John Horton Conway

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 now holds the title John von Neumann Professor Emeritus.

Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age; his mother has recalled that he could recite the powers of two when he was four years old. 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 the University of Cambridge.

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

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'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.

Combinatorial game theory

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

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, 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, son of Richard 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 has 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 describes rules for deciding if a prototile will tile the plane.

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

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, nowadays 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. See Topology Proceedings 7 (1982) 118.

Group theory

He 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 in which 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.

Algebra

Conway has written textbooks and done original work in algebra, focusing particularly on quaternions and octonions.Together with Neil Sloane, he invented the icosians.

Analysis

He invented a 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.

Algorithmics

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 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.

Theoretical physics

In 2004, Conway and Simon B. 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, 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. In Conway's provocative wording: "if experimenters have free will, then so do elementary particles."

Conway received the Berwick Prize (1971), was elected a Fellow of the Royal Society (1981), was the first recipient of the Pólya Prize (LMS) (1987), won the Nemmers Prize in Mathematics (1998) and received the Leroy P. Steele Prize for Mathematical Exposition (2000) of the American Mathematical Society.

His 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).

In 2017 Conway was given honorary membership of the British Mathematical Association.

• 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.

• Alpert, Mark (1999). Not Just Fun and Games Scientific American, April 1999
• Conway, John and Smith, Derek A. (2003). On quaternions and Octonions : their Geometry, Arithmetic, and Symmetry Bull. Amer. Math. Soc. 2005, vol=42, issue=2, pp. 229–243, ISBN 1568811349
• Boden, Margaret (2006). Mind As Machine, Oxford University Press, 2006, p. 1271
• Case, James (2014). Martin Gardner’s Mathematical GrapevineBook Reviews of Undiluted Hocus-Pocus: The Autobiography of Martin Gardner and Martin Gardner in the Twenty-First Century, By James Case, SIAM News, April 01, 2014
• Conway, John and Sigur, Steve (2005). The Triangle Book AK Peters, Ltd, 15 June 2005, ISBN 1568811659
• du Sautoy, Marcus (2008). Symmetry, HarperCollins, p. 308
• Guy, Richard K (1983). Conway's Prime Producing Machine Mathematics Magazine, Vol. 56, No. 1 (Jan. 1983), pp. 26–33
• Harris, Michael (2015). Review of Genius At Play: The Curious Mind of John Horton Conway Nature, 23 July 2015
• Mulcahy, Colm (2014). The Top 10 Martin Gardner Scientific American Articles Scientific American, October 21, 2014
• Roberts, Siobhan (2015). Genius at play: The curious mind of John Horton Conway. Bloomsbury. ISBN 978-1620405932.
• O'Connor, John J.; Robertson, Edmund F., "John Horton Conway", MacTutor History of Mathematics archive, University of St Andrews.
• John Horton Conway at the Mathematics Genealogy Project
• Princeton University (2009). Bibliography of John H. Conway Mathematics Department
• Rendell, Paul (2015). Turing Machine Universality of the Game of Life Springer, July 2015, ISBN 3319198416
• Seife, Charles (1994). Impressions of Conway The Sciences