mathematics
Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games).
The Mathematics Portal
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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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 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...) -
The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group S̃3
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...) -
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...) -
Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist, 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, as well as his work 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...) -
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...) -
Theodore John Kaczynski (/kəˈzɪnski/ ⓘ kə-ZIN-skee; May 22, 1942 – June 10, 2023), also known as the Unabomber (/ˈjuːnəbɒmər/ ⓘ YOO-nə-bom-ər), was an American mathematician and domestic terrorist. He was a mathematics prodigy, but abandoned his academic career in 1969 to pursue a primitive lifestyle.
Between 1978 and 1995, Kaczynski murdered three individuals and injured 23 others in a nationwide mail bombing campaign against people he believed to be advancing modern technology and the destruction of the natural environment. He authored Industrial Society and Its Future, a 35,000-word manifesto and social critique opposing industrialization, rejecting leftism, and advocating for a nature-centered form of anarchism. (Full article...) -
Leonhard Euler (/ˈɔɪlər/ OY-lər, German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] ⓘ, Swiss Standard German: [ˈleːɔnhart ˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.
Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is also widely considered to be the most prolific; his 866 publications as well as his correspondences are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quarto volumes. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. (Full article...) - In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. (Full article...) -
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...) -
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...) -
The manipulations of the Rubik's Cube form the Rubik's Cube group.
In mathematics, a group is a set with an operation that satisfies the following constraints: the operation is associative and 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 (these properties characterize the integers in a unique way). (Full article...) -
Josiah Willard Gibbs (/ɡɪbz/; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he created modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis.
In 1863, Yale University awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of mathematical physics from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history." In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics." (Full article...) -
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...) -
Robert Hues (1553 – 24 May 1632) was an English mathematician and geographer. He attended St. Mary Hall at Oxford, and graduated in 1578. Hues became interested in geography and mathematics, and studied navigation at a school set up by Walter Raleigh. During a trip to Newfoundland, he made observations which caused him to doubt the accepted published values for variations of the compass. Between 1586 and 1588, Hues travelled with Thomas Cavendish on a circumnavigation of the globe, performing astronomical observations and taking the latitudes of places they visited. Beginning in August 1591, Hues and Cavendish again set out on another circumnavigation of the globe. During the voyage, Hues made astronomical observations in the South Atlantic, and continued his observations of the variation of the compass at various latitudes and at the Equator. Cavendish died on the journey in 1592, and Hues returned to England the following year.
In 1594, Hues published his discoveries in the Latin work Tractatus de globis et eorum usu (Treatise on Globes and Their Use) which was written to explain the use of the terrestrial and celestial globes that had been made and published by Emery Molyneux in late 1592 or early 1593, and to encourage English sailors to use practical astronomical navigation. Hues' work subsequently went into at least 12 other printings in Dutch, English, French and Latin. (Full article...) -
An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. The name of the corresponding field is actuarial science which covers rigorous mathematical calculations in areas of life expectancy and life insurance. 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.
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 and manage programs that control risk. The actual steps needed to become an actuary are usually country-specific; however, almost all processes share a rigorous schooling or examination structure and take many years to complete. (Full article...)
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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 pth 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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Diagram of the three utilities problem showing lines in a plane. Can each house be connected to each utility, with no connection lines crossing?
The classical mathematical puzzle known as the three utilities problem or sometimes water, gas and electricity asks for non-crossing connections to be drawn between three houses and three utility companies in the plane. When posing it in the early 20th century, Henry Dudeney wrote that it was already an old problem. It is an impossible puzzle: it is not possible to connect all nine lines without crossing. Versions of the problem on nonplanar surfaces such as a torus or Möbius strip, or that allow connections to pass through other houses or utilities, can be solved.
This puzzle can be formalized as a problem in topological graph theory by asking whether the complete bipartite graph , with vertices representing the houses and utilities and edges representing their connections, has a graph embedding in the plane. The impossibility of the puzzle corresponds to the fact that is not a planar graph. Multiple proofs of this impossibility are known, and form part of the proof of Kuratowski's theorem characterizing planar graphs by two forbidden subgraphs, one of which is . The question of minimizing the number of crossings in drawings of complete bipartite graphs is known as Turán's brick factory problem, and for the minimum number of crossings is one. (Full article...) - In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) systems to multiple-input and multiple-output (MIMO) systems. The matrix relates the outputs of the system to its inputs. It is a particularly useful construction for linear time-invariant (LTI) systems because it can be expressed in terms of the s-plane.
In some systems, especially ones consisting entirely of passive components, it can be ambiguous which variables are inputs and which are outputs. In electrical engineering, a common scheme is to gather all the voltage variables on one side and all the current variables on the other regardless of which are inputs or outputs. This results in all the elements of the transfer matrix being in units of impedance. The concept of impedance (and hence impedance matrices) has been borrowed into other energy domains by analogy, especially mechanics and acoustics. (Full article...) -
In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares.
In geometry, Keller's conjecture is the conjecture that in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other. For instance, in any tiling of the plane by identical squares, some two squares must share an entire edge, as they do in the illustration.
This conjecture was introduced by Ott-Heinrich Keller (1930), after whom it is named. A breakthrough by Lagarias and Shor (1992) showed that it is false in ten or more dimensions, and after subsequent refinements, it is now known to be true in spaces of dimension at most seven and false in all higher dimensions. The proofs of these results use a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. (Full article...) -
Graphical demonstration of the convergence of the sum 1/2 + 1/3 + 1/7 + 1/43 + ... to 1. Each row of k squares of side length 1/k has total area 1/k, and all the squares together exactly cover a larger square with area 1. Squares with side lengths 1/1807 or smaller are too small to see in the figure and are not shown.
In number theory, Sylvester's sequence is an integer sequence in which each term is the product of the previous terms, plus one. Its first few terms are
:2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).
Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its terms. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms. (Full article...) - In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with colors, the same is true for the whole graph. The theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős (1951), after whom it is named.
The De Bruijn–Erdős theorem has several different proofs, all depending in some way on the axiom of choice. Its applications include extending the four-color theorem and Dilworth's theorem from finite graphs and partially ordered sets to infinite ones, and reducing the Hadwiger–Nelson problem on the chromatic number of the plane to a problem about finite graphs. It may be generalized from finite numbers of colors to sets of colors whose cardinality is a strongly compact cardinal. (Full article...) -
John von Neumann (/vɒn ˈnɔɪmən/ von NOY-mən; Hungarian: Neumann János Lajos [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist, engineer and polymath. He had perhaps the widest coverage of any mathematician of his time, integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including cellular automata, the universal constructor and the digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA.
During World War II, von Neumann worked on the Manhattan Project. He developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon. Before and after the war, he consulted for many organizations including the Office of Scientific Research and Development, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project and the Oak Ridge National Laboratory. At the peak of his influence in the 1950s, he chaired a number of Defense Department committees including the Strategic Missile Evaluation Committee and the ICBM Scientific Advisory Committee. He was also a member of the influential Atomic Energy Commission in charge of all atomic energy development in the country. He played a key role alongside Bernard Schriever and Trevor Gardner in the design and development of the United States' first ICBM programs. At that time he was considered the nation's foremost expert on nuclear weaponry and the leading defense scientist at the U.S. Department of Defense. (Full article...) -
Numbers written with Cistercian numerals. From left to right: 1 in units place, 2 in tens place (20), 3 in hundreds place (300), 4 in thousands place (4,000), then compound numbers 5,555, 6,789, 9,394.
The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single glyph able to indicate any integer from 1 to 9,999.
Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its place value (units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century. (Full article...) -
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:
:
The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. (Full article...) - The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2p−1(2p − 1), where 2p − 1 is a prime number. The theorem is named after mathematicians Euclid and Leonhard Euler, who respectively proved the "if" and "only if" aspects of the theorem.
It has been conjectured that there are infinitely many Mersenne primes. Although the truth of this conjecture remains unknown, it is equivalent, by the Euclid–Euler theorem, to the conjecture that there are infinitely many even perfect numbers. However, it is also unknown whether there exists even a single odd perfect number. (Full article...) -
An antiparallelogram
In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, each pair of sides is antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also called contraparallelograms or crossed parallelograms.
Antiparallelograms occur as the vertex figures of certain nonconvex uniform polyhedra. In the theory of four-bar linkages, the linkages with the form of an antiparallelogram are also called butterfly linkages or bow-tie linkages, and are used in the design of non-circular gears. In celestial mechanics, they occur in certain families of solutions to the 4-body problem. (Full article...) -
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.
The edges and vertices of the triaugmented triangular prism form a maximal planar graph with 9 vertices and 21 edges, called the Fritsch graph. It was used by Rudolf and Gerda Fritsch to show that Alfred Kempe's attempted proof of the four color theorem was incorrect. The Fritsch graph is one of only six graphs in which every neighborhood is a 4- or 5-vertex cycle. (Full article...) -
A set of 20 points in a 10 × 10 grid, with no three points in a line.
The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied geometric questions concerning lattice points".
At most points can be placed, because points in a grid would include a row of three or more points, by the pigeonhole principle. Although the problem can be solved with points for every up to , it is conjectured that fewer than points can be placed in grids of large size. Known methods can place linearly many points in grids of arbitrary size, but the best of these methods place slightly fewer than points, not . (Full article...)
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Did you know (auto-generated) –
- ... that ten-sided gaming dice have kite-shaped faces?
- ... that Arithmetic was the first mathematics text book written in the Russian language?
- ... that two members of the French parliament were killed when a delayed-action German bomb exploded in the town hall at Bapaume on 25 March 1917?
- ... that mathematician Daniel Larsen was the youngest contributor to the New York Times crossword puzzle?
- ... that Catechumen, a Christian first-person shooter, was funded only in the aftermath of the Columbine High School massacre?
- ... that museum director Alena Aladava rebuilt the Belarusian national art collection in the aftermath of the Second World War?
- ... that circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?
- ... that after Florida schools banned 54 mathematics books, Chaz Stevens petitioned that they also ban the Bible?
More did you know –
- ...that every natural number can be written as the sum of four squares?
- ...that the largest known prime number is nearly 25 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?
- ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
- ...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area, such as Gabriel's Horn?
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| Example of a four color map Image credit: User:Inductiveload |
The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. "Color by Number" worksheets and exercises, which combine learning art and math for people of young ages, are a good example of the four color theorem.
It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.
The four color theorem was the first major theorem to be proven using a computer, and the proof is disputed by some mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.
The lack of mathematical elegance was another factor, and to paraphrase comments of the time, "a good mathematical proof is like a poem — this is a telephone directory!" (Full article...)
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