supremum of this sequence. This number is equal to 1. In other words, "0.999..." is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent exactly the same number.
There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined. (Full article...) -
Zhang Heng (Chinese: 張衡; AD 78–139), formerly romanized Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, mathematician, seismologist, hydraulic engineer, inventor, geographer, cartographer, ethnographer, artist, poet, philosopher, politician, and literary scholar.
Zhang Heng began his career as a minor civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages, and then Palace Attendant at the imperial court. His uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator of Hejian Kingdom in present-day Hebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139. (Full article...) -
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...) -
Marian Adam Rejewski (Polish: [ˈmarjan rɛˈjɛfskʲi] ⓘ; 16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in late 1932 reconstructed the sight-unseen German military Enigma cipher machine, aided by limited documents obtained by French military intelligence.
Over the next nearly seven years, Rejewski and fellow mathematician-cryptologists Jerzy Różycki and Henryk Zygalski, working at the Polish General Staff's Cipher Bureau, developed techniques and equipment for decrypting the Enigma ciphers, even as the Germans introduced modifications to their Enigma machines and encryption procedures. Rejewski's contributions included the cryptologic card catalog and the cryptologic bomb. (Full article...) -
Plots of logarithm functions, with three commonly used bases. The special points logb b = 1 are indicated by dotted lines, and all curves intersect in logb 1 = 0.
In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.
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...) -
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": it 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...) -
Amalie Emmy Noether (US: /ˈnʌtər/, UK: /ˈnɜːtə/; German: [ˈnøːtɐ]; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations but instead studied mathematics at the University of Erlangen, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent. (Full article...) -
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 held to be one of the greatest 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." Euler is also widely considered to be the most prolific; his 866 publications as well as his correspondences are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quarto volumes. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. (Full article...) -
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...) -
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...) -
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...) -
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...) -
Theodore John Kaczynski (/kəˈzɪnski/ ⓘ kə-ZIN-skee; May 22, 1942 – June 10, 2023), also known as the Unabomber (/ˈjuːnəbɒmər/ ⓘ YOO-nə-bom-ər), was an American mathematician and domestic terrorist. He was a mathematics prodigy, but abandoned his academic career in 1969 to pursue a primitive lifestyle.
Between 1978 and 1995, Kaczynski murdered three individuals and injured 23 others in a nationwide mail bombing campaign against people he believed to be advancing modern technology and the destruction of the natural environment. He authored Industrial Society and Its Future, a 35,000-word manifesto and social critique opposing industrialization, rejecting leftism, and advocating for a nature-centered form of anarchism. (Full article...)
Credit: User:LucasVB
An animation showing how an obliquely cut torus reveals a pair of intersecting circles known as Villarceau circles, named after the French astronomer and mathematician Yvon Villarceau. These are two of the four circles that can be drawn through any given point on the torus. (The other two are oriented horizontally and vertically, and are the analogs of lines of latitude and longitude drawn through the given point.) The circles have no known practical application and seem to be merely a curious characteristic of the torus. However, Villarceau circles appear as the fibers in the Hopf fibration of the 3-sphere over the ordinary 2-sphere, and the Hopf fibration itself has interesting connections to fluid dynamics, particle physics, and quantum theory.
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Fleiss' kappa (named after Joseph L. Fleiss) is a statistical measure for assessing the reliability of agreement between a fixed number of raters when assigning categorical ratings to a number of items or classifying items. This contrasts with other kappas such as Cohen's kappa, which only work when assessing the agreement between not more than two raters or the intra-rater reliability (for one appraiser versus themself). The measure calculates the degree of agreement in classification over that which would be expected by chance.
Fleiss' kappa can be used with binary or nominal-scale. It can also be applied to ordinal data (ranked data): the MiniTab online documentation gives an example. However, this document notes: "When you have ordinal ratings, such as defect severity ratings on a scale of 1–5, Kendall's coefficients, which account for ordering, are usually more appropriate statistics to determine association than kappa alone." Keep in mind however, that Kendall rank coefficients are only appropriate for rank data. (Full article...) -
Hypatia (born c. 350–370; died 415 AD) was a Neoplatonist philosopher, astronomer, and mathematician who lived in Alexandria, Egypt, then part of the Eastern Roman Empire. She was a prominent thinker in Alexandria where she taught philosophy and astronomy. Although preceded by Pandrosion, another Alexandrian female mathematician, she is the first female mathematician whose life is reasonably well recorded. Hypatia was renowned in her own lifetime as a great teacher and a wise counselor. She wrote a commentary on Diophantus's thirteen-volume Arithmetica, which may survive in part, having been interpolated into Diophantus's original text, and another commentary on Apollonius of Perga's treatise on conic sections, which has not survived. Many modern scholars also believe that Hypatia may have edited the surviving text of Ptolemy's Almagest, based on the title of her father Theon's commentary on Book III of the Almagest.
Hypatia constructed astrolabes and hydrometers, but did not invent either of these, which were both in use long before she was born. She was tolerant toward Christians and taught many Christian students, including Synesius, the future bishop of Ptolemais. Ancient sources record that Hypatia was widely beloved by pagans and Christians alike and that she established great influence with the political elite in Alexandria. Toward the end of her life, Hypatia advised Orestes, the Roman prefect of Alexandria, who was in the midst of a political feud with Cyril, the bishop of Alexandria. Rumors spread accusing her of preventing Orestes from reconciling with Cyril and, in March 415 AD, she was murdered by a mob of Christians led by a lector named Peter. (Full article...) -
Wave functions of the electron in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations. The brighter areas represent a higher probability of finding the electron.
Quantum mechanics is a fundamental theory in physics that describes the behavior of nature at and below the scale of atoms. It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.
Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic/microscopic) scale. (Full article...) -
Maurits Cornelis Escher (Dutch pronunciation: [ˈmʌurɪt͡s kɔrˈneːlɪs ˈɛʃər]; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithographs, and mezzotints, many of which were inspired 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 including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, and Donald Coxeter, and the crystallographer Friedrich Haag, and conducted his own research into tessellation. (Full article...) -
Walton in Pomona College's Walker Lounge c. 1957
Jean Brosius Walton (March 6, 1914 – July 5, 2006) was an American academic administrator and women's studies scholar. She spent the bulk of her career at Pomona College in Claremont, California.
Born to a Pennsylvania Quaker family, Walton grew up at George School and studied mathematics at Swarthmore College, Brown University and the University of Pennsylvania. She joined Pomona College in 1949 as the Dean of Women, and was promoted to dean of students in 1969 and vice president for student affairs in 1976, three years before her formal retirement. During her tenure, she advocated for women's education, engaged with student protests against the Vietnam War, oversaw reform of residential life policies to eliminate parietal rules, and co-founded the Claremont Colleges' Intercollegiate Women's Studies Program. She earned widespread recognition for her work and was praised by colleagues for her independent and dignified personality. (Full article...) -
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential 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 the right kites, with two opposite right angles; the rhombi, with two diagonal axes of symmetry; and the squares, which are also special cases of both right kites and rhombi. (Full article...) -
Arnold Ephraim Ross (August 24, 1906 – September 25, 2002) was a mathematician and educator who founded the Ross Mathematics Program, a number theory summer program for gifted high school students. He was born in Chicago, but spent his youth in Odesa, Ukraine, where he studied with Samuil Shatunovsky. Ross returned to Chicago and enrolled in University of Chicago graduate coursework under E. H. Moore, despite his lack of formal academic training. He received his Ph.D. and married his wife, Bee, in 1931.
Ross taught at several institutions including St. Louis University before becoming chair of University of Notre Dame's mathematics department in 1946. He started a teacher training program in mathematics that evolved into the Ross Mathematics Program in 1957 with the addition of high school students. The program moved with him to Ohio State University when he became their department chair in 1963. Though forced to retire in 1976, Ross ran the summer program until 2000. He had worked with over 2,000 students during more than forty summers. (Full article...) -
Measuring the width of a Reuleaux triangle as the distance between parallel supporting lines. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width.
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve.
Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's theorem, the body's perimeter is exactly π times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest possible area for its width and the circle the largest. Every superset of a body of constant width includes pairs of points that are farther apart than the width, and every curve of constant width includes at least six points of extreme curvature. Although the Reuleaux triangle is not smooth, curves of constant width can always be approximated arbitrarily closely by smooth curves of the same constant width. (Full article...) -
James Clerk Maxwell FRSE FRS (13 June 1831 – 5 November 1879) was a Scottish physicist with broad interests who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" where the first one had been realised by Isaac Newton.
With the publication of "A Dynamical Theory of the Electromagnetic Field" in 1865, Maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light. He proposed that light is an undulation in the same medium that is the cause of electric and magnetic phenomena. The unification of light and electrical phenomena led to his prediction of the existence of radio waves. Maxwell is also regarded as a founder of the modern field of electrical engineering. (Full article...) -
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...) -
In geometry, the complete or final stellation of the icosahedron[1] is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, simple, and star polyhedron. (Full article...) -
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...)
- ... that circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?
- ... that record-setting airplane spinner Catherine Cavagnaro is also a professional mathematician?
- ... that the 1914 Lubin vault fire in Philadelphia destroyed several thousand unique early silent films?
- ... that mathematician Daniel Larsen was the youngest contributor to the New York Times crossword puzzle?
- ... that according to one critic, the math rock album Cryptooology by Yowie "sounds like an explosion in a Slinky factory"?
- ... that owner Matthew Benham influenced both Brentford FC in the UK and FC Midtjylland in Denmark to use mathematical modelling to recruit undervalued football players?
- ... that Arithmetic was the first mathematics text book written in the Russian language?
- ... that despite a mathematical model deeming the ice cream bar flavour Goody Goody Gum Drops impossible, it was still created?
- ...that in graph theory, a pseudoforest can contain trees and pseudotrees, but cannot contain any butterflies, diamonds, handcuffs, or bicycles?
- ...that it is not possible to configure two mutually inscribed quadrilaterals in the Euclidean plane, but the Möbius–Kantor graph describes a solution in the complex projective plane?
- ...that the six permutations of the vector (1,2,3) form a hexagon in 3D space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
- ...that the Rule 184 cellular automaton can simultaneously model the behavior of cars moving in traffic, the accumulation of particles on a surface, and particle-antiparticle annihilation reactions?
- ...that a cyclic cellular automaton is a system of simple mathematical rules that can generate complex patterns mixing random chaos, blocks of color, and spirals?
- ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
- ...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
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| In this shear transformation of the Mona Lisa, the central vertical axis (red vector) is unchanged, but the diagonal vector (blue) has changed direction. Hence the red vector is said to be an eigenvector of this particular transformation and the blue vector is not. Image credit: User:Voyajer |
In mathematics, an eigenvector of a transformation is a vector, different from the zero vector, which that transformation simply multiplies by a constant factor, called the eigenvalue of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. The eigenspace for a factor is the set of eigenvectors with that factor as eigenvalue, together with the zero vector.
In the specific case of linear algebra, the eigenvalue problem is this: given an n by n matrix A, what nonzero vectors x in exist, such that Ax is a scalar multiple of x?
The scalar multiple is denoted by the Greek letter λ and is called an eigenvalue of the matrix A, while x is called the eigenvector of A corresponding to λ. These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear situations.
It is common to prefix any natural name for the vector with eigen instead of saying eigenvector. For example, eigenfunction if the eigenvector is a function, eigenmode if the eigenvector is a harmonic mode, eigenstate if the eigenvector is a quantum state, and so on. Similarly for the eigenvalue, e.g. eigenfrequency if the eigenvalue is (or determines) a frequency. (Full article...)
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- ^ Coxeter et al. (1999), p. 30–31; Wenninger (1971), p. 65.