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

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Image 1

Noether c. 1900–1910

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...)
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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 $1000 = 10 3$ , the logarithm base $10$ of $1000$ is $3$ , or $log 10 (1000) = 3$ . The logarithm of $x$ to base $b$ is denoted as $log b (x)$ , or without parentheses, $log b x$ . When the base is clear from the context or is irrelevant it is sometimes written $log x$ .

The logarithm base $10$ is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number $e \approx 2.718$ as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base $2$ and is frequently used in computer science. (Full article...)
$Image 3 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, 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, 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 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. (Full article...)$

Image 3
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, 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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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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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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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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Damage from Hurricane Katrina in 2005. Actuaries need to estimate long-term levels of such damage in order to accurately price property insurance, set appropriate reserves, and design appropriate reinsurance and capital management strategies.

An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require asset management, liability management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms. The name of the corresponding academic discipline is actuarial science.

While the concept of insurance dates to antiquity, the concepts needed to scientifically measure and mitigate risks have their origins in the 17th century studies of probability and annuities. Actuaries of the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems to design programs that manage risk, by determining if the implementation of strategies proposed for mitigating potential risks, does not exceed the expected cost of those risks actualized. The steps needed to become an actuary, including education and licensing, are specific to a given country, with various additional requirements applied by regional administrative units; however, almost all processes impart universal principles of risk assessment, statistical analysis, and risk mitigation, involving rigorously structured training and examination schedules, taking many years to complete. (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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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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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...)
$Image 11 The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as '"`UNIQ--postMath-00000005-QINU`"' are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. (Full article...)$

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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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Elementary algebra studies which values solve equations formed using arithmetical operations.

Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of statements within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.

Elementary algebra is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. (Full article...)
Image 13

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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The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... The graph is situated with positive integers to the right and negative integers to the left.

In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
$\sum _{n=1}^{m}n(-1)^{n-1}.$

The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:
$1-2+3-4+\cdots ={\frac {1}{4}}.$ (Full article...)
Image 15

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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Gamma function

Credit: TakuyaMurata (uploader)

This is a hand-drawn graph of the absolute value (or modulus) of the gamma function on the complex plane, as published in the 1909 book Tables of Higher Functions, by Eugene Jahnke and Fritz Emde. Such three-dimensional graphs of complicated functions were rare before the advent of high-resolution computer graphics (even today, tables of values are used in many contexts to look up function values instead of consulting graphs directly). Published even before applications for the complex gamma function were discovered in theoretical physics in the 1930s, Jahnke and Emde's graph "acquired an almost iconic status", according to physicist Michael Berry. See a similar computer-generated image for comparison. When restricted to positive integers, the gamma function generates the factorials through the relation

Γ(n) = (n - 1)!

, which is the product of all positive integers from

n - 1

down to 1 (

0!

is defined to be equal to 1). For real and complex numbers, the function is defined by the improper integral

\textstyle \Gamma (t)=\int _{0}^{\infty }x^{t-1}e^{-x}\,dx

. This integral diverges when t is a negative integer, which is causing the spikes in the left half of the graph (these are simple poles of the function, where its values increase to infinity, analogous to asymptotes in two-dimensional graphs). The gamma function has applications in quantum physics, astrophysics, and fluid dynamics, as well as in number theory and probability.

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Image 1
The convex hull of the red set is the blue and red convex set.

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

Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points.
The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. They can be solved in time $O(n\log n)$ for two or three dimensional point sets, and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions. (Full article...)
Image 2
Ronald Paul "Ron" Fedkiw (born February 27, 1968) is a full professor in the Stanford University department of computer science and a leading researcher in the field of computer graphics, focusing on topics relating to physically based simulation of natural phenomena and machine learning. His techniques have been employed in many motion pictures. He has earned recognition at the 80th Academy Awards and the 87th Academy Awards as well as from the National Academy of Sciences.

His first Academy Award was awarded for developing techniques that enabled many technically sophisticated adaptations including the visual effects in 21st century movies in the Star Wars, Harry Potter, Terminator, and Pirates of the Caribbean franchises. Fedkiw has designed a platform that has been used to create many of the movie world's most advanced special effects since it was first used on the T-X character in Terminator 3: Rise of the Machines. His second Academy Award was awarded for computer graphics techniques for special effects for large scale destruction. Although he has won an Oscar for his work, he does not design the visual effects that use his technique. Instead, he has developed a system that other award-winning technicians and engineers have used to create visual effects for some of the world's most expensive and highest-grossing movies. (Full article...)
Image 3
Two greedy colorings of the same crown graph using different vertex orders. The right example generalises to 2-colorable graphs with $n$ vertices, where the greedy algorithm expends $n /2$ colors.

In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not, in general, use the minimum number of colors possible.

Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering. There always exists an ordering that produces an optimal coloring, but although such orderings can be found for many special classes of graphs, they are hard to find in general. Commonly used strategies for vertex ordering involve placing higher-degree vertices earlier than lower-degree vertices, or choosing vertices with fewer available colors in preference to vertices that are less constrained. (Full article...)
Image 4
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...)
Image 5
A polyhedron and its midsphere. The red circles are the boundaries of spherical caps within which the surface of the sphere can be seen from each vertex.

In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals (Catalan solids) all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.

When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their corresponding circles in this circle packing. (Full article...)
Image 6
A simplicial line arrangement (left) and a simple line arrangement (right).

In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons, the cells of the arrangment, line segments and rays, the edges of the arrangement, and points where two or more lines cross, the vertices of the arrangement. When considered in the projective plane rather than in the Euclidean plane, every two lines cross, and an arrangement is the projective dual to a finite set of points. Arrangements of lines have also been considered in the hyperbolic plane, and generalized to pseudolines, curves that have similar topological properties to lines. The initial study of arrangements has been attributed to an 1826 paper by Jakob Steiner.

An arrangement is said to be simple when at most two lines cross at each vertex, and simplicial when all cells are triangles (including the unbounded cells, as subsets of the projective plane). There are three known infinite families of simplicial arrangements, as well as many sporadic simplicial arrangements that do not fit into any known family. Arrangements have also been considered for infinite but locally finite systems of lines. Certain infinite arrangements of parallel lines can form simplicial arrangements, and one way of constructing the aperiodic Penrose tiling involves finding the dual graph of an arrangement of lines forming five parallel subsets. (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...)
Image 8
Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.

In mathematical analysis, the Dirac delta function (or $δ$ distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as

$\delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}$ (Full article...)
Image 9
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. For instance, Minkowski's question-mark function produces an isomorphism (a one-to-one order-preserving correspondence) between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals.

The theorem is named after Georg Cantor, who first published it in 1895, using it to characterize the (uncountable) ordering on the real numbers. It can be proved by a back-and-forth method that is also sometimes attributed to Cantor but was actually published later, by Felix Hausdorff. The same back-and-forth method also proves that countable dense unbounded orders are highly symmetric, and can be applied to other kinds of structures. However, Cantor's original proof only used the "going forth" half of this method. In terms of model theory, the isomorphism theorem can be expressed by saying that the first-order theory of unbounded dense linear orders is countably categorical, meaning that it has only one countable model, up to logical equivalence. (Full article...)
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Title page of De quinque corporibus regularibus

De quinque corporibus regularibus (sometimes called Libellus de quinque corporibus regularibus) is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is a manuscript, in the Latin language; its title means [the little book] on the five regular solids. It is one of three books known to have been written by della Francesca.

Along with the Platonic solids, De quinque corporibus regularibus includes descriptions of five of the thirteen Archimedean solids, and of several other irregular polyhedra coming from architectural applications. It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra, including Luca Pacioli's 1509 Divina proportione (which incorporated without credit an Italian translation of della Francesca's work). (Full article...)
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Bernt Michael Holmboe

Bernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. He was home-tutored from an early age, and was not enrolled in school until 1810. Following a short period at the Royal Frederick University, which included a stint as assistant to Christopher Hansteen, Holmboe was hired as a mathematics teacher at the Christiania Cathedral School in 1818, where he met the future renowned mathematician Niels Henrik Abel. Holmboe's lasting impact on mathematics worldwide has been said to be his tutoring of Abel, both in school and privately. The two became friends and remained so until Abel's early death. Holmboe moved to the Royal Frederick University in 1826, where he worked until his own death in 1850.

Holmboe's significant impact on mathematics in the fledgling Norway was his textbook in two volumes for secondary schools. It was widely used, but faced competition from Christopher Hansteen's alternative offering, sparking what may have been Norway's first debate about school textbooks. (Full article...)
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Huygens by Caspar Netscher (1671), Museum Boerhaave, Leiden

Christiaan Huygens, Lord of Zeelhem, (/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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...that the sum of the first n odd numbers divided by the sum of the next n odd numbers is always equal to one third?
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e is the unique number such that the slope of y=e^x (blue curve) is exactly 1 when x=0 (illustrated by the red tangent line). For comparison, the curves y=2^x (dotted curve) and y=4^x (dashed curve) are shown.
Image credit: Dick Lyon

The mathematical constant e is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions. One is given in the caption of the image to the right, and three more are:

The sum of the infinite series
${\begin{aligned}e&=\sum _{n=0}^{\infty }{\frac {1}{n!}}\\&={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots \\\end{aligned}}$

where n! is the factorial of n, and 0! is defined to be 1 by convention.
The global maximizer of the function
$f(x)=x^{1 \over x}.$
The limit:
$e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

The number e is also the base of the natural logarithm. Since e is transcendental, and therefore irrational, its value can not be given exactly. The numerical value of e truncated to 20 decimal places is 2.71828 18284 59045 23536. (Full article...)

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