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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
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 2

Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and in particle physics, for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.

Feynman developed a widely used pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World, he was ranked the seventh-greatest physicist of all time. (Full article...)
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Portrait by August Köhler, c. 1910, after 1627 original

Johannes Kepler (/ˈkɛplər/; German: [joˈhanəs ˈkɛplɐ, -nɛs -] ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonice Mundi, and Epitome Astronomiae Copernicanae, influencing among others Isaac Newton, providing one of the foundations for his theory of universal gravitation. The variety and impact of his work made Kepler one of the founders and fathers of modern astronomy, the scientific method, natural and modern science. He has been described as the "father of science fiction" for his novel Somnium.

Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague, and eventually the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He also taught mathematics in Linz, and was an adviser to General Wallenstein.
Additionally, he did fundamental work in the field of optics, being named the father of modern optics, in particular for his Astronomiae pars optica. He also invented an improved version of the refracting telescope, the Keplerian telescope, which became the foundation of the modern refracting telescope, while also improving on the telescope design by Galileo Galilei, who mentioned Kepler's discoveries in his work. (Full article...)
$Image 4 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 4
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...)
Image 5
The manipulations of the Rubik's Cube form the Rubik's Cube group.

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation form an infinite group, which is generated by a single element called ⁠ $1$ ⁠ (these properties characterize the integers in a unique way). (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...)
$Image 7 The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as '"`UNIQ--postMath-00000004-QINU`"' are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. (Full article...)$

Image 7
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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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...)
Image 10
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...)
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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...)
Image 13

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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The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group $S̃ 3$

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections of permutations (rearrangements) of the integers ( $..., -2, -1, 0, 1, 2, ...$ ) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in combinatorics and representation theory.

A finite symmetric group consists of all permutations of a finite set. Each affine symmetric group is an infinite extension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as descents and inversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation. (Full article...)
Image 15

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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Squared triangular number

Credit: Cmglee

Nicomachus's theorem states that the sum of the cubes of the first

n

natural numbers is the square of the sum of the first

n

natural numbers. This result is generalized by Faulhaber's formula, which gives the sum of

p

th powers of the first

n

natural numbers. The special case of Nicomachus's theorem can be proved by mathematical induction, but a more direct proof can be given which is illustrated by a proof without words, pictured here.

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Image 1
The 13 possible strict weak orderings on a set of three elements ${a, b, c$ }

In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the weak orderings on a set of $n$ elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race.

The ordered Bell numbers were studied in the 19th century by Arthur Cayley and William Allen Whitworth. They are named after Eric Temple Bell, who wrote about the Bell numbers, which count the partitions of a set; the ordered Bell numbers count partitions that have been equipped with a total order. Their alternative name, the Fubini numbers, comes from a connection to Guido Fubini and Fubini's theorem on equivalent forms of multiple integrals. Because weak orderings have many names, ordered Bell numbers may also be called by those names, for instance as the numbers of preferential arrangements or the numbers of asymmetric generalized weak orders. (Full article...)
Image 2
Two right triangles with the two legs of the top one equal to the leg and hypotenuse of the bottom one. For these lengths, $a^{2}$ , $b^{2}$ , and $c^{2}$ form an arithmetic progression separated by a gap of $d^{2}$ . It is not possible for all four lengths $a$ , $b$ , $c$ , and $d$ to be integers.

Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has many equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci. In its geometric forms, it states:
- A right triangle in the Euclidean plane for which all three side lengths are rational numbers cannot have an area that is the square of a rational number. The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square.
- A right triangle and a square with equal areas cannot have all sides commensurate with each other.
- There do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle.
More abstractly, as a result about Diophantine equations (integer or rational-number solutions to polynomial equations), it is equivalent to the statements that:
- If three square numbers form an arithmetic progression, then the gap between consecutive numbers in the progression (called a congruum) cannot itself be square.
- The only rational points on the elliptic curve $y^{2}=x(x-1)(x+1)$ are the three trivial points with $x\in \{-1,0,1\}$ and $y=0$ .
- The quartic equation $x^{4}-y^{4}=z^{2}$ has no nonzero integer solution.
An immediate consequence of the last of these formulations is that Fermat's Last Theorem is true in the special case that its exponent is 4. (Full article...)
Image 3
A Doyle spiral of type (8,16) printed in 1911 in Popular Science as an illustration of phyllotaxis. One of its spiral arms is shaded.

In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane in which each circle is surrounded by a ring of six tangent circles. These patterns contain spiral arms formed by circles linked through opposite points of tangency, with their centers on logarithmic spirals of three different shapes.

Doyle spirals are named after mathematician Peter G. Doyle, who made an important contribution to their mathematical construction in the late 1980s or early 1990s. However, their study in phyllotaxis (the mathematics of plant growth) dates back to the early 1900s. (Full article...)
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Turing in 1936

Alan Mathison Turing (/ˈtjʊərɪŋ/; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science.

Born in London, Turing was raised in southern England. He graduated from King's College, Cambridge, and in 1938, earned a doctorate degree from Princeton University. During World War II, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bomba method, an electromechanical machine that could find settings for the Enigma machine. He played a crucial role in cracking intercepted messages that enabled the Allies to defeat the Axis powers in many crucial engagements, including the Battle of the Atlantic. (Full article...)
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69 (sixty-nine; LXIX) is the natural number following 68 and preceding 70. An odd number and a composite number, 69 is divisible by 1, 3, 23 and 69.

The number and its pictograph give its name to the sexual position of the same name. The association of the number with this sex position has resulted in it being associated in meme culture with sex. People knowledgeable of the meme may respond "nice" in response to the appearance of the number, whether intentionally an innuendo or not. (Full article...)
Image 6

The Herschel graph.

In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. It is named after British astronomer Alexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs (but not of this graph). (Full article...)
Image 7
Fig. 1: A binary search tree of size 9 and depth 3, with 8 at the root.

In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is linear with respect to the height of the tree.

Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler. (Full article...)
Image 8
In this graph, an even number of vertices (the four vertices numbered 2, 4, 5, and 6) have odd degrees. The sum of degrees of all six vertices is 2 + 3 + 2 + 3 + 3 + 1 = 14, twice the number of edges.

In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. The handshaking lemma is a consequence of the degree sum formula, also sometimes called the handshaking lemma, according to which the sum of the degrees (the numbers of times each vertex is touched) equals twice the number of edges in the graph. Both results were proven by Leonhard Euler (1736) in his famous paper on the Seven Bridges of Königsberg that began the study of graph theory.

Beyond the Seven Bridges of Königsberg Problem, which subsequently formalized Eulerian Tours, other applications of the degree sum formula include proofs of certain combinatorial structures. For example, in the proofs of Sperner's lemma and the mountain climbing problem the geometric properties of the formula commonly arise. The complexity class PPA encapsulates the difficulty of finding a second odd vertex, given one such vertex in a large implicitly-defined graph. (Full article...)
Image 9
In this example, the alternating sum of angles (clockwise from the bottom) is 90° − 45° + 22.5° − 22.5° + 45° − 90° + 22.5° − 22.5° = 0°. Since it adds to zero, the crease pattern may be flat-folded.

Kawasaki's theorem or Kawasaki–Justin theorem is a theorem in the mathematics of paper folding that describes the crease patterns with a single vertex that may be folded to form a flat figure. It states that the pattern is flat-foldable if and only if alternatingly adding and subtracting the angles of consecutive folds around the vertex gives an alternating sum of zero.
Crease patterns with more than one vertex do not obey such a simple criterion, and are NP-hard to fold.

The theorem is named after one of its discoverers, Toshikazu Kawasaki. However, several others also contributed to its discovery, and it is sometimes called the Kawasaki–Justin theorem or Husimi's theorem after other contributors, Jacques Justin and Kôdi Husimi. (Full article...)
Image 10
A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio.

A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is ${\sqrt {\varphi }}$ where $\varphi =(1+{\sqrt {5}})/2$ is the golden ratio, and the progression can be written: $1:{\sqrt {\varphi }}:\varphi$ , or approximately $1:1.272:1.618$ . Squares on the edges of this triangle have areas in another geometric progression, $1:\varphi :\varphi ^{2}$ . Alternative definitions of the same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles.

This triangle is named after Johannes Kepler, but can be found in earlier sources. Although some sources claim that ancient Egyptian pyramids had proportions based on a Kepler triangle, most scholars believe that the golden ratio was not known to Egyptian mathematics and architecture. (Full article...)
Image 11
A spiral staircase in the Cathedral of St. John the Divine. Several helical curves in the staircase project to hyperbolic spirals in its photograph.

A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals or decreasing angles of Archimedean spirals. As this curve widens, it approaches an asymptotic line. It can be found in the view up a spiral staircase and the starting arrangement of certain footraces, and is used to model spiral galaxies and architectural volutes.

As a plane curve, a hyperbolic spiral can be described in polar coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (r,\varphi)} by the equation
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Image 12

Berlin, 1959

Andrew Mattei Gleason (1921–2008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in mathematics teaching at all levels. Gleason's theorem in quantum logic and the Greenwood–Gleason graph, an important example in Ramsey theory, are named for him.

As a young World War II naval officer, Gleason broke German and Japanese military codes. After the war he spent his entire academic career at Harvard University, from which he retired in 1992. His numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and the Harvard Society of Fellows, and presidency of the American Mathematical Society. He continued to advise the United States government on cryptographic security, and the Commonwealth of Massachusetts on mathematics education for children, almost until the end of his life. (Full article...)

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... that Ukrainian baritone Danylo Matviienko, who holds a master's degree in mathematics, appeared as Demetrius in Britten's opera A Midsummer Night's Dream at the Oper Frankfurt?
... that Latvian-Soviet artist Karlis Johansons exhibited a skeletal tensegrity form of the Schönhardt polyhedron seven years before Erich Schönhardt's 1928 paper on its mathematics?
... that the prologue to The Polymath was written by Martin Kemp, a leading expert on Leonardo da Vinci?
... that the music of math rock band Jyocho has been alternatively described as akin to "madness" or "contemplative and melancholy"?
... that mathematics professor Ari Nagel has fathered more than a hundred children?
... that despite a mathematical model deeming the ice cream bar flavour Goody Goody Gum Drops impossible, it was still created?

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... that the Life without Death cellular automaton, a mathematical model of pattern formation, is a variant of Conway's Game of Life in which cells, once brought to life, never die?
... that one can list every positive rational number without repetition by breadth-first traversal of the Calkin–Wilf tree?
... that the Hadwiger conjecture implies that the external surface of any three-dimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?
... that an equitable coloring of a graph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than a graph coloring without this constraint?
... that no matter how biased a coin one uses, flipping a coin to determine whether each edge is present or absent in a countably infinite graph will always produce the same graph, the Rado graph?
...that it is possible to stack identical dominoes off the edge of a table to create an arbitrarily large overhang?
...that in Floyd's algorithm for cycle detection, the tortoise and hare move at very different speeds, but always finish at the same spot?

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A labeled graph on 6 vertices and 7 edges
Image credit: User:Booyabazooka

Informally speaking, a graph is a set of objects called points, nodes, or vertices connected by links called lines or edges. In a proper graph, which is by default undirected, a line from point A to point B is considered to be the same thing as a line from point B to point A. In a digraph, short for directed graph, the two directions are counted as being distinct arcs or directed edges. Typically, a graph is depicted in diagrammatic form as a set of dots (for the points, vertices, or nodes), joined by curves (for the lines or edges). Graphs have applications in both mathematics and computer science, and form the basic object of study in graph theory.

Applications of graph theory are generally concerned with labeled graphs and various specializations of these. Many problems of practical interest can be represented by graphs. The link structure of a website could be represented by a directed graph: the vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B. A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. For example if a graph represents a road network, the weights could represent the length of each road. A digraph with weighted edges in the context of graph theory is called a network. Networks have many uses in the practical side of graph theory, network analysis (for example, to model and analyze traffic networks). (Full article...)

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