John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics. December 26 is the 360th day of the year in the Gregorian Calendar, 361st in leap years. ... 1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ... Liverpool waterfront by night, as seen from the Wirral. ... Motto: (French for God and my right) Anthem: Multiple unofficial anthems Capital London Largest city London Official language(s) English Government Constitutional monarchy  - Queen Queen Elizabeth II  - Prime Minister Tony Blair MP Unification    - by Athelstan AD927  Area    - Total 130,395 km² (1st in UK)   50,346 sq mi   - Water (%) Population... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Trefoil knot, the simplest non-trivial knot. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Combinatorial game theory (CGT) is a mathematical theory that only studies two-player games which have a position which the players take turns changing in defined ways or moves to achieve a defined winning condition. ... Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ... Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ...

John Horton Conway
John Horton Conway
Born	December 26, 1937 Liverpool, England

Conway is currently professor of mathematics at Princeton University. He studied at University of Cambridge, where he started research under Harold Davenport. In 1981 he was elected a Fellow of the Royal Society. Image File history File links John_Horton_Conway. ... December 26 is the 360th day of the year in the Gregorian Calendar, 361st in leap years. ... 1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ... Liverpool waterfront by night, as seen from the Wirral. ... Motto: (French for God and my right) Anthem: Multiple unofficial anthems Capital London Largest city London Official language(s) English Government Constitutional monarchy  - Queen Queen Elizabeth II  - Prime Minister Tony Blair MP Unification    - by Athelstan AD927  Area    - Total 130,395 km² (1st in UK)   50,346 sq mi   - Water (%) Population... Princeton University is a coeducational private university located in Princeton, New Jersey. ... The University of Cambridge (often called Cambridge University, or just Cambridge), located in Cambridge, England, is the second-oldest university in the English-speaking world. ... Harold Davenport (30 October 1907 - 9 June 1969) was an English mathematician, known for his extensive work in number theory. ... 1981 (MCMLXXXI) was a common year starting on Thursday of the Gregorian calendar. ... The premises of the Royal Society in London (first four properties only). ...

Biography

Conway's parents were Agnes Boyce and Cyril Horton Conway. John had two older sisters, Sylvia and Joan. Cyril Conway was a chemistry laboratory assistant. John became interested in mathematics at a very early age and his mother Agnes recalled that he could recite the powers of two when aged four years. John's young years were difficult for he grew up in Britain at a time of wartime shortages. At primary school John was outstanding and he topped almost every class. At the age of eleven his ambition was to become a mathematician.

After leaving seconday school, Conway entered Gonville and Caius College Cambridge to study mathematics. He was awarded his BA in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying at Cambridge, where he became an avid backgammon player spending hours playing the game in the common room. He was awarded his doctorate in 1964 and was appointed as Lecturer in Pure Mathematics at the University of Cambridge. Harold Davenport (30 October 1907 - 9 June 1969) was an English mathematician, known for his extensive work in number theory. ... The University of Cambridge (often called Cambridge University, or just Cambridge), located in Cambridge, England, is the second-oldest university in the English-speaking world. ...

Conway has been married to wife Diana since 2001 and has a son Gareth (b. 2001). Their home is in Princeton, New Jersey, USA. With previous wives he has sons Oliver (b. 1988) and Alex (b. 1983); daughters Susan (b. 1962), (Rose b. 1963), Elena (b. 1965) and Ann-Louise (b. 1968). He has three grandchildren: John, Ellen and Joseph Wayman. Princeton, New Jersey, is the name of a section of Mercer County, New Jersey, United States. ... This article does not cite its references or sources. ...

Game theory

Among amateur mathematicians, he is perhaps most widely known for his contributions to combinatorial game theory, a theory of partizan games. This he developed with Elwyn Berlekamp and Richard Guy. In combinatorial game theory, a game is partisan or partizan if it is not impartial. ... Elwyn Ralph Berlekamp is professor of mathematics at University of California, Berkeley. ... Richard K. Guy is a Professor Emeritus in the Department of Mathematics at the University of Calgary. ...

He is also one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, and peg solitaire. He came up with the still unsolved Angel problem. Sprouts is a pencil-and-paper game with interesting mathematical properties. ... Phutball (short for philosophers football) is a two-player board game described in Elwyn Berlekamp, John Conway, and Richard Guys Winning Ways for your Mathematical Plays. ... The pieces of a Soma cube (with extra coloring) The same puzzle, assembled into a cube The Soma cube is a solid dissection puzzle created by Piet Hein during a lecture on quantum mechanics by Werner Heisenberg. ... English peg solitaire board European peg solitaire board Peg Solitaire is a board game for one player involving movement of pegs on a board with holes. ... The blue dotted region shows where an angel of power 3 could reach The Angel problem is a question in game theory proposed by John Conway. ...

He invented a new system of numbers, the surreal numbers, which are closely related to certain games and have been the subject of a mathematical novel by Donald Knuth. He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similiar to superreal numbers and hyperreal numbers. ... Donald Knuth at a reception for the Open Content Alliance. ... For information on how large numbers are named in English, see names of large numbers. ... Conway chained arrow notation, created by mathematician John Conway, is a means of expressing certain extremely large numbers. ...

He is also known for the invention of the game of life. This is one of the early and still celebrated examples of a cellular automaton. Gospers Glider Gun creating gliders. The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. ... A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory, mathematics, and theoretical biology. ...

Geometry

In the mid-1960s with Michael Guy, son of Richard Guy, he established that there are sixty-four convex nonprismatic uniform polychora. Richard K. Guy is a Professor Emeritus in the Department of Mathematics at the University of Calgary. ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ...

Geometric topology

Conway's approach to computing the Alexander polynomial of knot theory, in a variant now called the Alexander-Conway polynomial, involved a skein relation. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials. Conway further developed tangle theory. Conway has also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. This article needs cleanup. ... Skein relations are a piece of knot theory usually used to recursively define knot polynomials using knot diagrams as bookkeeping (compare Stückelberg-Feynman diagrams). ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... Conway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operators. ...

Group theory

He worked on the classification of finite simple groups and discovered the Conway groups. He was the primary author of the Atlas of Finite Groups giving properties of many finite simple groups. He with collaborators constructed the first concrete representations of some of the sporadic groups. The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ... In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...

With Simon Norton he formulated the complex of conjectures relating the monster group with modular functions, christened by them monstrous moonshine. Simon P. Norton is a mathematician in Cambridge, England, who works on finite simple groups. ... In mathematics, the Monster group M is a group of order    246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ... In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ... In mathematics, monstrous moonshine is a term devised by John Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function). ...

Algorithmics

For calculating the day of the week, he invented the Doomsday algorithm. The algorithm is simple enough for Conway to do the calculations in his head. He can usually give the correct answer in under two seconds. To improve his speed, he practices his calendrical calculations on his computer, which is programmed to quiz him with random dates every time he logs on. One of his early books was on finite state machines. This article details various mathematical algorithms to calculate the day of the week for any particular date in the past or future. ... The Doomsday algorithm is a way of calculating the day of the week of a given date. ... Fig. ...

Theoretical physics

In 2004, Conway and Simon Kochen, another Princeton mathematician, proved the Free will theorem, a startling version of the No Hidden Variables principle of Quantum Mechanics. It states that given certain conditions (that almost every physicist agrees are true), if an experimenter can freely decide what quantities to measure in a particular experiment, then elementary particles must be free to choose their spins in order to make the measurements consistent with physical law. Or, in Conway's provocative wording, if experimenters have free will, then so do elementary particles. The free will theorem of J. H. Conway and Simon Kochen states that, if we have a certain amount of free will, then, subject to certain assumptions, so do some elementary particles. ... In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ... Fig. ...

Books

He has (co-)written several books including the Atlas of Finite Groups, Regular Algebra and Finite Machines, Sphere Packings, Lattices and Groups, The Sensual (Quadratic) Form, On Numbers and Games, Winning Ways for your Mathematical Plays, The Book of Numbers, and On Quaternions and Octonions. On Numbers and Games is a mathematics book by John Horton Conway. ... Winning Ways for your Mathematical Plays (ISBN 1568811306) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. ...

See also

• Conway's LUX method for magic squares
• Conway's orbifold notation
• Conway chained arrow notation
• Conway's Game of Life
• Phutball
• Look-and-say sequence
• 15 theorem

Conways LUX method for magic squares is an algorithm for creating magic squares of order 4n+2, where n is a natural number. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Conway chained arrow notation, created by mathematician John Conway, is a means of expressing certain extremely large numbers. ... Gospers Glider Gun creating gliders. The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. ... Phutball (short for philosophers football) is a two-player board game described in Elwyn Berlekamp, John Conway, and Richard Guys Winning Ways for your Mathematical Plays. ... In mathematics, the look-and-say sequence is the sequence of integers beginning as follows: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of... The 15 theorem of John H. Conway and W. A. Schneeberger, proved in 1993, states that if an integral quadratic form with integral matrix represents all positive integers up to 15, then it represents all positive integers. ...

External links and references

• O'Connor, John J., and Edmund F. Robertson. "John Horton Conway". MacTutor History of Mathematics archive. by O'Connor and Robertson
• Charles Seife, "Impressions of Conway", The Sciences
• Mark Alpert, "Not Just Fun and Games", Scientific American April 1999. (official online version; registration-free online version)
• Jasvir Nagra, "Conway's Proof Of The Free Will Theorem" [1]
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
• John Horton Conway at the Mathematics Genealogy Project
• Video of Conway leading a tour of brickwork patterns in Princeton, lecturing on the ordinals, and lecturing on sums of powers and Bernoulli numbers.