John Horton Conway

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Prolific mathematician known for the Game of Life whose inventions ranged from the surreal numbers to the Doomsday Rule


John Horton Conway, who has died aged 82 after contracting Covid-19, was one of the most prolific and charismatic British mathematicians of the 20th century. He was active in many branches of mathematics, including group theory, coding theory, knot theory, geometry, number theory and quadratic forms, as well as in recreational mathematics.

An iconoclastic academic, he held court for over half a century in mathematics departments worldwide, notably at Cambridge and Princeton universities. Although he thought it one of his lesser accomplishments, John is best known for his late 1960s invention of the Game of Life. It is an "organic life"simulation carried out on a square grid of cells, each of which is alive or dead according to how many living neighbours it has. Despite the simplicity of John's defining rules, it turned out that anything that can be algorithmically computed can be done so within the zero-player Game of Life. While John's approach was done with pen and paper, the game was a catalyst for computer programmers in the 70s and is now viewed as a watershed development in the field of cellular automata.

John, together with Elwyn Berlekamp and Richard Guy, is credited with co-founding the field of combinatorial game theory. That is the mathematical analysis of games such as noughts and crosses, draughts, chess and Go, as well as a wealth of original games that John and assorted collaborators devised over the decades, such as Phutball (short for Philosopher's Football), a two-person board game played on a grid using one white stone (the ball) and numerous black stones (representing men), Hackenbush, and Sprouts. This led to a joint book, Winning Ways for Your Mathematical Plays (1982).

In the early 70s, John came up with the definitive refinement of what is now known as the Monster Group. Groups arise from the study of symmetry of objects in mathematics and in nature. For instance, the group of symmetries of an equilateral triangle has six things in it: three rotations and three flips. There are standard "atomic building blocks"of all finite groups of symmetries, which are well understood, and 26 strange additional ones, called sporadic groups, of which the Monster Group, predicted independently by two mathematicians, Bernd Fischer and Bob Griess, is the largest. In 1973 Griess finally constructed the Monster itself, and John then came up with a simpler construction. This group has
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 symmetries in it. That is vast beyond comprehension, and about 1,000 times the current estimate for the number of atoms in the Earth.

A few years later, John went further. Along with his former PhD student Simon Norton, he came up with the Monstrous Moonshine conjecture, connecting group theory with the seemingly unrelated study of modular forms. Another of John's former students, Richard Borcherds, got a Fields medal (one of the two highest honours in mathematics) in 1998 for his successful proof of the conjecture.

John often said that his proudest invention was the surreal numbers, a unifying number system that he conceived in the late 60s, which encompassed ordinary numbers as well as those that are infinitely small or large. He was also particularly pleased with the free will theorem in quantum mechanics from early in the current century, which he formulated with Simon Kochen. In his own words, it says that "if experimenters have free will, then so do elementary particles".

John was a prodigious mental calculator too. He delighted in being challenged to shout out the day of the week for any randomly chosen date in history, which he could do using his own so-called Doomsday Rule; his methods were adapted and streamlined from Lewis Carroll's 1887 algorithm. He attributed his success in so many different arenas to his habit of always working simultaneously on several unrelated problems: he might be stuck on most of them but suddenly have an idea leading to a breakthrough on another one.

While many of John's creations were the products of his own fertile mind, he thrived on collaboration. His best known book, the Atlas of Finite Groups (1985), was co-authored by Norton, Robert Curtis, Richard Parker and Robert Wilson. Around 1976, he helped Roger Penrose refine the analysis of what we now know as Penrose tiles, and he came up with the catchy terms kite and dart for the two basic shapes from which they are made.

John left his mark on the groundbreaking book series The Art of Computer Programming. The computer scientist and author of that work, Donald Knuth, who also wrote a book about Conway's surreal numbers, said: "Although John was a pure mathematician, he covered so many bases that I've cited him more than 25 times (so far) for different contributions to The Art of Computer Programming."

The American writer Martin Gardner helped to popularise much of John's output in his Scientific American columns, going back to the late 50s. A decade ago, John worked with the editor Peter Renz on updated versions of some of Gardner's books of collected columns, again contributing fresh results of significance, including a new proof of Morley's theorem about triangles. "Anyone who has sat with John has been touched, changed for ever,"said Renz. "He had a way of refining things to their purest forms."

Born in Liverpool, John was the son of Agnes (née Boyce) and Cyril Horton Conway, a chemistry lab technician at the Liverpool Institute high school for boys. John got his secondary education at the Holt high school for boys, Childwall, then went to Gonville and Caius College, Cambridge, gaining his BA in 1959. A PhD (1964) followed, under Harold Davenport.

He became an assistant lecturer at Cambridge, later rising to be professor of mathematics. In 1987 he took up the position of John von Neumann professor of applied and computational mathematics at Princeton University, New Jersey.

While he was reportedly shy as a young man, over time John developed a disarming charm and an extrovert Pied Piper persona. He earned a reputation for delivering one brilliant lecture after another, and his classes at both Cambridge and Princeton were invariably oversubscribed. His infectious enthusiasm turned on generations of young people to the joy of research mathematics.

His biographer, Siobhan Roberts, dubbed him "the world's most lovable egomaniac", adding, "He is Archimedes, Mick Jagger, Salvador Dalí and Richard Feynman, all rolled into one."

John was elected FRS, a fellow of the Royal Society, in 1981 -following which he told people he was now officially a Filthy Rotten Swine -and was the first recipient of the London Mathematical Society's Pólya prize in 1987. He was awarded the Nemmers prize in mathematics (1998), the Leroy P Steele prize for mathematical exposition (2000), and the Joseph Priestley award (2001-02), and received honorary doctorates from the University of Liverpool (2001) and Jacobs University, Bremen (2015). He retired in 2013, when he became emeritus professor at Princeton.

In a 2014 video for the online channel Numberphile, John discussed getting over his feelings of inadequacy in his 20s, facing mortality in his 70s, and his lingering mathematical wish to really understand why the extraordinarily large Monster Group exists.

His first two marriages, to Eileen Howe and Larissa Queen, ended in divorce. He is survived by his third wife, Diana (nee Cutsogeorge), whom he married in 2001, and their son, Gareth; four daughters, Susie, Rosie, Ellie and Annie, from his first marriage; two sons, Alex and Oliver, from his second marriage; three grandchildren; and six great-grandchildren.

John Horton Conway, mathematician, born 26 December 1937; died 11 April 2020

Colm Mulcahy

29 Apr 2020 © Guardian Newspapers Limited 2020