Portal:Mathematics

The Mathematics Portal

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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Bust of Shen at the Beijing Ancient Observatory

Shen Kuo (Chinese: 沈括; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁), was a Chinese polymath, scientist, and statesman of the Northern Song dynasty. Shen was a master in many fields of study including mathematics, optics, and horology. In his career as a civil servant, he became a finance minister, governmental state inspector, head official for the Bureau of Astronomy in the Song court, Assistant Minister of Imperial Hospitality, and also served as an academic chancellor. At court his political allegiance was to the Reformist faction known as the New Policies Group, headed by Chancellor Wang Anshi (1021–1085).

In his Dream Pool Essays or Dream Torrent Essays (夢溪筆談; Mengxi Bitan) of 1088, Shen was the first to describe the magnetic needle compass, which would be used for navigation (first described in Europe by Alexander Neckam in 1187). Shen discovered the concept of true north in terms of magnetic declination towards the north pole, with experimentation of suspended magnetic needles and "the improved meridian determined by Shen's [astronomical] measurement of the distance between the pole star and true north". This was the decisive step in human history to make compasses more useful for navigation, and may have been a concept unknown in Europe for another four hundred years (evidence of German sundials made circa 1450 show markings similar to Chinese geomancers' compasses in regard to declination). (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 3Euclid'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, 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...)$
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Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because the remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300).

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, 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, 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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Émile Michel Hyacinthe Lemoine (French: [emil ləmwan]; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most notably, the École Polytechnique. Lemoine taught as a private tutor for a short period after his graduation from the latter school.

Lemoine is best known for his proof of the existence of the Lemoine point (or the symmedian point) of a triangle. Other mathematical work includes a system he called Géométrographie and a method which related algebraic expressions to geometric objects. He has been called a co-founder of modern triangle geometry, as many of its characteristics are present in his work. (Full article...)
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Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He shared the 1965 Nobel Prize in Physics with Julian Schwinger and Shin'ichirō Tomonaga "for their fundamental work in quantum electrodynamics (QED), with deep-ploughing consequences for the physics of elementary particles".
He is also known for his work in the path integral formulation of quantum mechanics, the theory of the physics of the superfluidity of supercooled liquid helium, and the parton model. Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams and remains widely used.

He assisted in the development of the atomic bomb during World War II and became known to the wider public in the 1980s as a member of the Rogers Commission, the panel that investigated the Space Shuttle Challenger disaster. Along with his work in theoretical physics, Feynman has been credited with having pioneered the field of quantum computing and introducing the concept of nanotechnology. He held the Richard C. Tolman professorship in theoretical physics at the California Institute of Technology. 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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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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Archimedes Thoughtful (1620)

Archimedes of Syracuse (/ˌɑːrkɪˈmiːdiːz/ AR-kih-MEE-deez; c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the city of Syracuse in Sicily. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.

Archimedes' other mathematical achievements include deriving an approximation of pi ( $π$ ), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy known as Archimedes' principle. In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe. He is also said to have built a planetarium device that demonstrated the movements of the known celestial bodies, and may have been a precursor to the Antikythera mechanism. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. (Full article...)
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A statue depicting Zhang Heng at his tomb in Nanyang, Henan province, China

Zhang Heng (Chinese: 張衡; AD 78 – 139), courtesy name Pingzi, formerly romanized Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Eastern 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...)
$Image 9 1753 portraitLeonhard Euler (/ˈɔɪlər/ OY-lər; 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, music theorist and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.Euler is credited for popularizing the Greek letter π {\displaystyle \pi } (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle \Sigma } (capital sigma) to express summations, the Greek letter Δ {\displaystyle \Delta } (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant e {\displaystyle e} , the base of the natural logarithm, now known as Euler's number. Euler made contributions to applied mathematics and engineering, such as his study of ships helping navigation, his three volumes on optics, which contributed to the design of microscopes and telescopes, and his studies of beam bending and column critical loads. (Full article...)$
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1753 portrait

Leonhard Euler (/ˈɔɪlər/ OY-lər; 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, music theorist and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.

Euler is credited for popularizing the Greek letter $\pi$ (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation $f(x)$ for the value of a function, the letter $i$ to express the imaginary unit ${\sqrt {-1}}$ , the Greek letter $\Sigma$ (capital sigma) to express summations, the Greek letter $\Delta$ (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant $e$ , the base of the natural logarithm, now known as Euler's number. Euler made contributions to applied mathematics and engineering, such as his study of ships helping navigation, his three volumes on optics, which contributed to the design of microscopes and telescopes, and his studies of beam bending and column critical loads. (Full article...)
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In classical mechanics, the Laplace–Runge–Lenz vector (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.

Thus the hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. (Full article...)
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The number π (/paɪ/ ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.

The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as ⁠22/7⁠ are commonly used to approximate it. Consequently, its decimal representation never ends, nor does it enter a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of π appear to be evenly distributed, but no proof of this conjecture has been found. (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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The weighing pans of this balance scale contain zero objects, divided into two equal groups.

In mathematics, zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": zero is an integer multiple of 2, specifically 0 × 2. As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if $y$ is even then $y + x$ has the same parity as $x$ —indeed, $0 + x$ and $x$ always have the same parity.

Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as even − even = even, require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all. (Full article...)
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Elementary algebra studies which values solve equations formed using arithmetical operations.

Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions 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 schools. It examines mathematical statements using variables for unspecified values and 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...)
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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/ YOU-na-bom-ər), was an American mathematician and domestic terrorist. A mathematics prodigy, he abandoned his academic career to pursue a reclusive primitive lifestyle and lone wolf terrorism campaign.

Kaczynski murdered three people and injured 23 others between 1978 and 1995 in a nationwide mail bombing campaign against those he believed to be advancing modern technology and the destruction of the natural environment. He authored a roughly 35,000-word manifesto and social critique, Industrial Society and Its Future (1995), which opposes all forms of technology, rejects leftism, advocates cultural primitivism, and ultimately suggests violent revolution. (Full article...)

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Riemann zeta function

Credit: Linas Vepstas

This is a graph of a portion of the complex-valued Riemann zeta function along the critical line (the set of complex numbers having real part equal to ⁠12⁠). More specifically, it is a graph of

Im ζ (⁠ 1 / 2 ⁠ + it)

versus

Re ζ (⁠ 1 / 2 ⁠ + it)

(the imaginary part vs. the real part) for values of the real variable

t

running from 0 to 34 (the curve starts at its leftmost point, with real part approximately −1.46 and imaginary part 0). The first five zeros along the critical line are visible in this graph as the five times the curve passes through the origin (which occur at

t

≈ 14.13, 21.02, 25.01, 30.42, and 32.93 — for a different perspective, see a graph of the real and imaginary parts of this function plotted separately over a wider range of values). In 1914, G. H. Hardy proved that

ζ (⁠ 1 / 2 ⁠ + it)

has infinitely many zeros. According to the Riemann hypothesis, zeros of this form constitute the only non-trivial zeros of the full zeta function,

ζ (s)

, where

s

varies over all complex numbers. Riemann's zeta function grew out of Leonhard Euler's study of real-valued infinite series in the early 18th century. In a famous 1859 paper called "On the Number of Primes Less Than a Given Magnitude", Bernhard Riemann extended Euler's results to the complex plane and established a relation between the zeros of his zeta function and the distribution of prime numbers. The paper also contained the previously mentioned Riemann hypothesis, which is considered by many mathematicians to be the most important unsolved problem in pure mathematics. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

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Image 1
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...)
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3 + 2 = 5 with apples, a popular choice in textbooks

Addition, usually denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two equals five".

Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, and elements of additive groups. (Full article...)
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69 (sixty-nine) 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 the name to the sexual position of the same name. The association of the number with this particular sex position has resulted in it being associated in meme culture also with sex. (Full article...)
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Two random distributions on three-vertex binary trees, the binary search trees on three keys $a$ , $b$ , and $c$ . These five trees are each assigned probability 1/5 by the uniform distribution (top). The distribution generated by random insertion orderings (bottom) assigns the center tree probability 1/3, because two of the six possible insertion orderings generate the same tree; the other four trees have probability 1/6.

In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees. Different distributions have been used, leading to different properties for these trees.

Random binary trees have been used for analyzing the average-case complexity of data structures based on binary search trees. For this application it is common to use random trees formed by inserting nodes one at a time according to a random permutation. The resulting trees are very likely to have logarithmic depth and logarithmic Strahler number. The treap and related balanced binary search trees use update operations that maintain this random structure even when the update sequence is non-random. (Full article...)
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Rosa Margaret Morris (16 July 1914 – 15 October 2011) was a Welsh applied mathematician, working in potential theory and aerodynamics. When she was 23, her research and examination results made national and international news. She received several fellowships and awards before graduating with a PhD in mathematics from the University of Cambridge in 1940. In her later career, she taught at the University College of South Wales and Monmouthshire (now Cardiff University), where she co-authored a successful textbook on Mathematical Methods of Physics with Roy Chisholm and became one of the first female Heads of School of Mathematics in the United Kingdom. (Full article...)
Image 6
An instance of the matroid parity problem: given a graph with colored edges, having exactly two edges per color, find a forest (an independent set of the graphic matroid) with as many edges as possible that again has exactly two edges per color.

In combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid, a structure that abstracts and generalizes the notion of linear independence in vector spaces. The problem was formulated by Lawler (1976) as a common generalization of graph matching and matroid intersection. It is also known as polymatroid matching, or the matchoid problem.

Matroid parity can be solved in polynomial time for linear matroids. However, it is NP-hard for certain compactly-represented matroids, and requires more than a polynomial number of steps in the matroid oracle model. (Full article...)
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In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.

This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the vertex-edge graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construct three-dimensional visualizations of abstract graphs. Branko Grünbaum has called this theorem "the most important and deepest known result on 3-polytopes." (Full article...)
Image 8

Two symmetric orthographic projections

In geometry, the complete, final stellation of the icosahedron^[1], or echidnahedron 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...)
$Image 9In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n {\displaystyle n} points on a circle, at angles of θ , 2 θ , 3 θ , . . . {\displaystyle \theta ,2\theta ,3\theta ,...} from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Unless θ {\displaystyle \theta } is a rational multiple of π {\displaystyle \pi } , there will also be at least two distinct distances. This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi [hu], and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square. (Full article...)$
Image 9
In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places $n$ points on a circle, at angles of $\theta ,2\theta ,3\theta ,...$ from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Unless $\theta$ is a rational multiple of $\pi$ , there will also be at least two distinct distances.

This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi [hu], and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square. (Full article...)
$Image 10The Lambek–Moser theorem is a mathematical description of the partitions of natural numbers into two complementary sets. For instance, it applies to the partition of numbers into even and odd, or into prime and non-prime (one and the composite numbers). There are two parts to the Lambek–Moser theorem. One part states any two non-decreasing integer functions that are inverse, in a certain sense, can be used to split the natural numbers into two complementary subsets, and the other part states every complementary partition can be constructed in this way. When a formula is known for the n {\displaystyle n} th natural number in a set, the Lambek–Moser theorem can be used to obtain a formula for the n {\displaystyle n} th number not in the set.The Lambek–Moser theorem belongs to combinatorial number theory. It is named for Joachim Lambek and Leo Moser, who published it in 1954, and should be distinguished from an unrelated theorem of Lambek and Moser, later strengthened by Roy Wild, on the number of primitive Pythagorean triples. It extends Rayleigh's theorem, which describes complementary pairs of Beatty sequences, the sequences of rounded multiples of irrational numbers. (Full article...)$
Image 10
The Lambek–Moser theorem is a mathematical description of the partitions of natural numbers into two complementary sets. For instance, it applies to the partition of numbers into even and odd, or into prime and non-prime (one and the composite numbers). There are two parts to the Lambek–Moser theorem. One part states any two non-decreasing integer functions that are inverse, in a certain sense, can be used to split the natural numbers into two complementary subsets, and the other part states every complementary partition can be constructed in this way. When a formula is known for the $n$ th natural number in a set, the Lambek–Moser theorem can be used to obtain a formula for the $n$ th number not in the set.

The Lambek–Moser theorem belongs to combinatorial number theory. It is named for Joachim Lambek and Leo Moser, who published it in 1954, and should be distinguished from an unrelated theorem of Lambek and Moser, later strengthened by Roy Wild, on the number of primitive Pythagorean triples. It extends Rayleigh's theorem, which describes complementary pairs of Beatty sequences, the sequences of rounded multiples of irrational numbers. (Full article...)
Image 11
Convergence of a convex curve to a circle under the curve-shortening flow. Inner curves (lighter color) are flowed versions of the outer curves. Time steps between curves are not uniform.

In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, and arc length evolution.

As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth. It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of the curve decreases as the curve converges to a circular shape, before collapsing to a singularity. If two disjoint simple smooth closed curves evolve, they remain disjoint until one of them collapses to a point. (Full article...)
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In geometry, a square pyramid is a pyramid with a square base and four triangles, having a total of five faces. If the apex of the pyramid is directly above the center of the square, it becomes a form of right pyramid with four isosceles triangles. When all of the pyramid's edges are equal in length, its triangles are all equilateral, an example of a Johnson solid.

Square pyramids have appeared throughout the history of architecture, with examples being Egyptian pyramids and many other similar buildings. They also occur in chemistry in square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra and as the cell of a four-dimensional polytope called cubic pyramid. Square pyramidal number is a natural number that counts the number of spheres stacked into a square pyramid. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches. (Full article...)

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6 October 2026 – Mathematics and artificial intelligence, List of mathematical discoveries by artificial intelligence: OpenAI releases 377 results on open problems in mathematics in areas such as algebra, number theory, combinatorics and theoretical computer science. (The New York Times)

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... that the symbol for equality in mathematics was not used for 61 years after its introduction, and was later popularized by Isaac Newton?
... that Pamela Ball was prouder of her mathematics degrees than of being the first Jamaican woman to be a fellow of the Royal College of Surgeons of England?
... that Eugene Parker described the mathematics behind his theory of solar wind as just "four lines of algebra"?
... that Peter Nagy based his song "Profesor Indigo" on his former math teacher?
... that the Prison Mathematics Project brought a professor from Italy to an American prison to celebrate Pi Day?
... that Livingstone Luboobi claimed that he chose to teach himself double mathematics at A-level because there was no teacher available?
... that the father of Lithuanian scientific forestry also directed theatre, organized a school, compiled maps, and wrote a dictionary of plants, a mathematics textbook, and a collection of poetry?
... that people in Madagascar perform algebra on tree seeds in order to tell the future?

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...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
...that every natural number can be written as the sum of four squares?
...that the largest known prime number is over 41 million digits long?
...that the set of rational numbers is equal in size to the set of integers; that is, they can be put in one-to-one correspondence?
...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
...that it is unknown whether π and e are algebraically independent?

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Fourteen ways of triangulating a hexagon
Image credit: User:Dmharvey

The Catalan numbers, named for the Belgian mathematician Eugène Charles Catalan, are a sequence of natural numbers that are important in combinatorial mathematics. The sequence begins:

1, 1, 2, 5, 14, 42, 132, ...

The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n^th Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves. It is also the number of ways of associating n applications of a binary operator as well as the number of ways that a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines. (Full article...)

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