Significant figures: John Conway

Thane Plambeck, CC BY 2.0

Jamie Handitye and Jakob Stein

Mymost memorable encounter with the work of late mathematician John Horton Conway came from a friend of mine I met as a first year graduate student. As we sat across from each other in the department common room, each having made little progress with our research, he slid me a piece of paper with five dots drawn on it. This game, he explained, consisted of us each taking turns to draw a line between any two dots, with the midpoint of the line we drew then counting as an additional dot. Although the lines could bend in any direction, they were not allowed to intersect each other, and each dot could join at most three line segments. The game was over when one player could not make any more moves, and the other player was declared the winner. At first, I was quickly defeated, and I spent quite some time trying to come up with the best strategies against my skilled opponent.

The game that we spent our lunchtime playing was Sprouts, invented by Conway and his friend Michael Paterson during their time at the University of Cambridge, and was later popularised by Martin Gardner in his Scientific American columnMathematical Games. Conway is perhaps best known for his interest in games: he invented many, and his two books on the subjectOn numbers and gamesandWinning ways for mathematical plays include detailed analyses of many two-player games. He was a regular contributor to Gardner’s column, and was a major figure in the world of recreational mathematics in his own right. As a graduate student, Conway proved that every positive whole number can be written as the sum of at most 37 fifth powers.

Born in 1937, Conway grew up in Liverpool, One of Conway’s most famous inventions was and attended Cambridge as an undergraduthegame of life, a very simple type of ‘game’, ate, staying on for his postdoctoral research in that takes place on a grid of pixels. Each pixel number theory, and eventual appointment as starts either switched on or off, and each sec- fellow and lecturer. He moved to Princeton in ond, any off pixel with exactly three on neigh- 1986, where he remained for the rest of his cabours will also switch on, and only on pix- reer. According to those who knew him, he was els with two or three on neighbours will re- always ready to play: he would carry around main on. Despite these simple rules, the game puzzles, pennies, coat-hangers, and dice on of life is actually so-called Turing complete, him, ready to stoke the imagination of some meaning that in theory, any computer pro- unwitting colleague with a lively demonstragramme could be run using these pixels. This tion or challenge. Often described as charisis an example of acellular automaton, for more matic, he certainly fulfilled certain stereotypes see pages35–38. of the eccentric mathematician, but was also an inspiration for many of those he taught and spoke to, and remains so even after his death in April 2020 from complications due to Covid-19.

Conway was a prominent mathematician, not only dedicated to his work on popular games: on the contrary, his willingness to approach any topic with the same enthusiasm led to him contributing to research fields across mathematics. His interests included number theory, topology, analysis, group theory, classical geometry, even theoretical physics. Analysts, for example, may be familiar with his base 13 function, a function that takes every value between 0 and 1, but is discontinuous everywhere. Among academics, he is better known for his work in group theory, in particular on sporadic simple groups and the Monstrous Moonshine conjecture: a mathematical theory that connects the sporadic groups, mysterious algebraic structures coming from group theory, with functions called modular forms, coming from analysis. His name continues to be relevant, not only through his own considerable research, but also through those who took inspiration from him. In 2018, in a branch of topology called knot theory, a long-standing conjecture was solved by then-graduate student Lisa Piccirillo, which involved the classification of a knot which bears Conway’s name.

But for all his contributions, it was Conway’s willingness to collaborate, and share his love of ideas, that are an example for all those interested in mathematics. So, to live by that example, I encourage you to pick up some pencils and sheet of paper, find a friend, and go play a game ofSprouts.

Jamie Handitye Jamie is a second year mathematics student at Christ’s College Cambridge. His main interests are in group theory, number theory, and a touch of algebraic geometry.

Jakob Stein Jakob is a PhD student and mathematician from London and works mainly in differential geometry. In his spare time, he likes to draw, and think about mathematics in art. a @jakob_media