John Horton Conway (1937-2020)

John Horton Conway, renowned mathematician of legendary creativity, died on 11 April at age 82. Conway's playful approach to mathematics is visible in his game-changing contributions to a wide variety of mathematical 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 mathematics 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 lecturer, eventually rising to the rank of professor. 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 breakthrough, in 1968, was the determination of the more than eight quintillion symmetries of the Leech lattice, an astonishingly symmetric 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 several additional groups, including 3 of the 26 so-called sporadic groups, which play an important 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 variables and number theory. These conjectures became known as “Monstrous Moonshine,” and Conway's former Ph.D. student Richard Borcherds won the prestigious Fields Medal for verifying them. Conway's most famous invention was undoubtedly the Game of Life. An example of what is now called a cellular automaton, this game uses simple rules that lead to wildly unpredictable 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 computer 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-player games into the invention of surreal numbers. These numbers contain not only all real numbers (i.e., infinite decimals such as pi or the square root of 2) but also a staggering cornucopia of new numbers, some infinitely large and some infinitely small. The unification of infinite set theory and the theory of combinatorial games that resulted from this work was hailed by Gardner as “an astonishing feat of legerdemain.” Conway made numerous other contributions 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 William Schneeberger 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 striking result in quantum mechanics that they christened the “free will theorem.” Conway phrased the result in layman's terms as follows: “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 spontaneous lectures on whatever mathematical subject the students requested. His lectures at Cambridge, and later Princeton, were famously idiosyncratic. He would balance objects 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 regale students with his astounding knowledge of etymology (a lifelong passion). I knew Conway through the biennial Gathering 4 Gardner Conference. His notoriously 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 different methods for mentally calculating the day of the week for any given date. Conway's astounding 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 nonunderstander,” claiming that he only comprehended things after thinking “for ages and [making] them very, very simple.” He followed his insatiably peripatetic curiosity no matter its direction, and the professional success 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 principles 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 mathematical discoveries.