John Conway died on April 11th

AS HE STROLLED in his dishevelled, friendly way through the hallways and common rooms of Cambridge and, later, Princeton, John Conway liked to feel he had about him everything he might require. Pennies in his pockets, to set spinning on the edge of a table to prove that more would fall heads than tails. A pen and paper for his game of Sprouts, which required spots to be joined up with curves that passed neither through old spots nor old curves. A board with a grid and stones, or peanuts at a pinch, to plot more of the games that fizzed from him continually. A couple of bits of rope for Twists and Turns, where four players, each holding an end, would change places turn by turn into nicely tangled permutations. And possibly as a party piece a wire coat-hanger, to bend into a square and whirl round his head, while ensuring that a coin balanced on the hook did not fly away. With any of these he would waylay the unwary, and challenge them to play.

The game for which he had become world-famous, though, needed no players and never ended. It was called the Game of Life, and was played out on a grid where “live” or “dead” cells interacted with their neighbours, second by second, according to three rules. If a cell had four or more neighbours, it died of overcrowding. If it had no neighbours, or only one, it died of isolation. If a dead cell had three live neighbours, it became a live cell. As cells lived and died, the formation moved. The game had taken 18 months of coffee-times to think up, plotting with pen and paper, but when it was described in 1970 in Scientific American it became a sensation. Legend had it that in the early 1970s a quarter of the world’s computers were playing it. A whole new field of mathematics, cellular automata, also sprang out of it, and celebrity descended.

That didn’t please him. The fame was all right; he was a natural show-off, roaring like a lion to get his students’ attention, throwing off his sandals at the start of a lecture, swinging from pipes when the mood took him, reciting pi to 1,111+ places and devising a Doomsday algorithm that allowed him to increase the speed at which he could tell, for any date, what day of the week it was. (To keep his mind agile, he programmed his computer to ask him ten random dates before he could log on.) But the Game of Life came to overshadow the more important things he had done in mathematics. He had made contributions to algebra, geometry, knot theory and coding theory, as well as game theory, and in two respects he had certainly got further than anyone had before.

The first was his discovery of surreal numbers, a universal ordered field that included the infinitely large and the infinitesimally small, and contained all the reals, fractions, rationals, superreals and hyperreals. He found ways to use them in arithmetic, adding, subtracting, multiplying and dividing with them. The second was his work in group theory. In 1966 he took on the challenge of finding the exquisite symmetry which was presumed to belong to the Leech lattice, a dense packing of spheres in 24 dimensions with the lattice formed by joining their central points. He deduced that the lattice contained 8, 315, 553, 613, 086, 720, 000 symmetries, a group which was given his name and made his reputation. It also led him further, to his “Atlas of Finite Groups”, written over 15 years, and his happy theorising about the Monster group, a “thing”—he could not find another simple name for it—which existed in 196,883 dimensions. It frustrated him that he couldn’t see the beauty of such symmetries, as he admired the almost-lattice-points of the stars, until he had done the calculations, often on rolls of wallpaper-lining that spooled for yards across the floor. But his work in the field earned him fellowship of the Royal Society, in the same big book as Newton and Einstein.

He was worthily there, he felt, and his route had been impressively single-minded: reciting the powers of two at the age of four, deciding to read maths at Cambridge at the age of 11, a doctorate in set theory, assistant lecturer, professor at Cambridge by 1983, lured to Princeton in 1987. But his approach was, as he admitted, lazy. He was poor at publishing his work, and simply liked to go wherever curiosity took him. In his younger years this bothered him, but the Leech-lattice work cleared his head, and he made “The Vow”: “Thou shalt stop worrying and…do whatever thou pleasest.”

Vow taken, he had fun. For his students he made abstruse theorems simple and everyday, such as by carving a turnip into a 20-triangular-faced icosahedron, snacking on it as he went, to illustrate Platonic solids. In his subject he became a magpie. While his colleagues laboured at research in their rooms, he would be folding bits of paper into flexagons or collecting pine cones, to see how many had a Fibonacci number of spirals (2, 3, 5, 8) and how many had a Lucas number, which would approach the golden ratio. Or, ensconced in some hallway nook, he would just observe a game. It had been while watching Go players that he realised each game contained many sub-games; and this had led him, first, to surreal numbers, and second to the light-bulb thought that playing games was not a distraction from mathematics. It wasmathematics.

As a magpie, though, hopping after any bit of plastic papered with gold, he drew back from some of the vaster ideas his “thinkering” touched on. Enthusiasts often said the Game of Life modelled not merely life but the universe, anything and everything. He doubted that. He hoped that surreal numbers might lead to something “greater”, but did not pursue that path himself. From 2004 he worked on the Free Will Theorem, which proposed that if experimenters had free will to decide what quantities to measure in an experiment, then elementary particles could choose how to spin. He threw out the provocation and left it there.

The question that dogged him most concerned the Monster group. Its enormous number of dimensions was not arbitrary. So what was it all about, and why was it there? On and off, he would have a think about it. He would like to have known. But meanwhile, no one had quite solved his Piano Problem: what was the largest object that could be manoeuvred round a right-angled corner in a fixed-width corridor? That would be good to know, too. ■