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82 Obituary John Conway
The game of maths
John Horton Conway, mathematician, died of covid-19 on
April 11th, aged 82
s he strolled in his dishevelled, friendly way through the
Ahallways and common rooms of Cambridge and, later, Prince-
ton, 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 angular-faced icosahedron, snacking on it as he went, to illustrate
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 bits of paper into flexagons or collecting pine cones, to see how
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 permuta-
tions. And possibly as a party piece a wire coat-hanger, to bend into had been while watching Go players that he realised each game
a square and whirl round his head, while ensuring that a coin bal-
anced 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 ering” touched on. Enthusiasts often said the Game of Life mod-
with their neighbours, second by second, according to three rules.
If a cell had four or more neighbours, it died of overcrowding. If it doubted that. He hoped that surreal numbers might lead to some-
had no neighbours, or only one, it died of isolation. If a dead cell thing “greater”, but did not pursue that path himself. From 2004 he
had three live neighbours, it became a live cell. As cells lived and worked on the Free Will Theorem, which proposed that if experi-
died, the formation moved. The game had taken 18 months of cof-
fee-times to think up, plotting with pen and paper, but when it was
described in 1970 in Scientific American it became a sensation. Le-
gend had it that in the early 1970s a quarter of the world’s comput-
ers were playing it. A whole new field of mathematics, cellular au-
tomata, 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, throw-
ing 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
The Economist April 25th 2020
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 mathemat-
ics. 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 or-
dered field that included the infinitely large and the infinitesimal-
ly small, and contained all the reals, fractions, rationals, super-
reals and hyperreals. He found ways to use them in arithmetic,
adding, subtracting, multiplying and dividing with them. The sec-
ond 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 beau-
ty of such symmetries, as he admired the almost-lattice-points of
the stars, until he had done the calculations, often on rolls of wall-
paper-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 impres-
sively 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 theo-
rems simple and everyday, such as by carving a turnip into a 20-tri-
Platonic solids. In his subject he became a magpie. While his col-
leagues laboured at research in their rooms, he would be folding
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
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 was mathematics.
As a magpie, though, hopping after any bit of plastic papered
with gold, he drew back from some of the vaster ideas his “think-
elled not merely life but the universe, anything and everything. He
menters 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. 7