SCIENCE sciencemag.orgPHOTO: DENISE APPLEWHITE/PRINCETON UNIVERSITY, OFFICE OF COMMUNICATIONS
By Matt BakerJ
ohn Horton Conway, renowned math-
ematician of legendary creativity,
died on 11 April at age 82. Conway’s
playful approach to mathematics is
visible in his game-changing contribu-
tions to a wide variety of mathemati-
cal fields. Conway was also a celebrated and
gifted educator whose enthusiasm, charisma,
and inventiveness captured the public’s
imagination.
Conway was born in Liverpool, England,
on 26 December 1937. He received his math-
ematics degrees from the University of
Cambridge: a bachelor’s degree in 1959 and
a Ph.D. 5 years later. He was subsequently
hired by the university as an assistant lec-
turer, eventually rising to the rank of pro-
fessor. In 1987, he accepted a position as the
John von Neumann Professor in Applied and
Computational Mathematics at Princeton
University in Princeton, New Jersey, where
he remained for the rest of his career.
Conway’s first big mathematical break-
through, in 1968, was the determination of
the more than eight quintillion symmetries
of the Leech lattice, an astonishingly sym-
metric mathematical structure that governs
the tightest and most efficient way to pack
spheres together in 24 dimensions. (Think
of an array of stacked oranges in a grocery
store, except the oranges are 24-dimensional
and each one touches exactly 196,560 other
oranges.) Through these investigations in the
field of group theory, Conway discovered sev-
eral additional groups, including 3 of the 26
so-called sporadic groups, which play an im-
portant role in mathematical physics and the
theory of error-correcting codes.
Conway later spearheaded a collaboration
that resulted in the Atlas of Finite Groups,
one of the most important (and longest)
books ever written on group theory. Together
with Atlas coauthor Simon Norton, Conway
put forth a series of fascinating conjectures
relating the largest of the sporadic groups
(which Conway dubbed “The Monster”) to
the theory of modular forms, a previously
unrelated subject arising in complex vari-
ables and number theory. These conjectures
became known as “Monstrous Moonshine,”
and Conway’s former Ph.D. student RichardBorcherds won the prestigious Fields Medal
for verifying them.
Conway’s most famous invention was un-
doubtedly the Game of Life. An example of
what is now called a cellular automaton, this
game uses simple rules that lead to wildly un-
predictable behavior. A Scientific American
column written by Conway’s lifelong friend
and champion Martin Gardner turbocharged
the game’s popularity and vaulted Conway
to international prominence. The simulation
became a frequent pastime for budding com-
puter programmers at the dawn of the age of
personal computers.
Conway’s proudest mathematical moment
came about when he parlayed his study of
the winning strategies for certain two-playergames into the invention of surreal numbers.
These numbers contain not only all real num-
bers (i.e., infinite decimals such as pi or the
square root of 2) but also a staggering cornu-
copia of new numbers, some infinitely large
and some infinitely small. The unification of
infinite set theory and the theory of combina-
torial games that resulted from this work was
hailed by Gardner as “an astonishing feat of
legerdemain.”
Conway made numerous other contribu-
tions to mathematics in a dizzying array of
fields. In topology, the Conway polynomial
is a fundamental tool for studying knots. In
geometry, Conway and mathematician and
computer scientist Michael Guy generalized
Archimedes’ classical enumeration of the 13
three-dimensional so-called “Archimedean
solids” to four dimensions. In number
theory, Conway and his student WilliamSchneeberger discovered an important and
unexpected result called the “15 theorem”
and formulated a conjectural generalization
(the “290 theorem”), which was later proved
correct. In theoretical physics, Conway and
mathematician Simon Kochen proved a strik-
ing result in quantum mechanics that they
christened the “free will theorem.” Conway
phrased the result in layman’s terms as fol-
lows: “If experimenters have free will, then so
do elementary particles.”
Conway was a captivating teacher. He
spent 2 weeks every summer with aspiring
young mathematicians at the Canada/USA
Mathcamp, where he would give sponta-
neous lectures on whatever mathematical
subject the students requested. His lectures
at Cambridge, and later Princeton, were fa-
mously idiosyncratic. He would balance ob-
jects on his chin, perform magic tricks with
a piece of rope, recite the digits of pi (he had
memorized more than 1000 of them), and re-
gale students with his astounding knowledge
of etymology (a lifelong passion).
I knew Conway through the biennial
Gathering 4 Gardner Conference. His no-
toriously outsized ego was occasionally off-
putting, but in the words of biographer
Siobhan Roberts, “Conway’s is a jocund
and playful egomania, sweetened by self-
deprecating charm.” Once, while visiting
Princeton, I ran into Conway, and we ended
up chatting for almost 2 hours about differ-
ent methods for mentally calculating the day
of the week for any given date. Conway’s as-
tounding speed (in his prime, he could do it
in under 2 seconds) made a convincing case
for his own doomsday rule, one of his famous
contributions to recreational mathematics.
Despite his considerable erudition and
numerous mathematical awards, Conway
referred to himself as a “professional non-
understander,” claiming that he only com-
prehended things after thinking “for ages
and [making] them very, very simple.” He fol-
lowed his insatiably peripatetic curiosity no
matter its direction, and the professional suc-
cess he achieved in this way led him to adopt
the dictum: “Thou shalt stop worrying and
feeling guilty; thou shalt do whatever thou
pleasest.” Another of Conway’s guiding prin-
ciples was to always go multiple steps beyond
what any reasonable person would do.
In addition to his scientific breakthroughs,
Conway transformed the public perception
of mathematics. His singularly creative ideas
have inspired generations of mathematicians.
Although “recreational mathematics” may
seem like an oxymoron to some, the term
perfectly captures Conway’s joyful approach
to both serious and recreational mathemati-
cal discoveries. j10.1126/science.abc5331RETROSPECTIVEJohn Horton Conway (1937–2020)
Innovative mathematician and passionate educator
School of Mathematics, Georgia Institute of Technology,
Atlanta, GA, USA. Email: [email protected]22 MAY 2020 • VOL 368 ISSUE 6493 831
Published by AAAS