John Horton Conway

Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Cambridge.

Conway was awarded a BA in 1959 and, supervised by Harold Davenport, began to undertake research in number theory. Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway became interested in infinite ordinals.

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

Conway and Martin Gardner Conway's career was intertwined with that of 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. For instance, he discussed Conway's game of Sprouts (July 1967), Hackenbush (January 1972), and his angel and devil problem (February 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 a prominent 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 kept him 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. it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics. The LifeWiki is 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. Conway came to dislike how discussions of him heavily focused on his Game of Life, feeling that it overshadowed deeper and more important things he had done, although he remained proud of his work on it. The game helped to launch a new branch of mathematics, the field of cellular automata. The Game of Life is known to be Turing complete.

Combinatorial game theory Conway contributed to combinatorial game theory (CGT), a theory of partisan games. He developed the theory with Elwyn Berlekamp and Richard Guy, and also co-authored the book Winning Ways for your Mathematical Plays with them. He also wrote On Numbers and Games (ONAG) which lays out the mathematical foundations of CGT.

He was also one of the inventors of the game sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conway's soldiers. He came up with the angel problem, which was solved in 2006.

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 novelette by Donald Knuth. He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG.

Geometry In the mid-1960s with Michael Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only non-Wythoffian uniform polychoron. Conway also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation.

In the theory of tessellations, he devised the Conway criterion which is a fast way to identify many prototiles that tile the plane.

He investigated lattices in higher dimensions and was the first to determine the symmetry group of the Leech lattice.

Together with Neil Sloane, he invented the icosians.

Geometric topology In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. 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 and invented a system of notation for tabulating knots, now known as Conway notation, while correcting a number of errors in the 19th-century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. The Conway knot is named after him.

Conway's conjecture that, in any thrackle, the number of edges is at most equal to the number of vertices, is still open.

Group theory Conway was the primary author of the ATLAS of Finite Groups giving properties of many finite simple groups. Working with his colleagues Robert Curtis and Simon P. Norton he constructed the first concrete representations of some of the sporadic groups. More specifically, he discovered three sporadic groups based on the symmetry of the Leech lattice, which have been designated the Conway groups. This work made him a key player in the successful classification of the finite simple groups.

Based on a 1978 observation by mathematician John McKay, Conway and Norton formulated the complex of conjectures known as monstrous moonshine. This subject, named by Conway, relates the monster group with elliptic modular functions, thus bridging two previously distinct areas of mathematics—finite groups and complex function theory. Monstrous moonshine theory has now been revealed to also have deep connections to string theory.

Conway introduced the Mathieu groupoid, an extension of the Mathieu group M12 to 13 points.

Number theory As a graduate student, he proved one case of a conjecture by Edward Waring, that every integer could be written as the sum of 37 numbers each raised to the fifth power, though Chen Jingrun solved the problem independently before Conway's work could be published. In 1972, Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable. Related to that, he developed the esoteric programming language FRACTRAN. While lecturing on the Collatz conjecture, Terence Tao (who was taught by him in graduate school) mentioned Conway's result and said that he was "always very good at making extremely weird connections in mathematics".

Algebra Conway published original work on algebraic structures, focusing particularly on quaternions and octonions.

Analysis He invented his base 13 function as a counterexample to the converse of the intermediate value theorem: the function takes on every real value in each interval on the real line, so it has a Darboux property but is not continuous.

Algorithms For calculating the day of the week, he invented the Doomsday algorithm. The algorithm is simple enough for anyone with basic arithmetic ability to do the calculations mentally. Conway could usually give the correct answer in under two seconds. To improve his speed, he practised his calendrical calculations on his computer, which was programmed to quiz him with random dates every time he logged on. One of his early textbooks was Stephen Kleene's theory of finite-state machines.

Theoretical physics In 2004, Conway and Simon B. Kochen, another Princeton mathematician, proved the free will theorem, a version of the "no hidden variables" principle of quantum mechanics. It states that given certain conditions, if an experimenter can freely decide what quantities to measure in a particular experiment, then elementary particles must be free to choose their spins to make the measurements consistent with physical law. Conway said that "if experimenters have free will, then so do elementary particles."

Personal life and death Conway was married three times. With his first two wives he had two sons and four daughters. He married Diana in 2001 and had another son with her. He had three grandchildren and two great-grandchildren. On 11 April, he died in New Brunswick, New Jersey, United States, at the age of 82.

Awards and honours Conway received the Berwick Prize (1971), was elected a fellow of the Royal Society (1981), became a fellow of the American Academy of Arts and Sciences in 1992, was the first recipient of the Pólya Prize (LMS) (1987), and in 2014 one from Alexandru Ioan Cuza University. His Fellow of the Royal Society nomination in 1981 reads: A versatile mathematician who combines a deep combinatorial insight with algebraic virtuosity, particularly in the construction and manipulation of "off-beat" algebraic structures which illuminate a wide variety of problems in completely unexpected ways. He has made distinguished contributions to the theory of finite groups, to the theory of knots, to mathematical logic (both set theory and automata theory) and to the theory of games (as also to its practice).

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

Select publications

1971 – Regular algebra and finite machines. Chapman and Hall, London, 1971, Series: Chapman and Hall mathematics series. 1976 – On numbers and games. Academic Press, New York, 1976, Series: L.M.S. monographs, 6. 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. 1979 – Monstrous Moonshine (with Simon P. Norton). Bulletin of the London Mathematical Society, vol. 11, issue 2, pp. 308–339. 1982 – Winning Ways for your Mathematical Plays (with Richard K. Guy and Elwyn Berlekamp). Academic Press. 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. 1988 – Sphere Packings, Lattices and Groups (with Neil Sloane). Springer-Verlag, New York, Series: Grundlehren der mathematischen Wissenschaften, 290. 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. 1996 – The Book of Numbers (with Richard K. Guy). Copernicus, New York, 1996. 1997 – The Sensual (quadratic) Form (with Francis Yein Chei Fung). Mathematical Association of America, Washington, DC, 1997, Series: Carus mathematical monographs, no. 26. 2002 – On Quaternions and Octonions (with Derek A. Smith). A. K. Peters, Natick, MA, 2002. 2008 – The Symmetries of Things (with Heidi Burgiel and Chaim Goodman-Strauss). A. K. Peters, Wellesley, MA, 2008.