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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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 17th-century studies of probability and annuities. Actuaries in the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems; actuaries use this knowledge 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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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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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...)
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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 of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of $1000$ to base $10$ is $3$ , because $1000$ is $10$ to the $3$ rd power: $1000 = 10 3 = 10 \times 10 \times 10$ . More generally, if $x = b y$ , then $y$ is the logarithm of $x$ to base $b$ , written $log b x = y$ , so $log 10 1000 = 3$ . As a single-variable function, the logarithm to base $b$ is the inverse of exponentiation with base $b$ .

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 widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written $log x$ . (Full article...)
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Rejewski, c. 1932

Marian Adam Rejewski (Polish: [ˈmarjan rɛˈjɛfskʲi] ; 16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in late 1932 reconstructed the sight-unseen German military Enigma cipher machine.

Over the next nearly seven years, Rejewski and fellow mathematician-cryptologists Jerzy Różycki and Henryk Zygalski, working at the Polish General Staff's Cipher Bureau, developed techniques and equipment for decrypting the Enigma ciphers, even as the Germans introduced modifications to their Enigma machines and encryption procedures. Rejewski's contributions included the cryptologic card catalog and the cryptologic bomb. (Full article...)
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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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Cantor, c. 1910

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp 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 of which he was well aware.

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 Konstantin Gutberlet [de; it] 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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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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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 May 1916 and is the accepted description of the gravitation of macroscopic objects in modern physics. General relativity generalizes special relativity and refines Isaac 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, momentum, and stress of whatever is present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations. John Archibald Wheeler summarized it: "Space-time tells matter how to move; matter tells space-time how to curve."

Newton's law of universal gravitation, which describes gravity in classical mechanics, 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, singularities and black holes. So far, all tests of general relativity have been in agreement with the theory. The time-dependent solutions of general relativity enable us to extrapolate the history of the universe into the past and future, 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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The title page of a 1634 version of Hues's 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's work subsequently went into at least 12 other printings in Dutch, English, French and Latin. (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...)
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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...)
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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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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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Hilary Putnam

The Quine–Putnam indispensability argument is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. It was named after the philosophers Willard Van Orman Quine and Hilary Putnam, and is one of the most important arguments in the philosophy of mathematics.

Although elements of the indispensability argument may have originated with thinkers such as Gottlob Frege and Kurt Gödel, Quine's development of the argument was unique for introducing to it a number of his philosophical positions such as naturalism, confirmational holism, and the criterion of ontological commitment. Putnam gave Quine's argument its first detailed formulation in his 1971 book Philosophy of Logic. He later came to disagree with various aspects of Quine's thinking, however, and formulated his own indispensability argument based on the no miracles argument in the philosophy of science. A standard form of the argument in contemporary philosophy is credited to Mark Colyvan; whilst being influenced by both Quine and Putnam, it differs in important ways from their formulations. It is presented in the Stanford Encyclopedia of Philosophy: (Full article...)

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colored ball with "hair" (representing a vector field on a sphere)

Hairy ball theorem

Credit: The Evil Midnight Uploader what Uploads at Midnight

This image illustrates a failed attempt to comb the "hair" on a ball flat, leaving a tuft sticking out at each pole. The hairy ball theorem of algebraic topology states that whenever one attempts to comb a hairy ball, there will always be at least one point on the ball at which a tuft of hair sticks out. More precisely, it states that there is no nonvanishing continuous tangent-vector field on an even-dimensional n‑sphere (an ordinary sphere in three-dimensional space is known as a "2-sphere"). This is not true of certain other three-dimensional shapes, such as a torus (doughnut shape) which can be combed flat. The theorem was first stated by Henri Poincaré in the late 19th century and proved in 1912 by L. E. J. Brouwer. If one idealizes the wind in the Earth's atmosphere as a tangent-vector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a cyclone somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound. (What is true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in computer modeling (including video game design), in which a common problem is to compute a non-zero 3-D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task.

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Thoralf Skolem, after whom the paradox is named

In mathematical logic and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from part of the Löwenheim–Skolem theorem; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox and was described as a "paradoxical state of affairs" by Skolem.

In model theory, a model corresponds to a specific interpretation of a formal language or theory. It consists of a domain (a set of objects) and an interpretation of the symbols and formulas in the language, such that the axioms of the theory are satisfied within this structure. The Löwenheim–Skolem theorem shows that any model of set theory in first-order logic, if it is consistent, has an equivalent model that is countable. This appears contradictory, because Georg Cantor proved that there exist sets which are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states "there are uncountable sets". (Full article...)
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Graph of $log 2 x$ as a function of a positive real number $x$

In mathematics, the binary logarithm ( $log 2 n$ ) is the power to which the number $2$ must be raised to obtain the value $n$ . That is, for any real number $x$ ,
: $x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.$
For example, the binary logarithm of $1$ is $0$ , the binary logarithm of $2$ is $1$ , the binary logarithm of $4$ is $2$ , and the binary logarithm of $32$ is $5$ .

The binary logarithm is the logarithm to the base $2$ and is the inverse function of the power of two function. There are several alternatives to the $log 2$ notation for the binary logarithm; see the Notation section below. (Full article...)
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In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.

In technical terms, QED can be described as a perturbation theory of the electromagnetic quantum vacuum. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen. It is the most precise and stringently tested theory in physics. However, the prediction of vacuum zero energy made by QED has 10^120 magnitude of difference compared to experimental observation value. This is called cosmological constant problem or vacuum catastrophe. (Full article...)
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Six points in the unit square, with the smallest triangles (red) having area 1/8, the optimal area for this number of points. Other larger triangles are colored blue. These points are an affine transformation of a regular hexagon, but for larger numbers of points the optimal solution does not form a convex polygon.

In discrete geometry and discrepancy theory, the Heilbronn triangle problem asks how to place $n$ points in a region of the plane, such as a square, so that the smallest of the triangles formed by triples of the points is as large as possible. It is named after Hans Heilbronn, who conjectured that this largest achievable area shrinks at least as quickly as the inverse square of the number of points. Komlós, Pintz, and Szemerédi disproved the conjecture in 1982, finding placements whose smallest triangle is larger by a logarithmic factor, but the asymptotic growth rate of the area remains unknown. (Full article...)
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Logo of the International Mathematical Olympiad

The International Mathematical Olympiad (IMO) is an annual mathematical competition for pre-university students, and is the oldest of the International Science Olympiads. It is widely regarded as the most prestigious mathematical competition in the world. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. More than 100 countries participate. Each country sends a team of up to six students, plus one team leader, one deputy leader, and observers.

The contest takes place over two days, and consists of six problems (three given on each day). The problems are extremely difficult, and may include topics not conventionally covered in school, but they avoid the use of calculus. Awards are given to approximately the top-scoring 50% of the individual contestants. Teams are not officially recognized—all scores are given only to individual contestants, but team scoring is unofficially compared more than individual scores. (Full article...)
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Isosceles triangle

In geometry, an isosceles triangle (/aɪˈsɒsəliːz/) is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.
Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. (Full article...)
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Composite numbers can be arranged into rectangles but prime numbers cannot.

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

The property of being prime is called primality. A simple but slow method of checking the primality of a given number ⁠ $n$ ⁠, called trial division, tests whether ⁠ $n$ ⁠ is a multiple of any integer between 2 and ⁠ ${\sqrt {n}}$ ⁠. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers, and these have been used to find large prime numbers. (Full article...)
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A double bubble. Note that the surface separating the small lower bubble from the large bubble bulges into the large bubble.

In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a standard double bubble: three spherical surfaces meeting at angles of 120° on a common circle. The double bubble theorem was formulated and thought to be true in the 19th century, and became a "serious focus of research" by 1989, but was not proven until 2002.

The proof combines multiple ingredients. Compactness of rectifiable currents (a generalized definition of surfaces) shows that a solution exists. A symmetry argument proves that the solution must be a surface of revolution, and it can be further restricted to having a bounded number of smooth pieces. Jean Taylor's proof of Plateau's laws describes how these pieces must be shaped and connected to each other, and a final case analysis shows that, among surfaces of revolution connected in this way, only the standard double bubble has locally-minimal area. (Full article...)
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An ideal regular octahedron in the Poincaré ball model of hyperbolic space (sphere at infinity not shown). All dihedral angles of this shape are right angles.

In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.

The Platonic solids and Archimedean solids have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra have ideal versions – a polyhedron can be ideal only when it can be represented in Euclidean space with all its vertices on a circumscribed sphere. Using linear programming, one can test whether a polyhedron has an ideal version, in polynomial time. (Full article...)
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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 computational complexity theory, Yao's principle (also called Yao's minimax principle or Yao's lemma) relates the performance of randomized algorithms to deterministic (non-random) algorithms. It states that, for certain classes of algorithms, and certain measures of the performance of the algorithms, the following two quantities are equal:
- The optimal performance that can be obtained by a deterministic algorithm on a random input (its average-case complexity), for a probability distribution on inputs chosen to be as hard as possible and for an algorithm chosen to work as well as possible against that distribution
- The optimal performance that can be obtained by a random algorithm on a deterministic input (its expected complexity), for an algorithm chosen to have the best performance on its worst case inputs, and the worst case input to the algorithm
Yao's principle is often used to prove limitations on the performance of randomized algorithms, by finding a probability distribution on inputs that is difficult for deterministic algorithms, and inferring that randomized algorithms have the same limitation on their worst case performance.

This principle is named after Andrew Yao, who first proposed it in a 1977 paper. It is closely related to the minimax theorem in the theory of zero-sum games, and to the duality theory of linear programs. (Full article...)
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Maxwell, c. 1870s

James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism achieved the second great unification in physics, where the first one had been realised by Isaac Newton. Maxwell was also key in the creation of statistical mechanics.

Maxwell graduated from Trinity College, Cambridge, in 1854, where he earned distinction in mathematics and the Smith's Prize. He remained at Cambridge briefly, publishing early mathematical work and investigations into optics, particularly the principles of colour combination and colour-blindness. He later held the Chair of Natural Philosophy at Marischal College in Aberdeen, where he studied the rings of Saturn and correctly proposed that they were composed of numerous small particles, work that earned him the Adams Prize in 1859. During this time he married Katherine Mary Dewar, who assisted him in his laboratory work. From 1860 to 1865, he served as the Professor of Natural Philosophy at King's College London, where he developed his theory of electromagnetic fields. His publication of "A Dynamical Theory of the Electromagnetic Field" in 1865 demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light, proposing that light is an undulation in the same medium that is the cause of electric and magnetic phenomena. His unification of light and electrical phenomena led to his prediction of the existence of radio waves. (Full article...)

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... that the identity of Cleo, who provided online answers to complex mathematics problems without showing any work, was revealed over a decade later in 2025?
... that Carmel Naughton, having been told that girls were "stupid and couldn't do maths", sponsored a STEM scholarship fund?
... 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 in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?
... that Hannah Fry used mathematics to compare Elizabeth II's Christmas messages with the lyrics of Snoop Dogg?
... that the word algebra is derived from an Arabic term for the surgical treatment of bonesetting?
... that the symbol for equality in mathematics was not used for 61 years after its introduction, and was later popularized by Isaac Newton?
... that the discovery of Descartes' theorem in geometry came from a too-difficult mathematics problem posed to a princess?

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Did you know...

...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?
...that you cannot knot strings in 4 dimensions, but you can knot 2-dimensional surfaces, such as spheres?
...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?

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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
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Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory. The Elements is one of the oldest extant axiomatic deductive treatments of geometry, and has proven instrumental in the development of logic and modern science.

It is considered one of the most successful textbooks ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (well over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today. (Full article...)

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