John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

He has also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge in England, and the second half at Princeton University in New Jersey, where he holds the title John von Neumann Professor Emeritus.

Education and early life

Conway was born in Liverpool, on 26 December 1937, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the age of eleven his ambition was to become a mathematician.

After leaving sixth form, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a "terribly introverted adolescent" in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person: an "extrovert".

He was awarded his Bachelor of Arts degree 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 the Cambridge Mathematical Tripos, 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 College Fellow and Lecturer in Mathematics at Sidney Sussex College, Cambridge.

After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University.

Conway's Game of Life

Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, long before personal computers existed.

Since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics. There is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days, it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. At times Conway has said he hates the Game of Life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the field of cellular automata.

The Game of Life is now known to be Turing complete.

Conway and Martin Gardner

Conway's career is intertwined with that of mathematics popularizer and Scientific American columnist Martin Gardner. When Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. 

Gardner and Conway had first corresponded in the late 1950s, and over the years Gardner had frequently written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts (Jul 1967), Hackenbush (Jan 1972), and his angel and devil problem (Feb 1974). In the September 1976 column he reviewed Conway's book On Numbers and Games and even managed to explain Conway's surreal numbers.

Conway was probably the most important member of Martin Gardner's Mathematical Grapevine. He regularly visited Gardner and often wrote him long letters summarizing his recreational research. In a 1976 visit Gardner pretty much held him prisoner for a week, pumping him for information on the Penrose tilings which had just been announced.  Conway had discovered many (if not most) of the major properties of the tilings. Gardner used these results when he introduced the world to Penrose tiles in his January 1977 column. The cover of that issue of Scientific American features the Penrose tiles and is based on a sketch by Conway.

Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, and Conway himself has often been a featured speaker at these events, discussing various aspects of recreational mathematics.

Publications

• 2008 The Symmetries of Things (with Heidi Burgiel and Chaim Goodman-Strauss). A. K. Peters, Wellesley, MA, 2008, ISBN 1568812205.
• 1997 The Sensual (quadratic) Form (with Francis Yein Chei Fung). Mathematical Association of America, Washington, DC, 1997, Series: Carus mathematical monographs, no. 26, ISBN 1614440255.
• 1996 The Book of Numbers (with Richard K. Guy). Copernicus, New York, 1996, ISBN 0614971667.
• 1995 Minimal-Energy Clusters of Hard Spheres (with Neil Sloane, R. H. Hardin, and Tom Duff). Discrete & Computational Geometry, vol. 14, no. 3, pp. 237–259.
• 1988 Sphere Packings, Lattices, and Groups (with Neil Sloane). Springer-Verlag, New York, Series: Grundlehren der mathematischen Wissenschaften, 290, ISBN 9780387966175.
• 1985 Atlas of finite groups (with Robert Turner Curtis, Simon Phillips Norton, Richard A. Parker, and Robert Arnott Wilson). Clarendon Press, New York, Oxford University Press, 1985, ISBN 0198531990.
• 1982 Winning Ways for your Mathematical Plays (with Richard K. Guy and Elwyn Berlekamp). Academic Press, ISBN 0120911507.
• 1979 Monstrous Moonshine (with Simon P. Norton). Bulletin of the London Mathematical Society, vol. 11, issue 2, pp. 308–339.
• 1979 On the Distribution of Values of Angles Determined by Coplanar Points (with Paul Erdős, Michael Guy, and H. T. Croft). Journal of the London Mathematical Society, vol. II, series 19, pp. 137–143.
• 1976 On numbers and games. Academic Press, New York, 1976, Series: L.M.S. monographs, 6, ISBN 0121863506.
• 1971 Regular algebra and finite machines. Chapman and Hall, London, 1971, Series: Chapman and Hall mathematics series, ISBN 0412106205.