John Horton Conway

Liverpool to Cambridge

Conway was born in Liverpool and, by his own account, knew he wanted to be a mathematician by the age of eleven. He read mathematics at Gonville and Caius College, Cambridge, completing a PhD in 1964 under Harold Davenport on number theory - work he was privately dissatisfied with. He stayed at Cambridge as a lecturer and, for a while, felt himself to be underachieving badly.

What changed things was a decision to stop worrying about whether a problem was important and simply work on whatever was interesting. It is a strategy that usually produces a dilettante. In his case it produced one of the broadest bodies of work in twentieth-century mathematics.

The Conway groups

In 1968 Conway spent a concentrated period - the story is that he set aside from six on Wednesday evening and midnight on Saturday, and finished in twelve hours - working out the symmetry group of the Leech lattice, an extraordinarily dense packing of spheres in 24 dimensions. Out of it came three previously unknown sporadic simple groups, now called Co₁, Co₂ and Co₃.

The result made his reputation overnight and drew him into the enormous international effort to classify all finite simple groups. He was a principal author of the ATLAS of Finite Groups (1985), the reference work that catalogues them, and later contributed to monstrous moonshine - the astonishing connection between the Monster group and modular functions, a phrase he coined and a conjecture whose proof won Richard Borcherds a Fields Medal.

The Game of Life

Around 1968-1969, working with a group of students and colleagues in the Cambridge common room, Conway set out to simplify von Neumann's self-reproducing automaton. The search was carried out largely by hand on a Go board, with counters pushed around and rules discarded when they produced boards that either died instantly or filled up with grey mush.

The criteria he settled on were aesthetic as much as mathematical: no obvious unbounded growth, but at least the suspicion of it; and small patterns that ran a long time before revealing their fate. B3/S23 satisfied all of them.

Martin Gardner published it in his Mathematical Games column in Scientific American in October 1970, and it escaped immediately. Within months people around the world were running it on institutional mainframes, usually without asking. Conway's $50 prize for a pattern with unbounded growth was claimed in November 1970 by Bill Gosper at MIT with the glider gun.

It's mathematically interesting, and I'm not ashamed of it, but I don't want it to be the only thing I'm known for.
Conway, on the Game of Life, more or less continuously from 1971 onwards

His relationship with it was genuinely ambivalent. He acknowledged that it had introduced more people to mathematics than anything else he did, and he could be charming about it. He also felt it had eclipsed work he regarded as far more significant, and he was capable of visible irritation at being introduced as "the Game of Life man". Late in his life he made a certain peace with it.

Surreal numbers

The work Conway himself rated most highly came out of studying the endgames of Go. Analysing them led him to construct - in a single, dazzling recursive definition - a class of numbers containing the reals, the ordinals, and infinitely many infinitesimals besides. Donald Knuth named them surreal numbers and, unusually, published them first, in a 1974 novella. Conway's own account came in On Numbers and Games (1976).

The construction is startling in its economy: a number is a pair of sets of previously constructed numbers, and everything else - arithmetic, ordering, infinity, infinitesimals - falls out. It is the closest thing mathematics has to creating a universe from nothing.

Games, sequences and everything else

A partial list of things named after him or discovered by him:

Combinatorial game theory, largely founded with Winning Ways for your Mathematical Plays (1982), co-written with Elwyn Berlekamp and Richard Guy.
Sprouts and Phutball (Philosopher's Football), both still played.
The Look-and-Say sequence (1, 11, 21, 1211, 111221…), whose lengths grow by a constant factor - Conway's constant, 1.303577…, the root of a degree-71 polynomial he derived.
The Doomsday algorithm for calculating the day of the week of any date in your head. He kept a program on his computer that quizzed him on it before letting him log in.
The Conway criterion for whether a shape tiles the plane.
Conway's Soldiers, a peg-solitaire problem with a beautiful impossibility proof.
Work on the Collatz conjecture, proving that a natural generalisation of it is undecidable.
The Free Will Theorem (2006), with Simon Kochen.

As a teacher

Conway moved to Princeton in 1987, where he was John von Neumann Professor in Applied and Computational Mathematics until his retirement. He was famous for doing mathematics in public - in corridors, on napkins, with whatever was to hand. He carried dice, rope, decks of cards, models of polyhedra and a coat hanger he used to demonstrate that any flat object has a balance point. Colleagues describe an office that was less a room than a geological formation.

He was also, by all accounts, magnetic to talk to and completely unbothered by hierarchy: an undergraduate with an interesting question got the same attention as a visiting professor.

Death

Conway died on 11 April 2020 in New Brunswick, New Jersey, of complications from COVID-19. He was 82. The mathematical world's response was immediate and enormous - and a great deal of it took the form of gliders.

The glider. Five cells, discovered in 1969 by Richard Guy while the group was tracking the R-pentomino across a Go board. It is the smallest thing in Life that goes anywhere, and it is what Conway is remembered by - whether he liked it or not.