1937 – 2020 • The Magical Mathematician
The most creative and playful mathematician of his era, who invented the Game of Life, surreal numbers, and made profound contributions to group theory, knot theory, and combinatorial game theory.
John Horton Conway was born on December 26, 1937 in Liverpool, England. His father Cyril was a chemistry laboratory assistant. From an early age, Conway was fascinated by mathematics and was able to recite powers of two as a child.
He entered Gonville and Caius College, Cambridge in 1956, initially interested in number theory. He described himself as a lazy student who "wanted to read interesting things," but his talent was obvious.
His PhD thesis (1964) was on number theory (Waring's problem for fifth powers). But he was already gravitating toward the combinatorial and recreational mathematics that would define his career.
He became a Fellow of Sidney Sussex College and then, from 1986 until his death, the John von Neumann Professor at Princeton — a fitting title for a polymath of extraordinary range.
Discovered three new sporadic simple groups (Co1, Co2, Co3) while studying the symmetries of the Leech lattice in 24 dimensions. This was deep, serious group theory that established his reputation.
Invented the cellular automaton "Life" with simple rules that produce astonishing complexity. Popularized by Martin Gardner's Scientific American column, it became the most famous cellular automaton and inspired decades of research.
Discovered the surreal numbers: a number system containing both the reals and the ordinals, arising naturally from combinatorial game theory. Donald Knuth wrote a novella about them: "Surreal Numbers."
With Norton, elaborated the "Monstrous Moonshine" conjectures connecting the Monster group to modular functions. Borcherds proved these conjectures (Fields Medal, 1998), opening deep connections between algebra and string theory.
Conway worked at the boundary between "recreational" and "serious" mathematics, showing that games, puzzles, and play can lead to deep theory. His work vindicated the tradition of Euler, Gauss, and Hilbert, who all valued mathematical play.
Martin Gardner's Scientific American column provided a unique platform for Conway's ideas, reaching millions of readers and inspiring a generation of mathematicians and computer scientists.
The 20th century's greatest collaborative project in pure mathematics was the classification of all finite simple groups, completed around 1981. Conway's discovery of three sporadic groups and his analysis of the Leech lattice were major contributions.
The Monster group, the largest sporadic simple group, has about 8 x 10^53 elements. Its mysterious connections to number theory (Moonshine) remain an active area of research.
Group Theory Recreational Mathematics Cellular Automata
The Game of Life (1970) is a cellular automaton on an infinite grid. Each cell is alive or dead. At each step:
These simple rules produce universal computation: Life can simulate any Turing machine. It generates gliders, oscillators, spaceships, and self-replicating patterns of extraordinary complexity.
Conway proved Life is Turing complete, meaning it can in principle compute anything computable.
Conway proved Life can simulate a universal Turing machine. Glider guns (discovered by Gosper, 1970) emit streams of gliders that carry information. Logic gates, memory, and programs can all be built from gliders, making Life a general-purpose computer.
In 2010, a self-replicating pattern ("Gemini") was constructed in Life, vindicating von Neumann's theoretical framework for self-reproducing automata. Conway's simple rules support the full complexity of universal construction.
Conway offered $50 for a pattern that grows without bound. Bill Gosper won by discovering the "glider gun" (1970), which fires a new glider every 30 generations. This was the first known infinite-growth pattern.
Life inspired research in cellular automata, artificial life, complex systems, and emergence. Wolfram's "A New Kind of Science" and the study of self-organization both trace partly to Conway's invention.
Conway's surreal numbers form the largest ordered field: they contain all real numbers, all ordinal numbers (including transfinite ones), and infinitesimal quantities.
A surreal number is defined recursively as a pair {L | R} where L and R are sets of previously constructed surreals, with every element of L less than every element of R.
Starting from {|} = 0, one gets {0|} = 1, {|0} = -1, {0|1} = 1/2, and so on, eventually generating all reals, all ordinals, and numbers like omega - 1 and 1/omega (an infinitesimal).
Surreal numbers arose from analyzing combinatorial games (Go, Nim, Hackenbush). Conway showed that every two-player perfect-information game has a "value" that is a surreal number (or a generalization called a "game"). This created an entire field.
Conway's book ONAG (1976) develops both the surreal numbers and combinatorial game theory in a characteristically playful style. "Winning Ways" (with Berlekamp and Guy) extends this to a comprehensive treatment.
Donald Knuth was so enchanted by surreal numbers that he wrote "Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness" (1974), the first mathematical research published as a work of fiction.
Surreal numbers provide the simplest construction of the reals, a natural framework for infinitesimal analysis (without non-standard model theory), and connections to algebra, set theory, and theoretical computer science.
The Leech lattice in 24 dimensions is the densest sphere packing known. Conway determined its full symmetry group (1968), discovering three new sporadic simple groups: Co1, Co2, Co3.
This work placed Conway at the center of the classification of finite simple groups, one of the 20th century's greatest achievements.
McKay (1978) noticed that 196884 = 196883 + 1: the coefficient of the j-function (number theory) is one more than the smallest dimension of the Monster group (algebra). Conway and Norton systematized this as the Monstrous Moonshine conjectures.
Borcherds proved the conjectures (Fields Medal, 1998) using vertex algebras from string theory, revealing a deep and unexpected connection between finite groups, modular forms, and physics.
Sporadic Groups Leech Lattice Moonshine
Conway's method was play: he explored mathematical structures with the joy and curiosity of a child, following whatever fascinated him.
Explore games, patterns, structures
Spot unexpected connections
Give memorable names to new concepts
Develop rigorous theory from playful origins
He was legendary for his ability to explain deep mathematics while playing games, doing card tricks, or tying knots in the Princeton common room. His charisma drew students and colleagues into collaborative mathematical adventures.
Conway suffered from periods of severe depression throughout his life. He attempted suicide in the 1990s. He was open about this, hoping to reduce stigma around mental health in academia.
Conway was ambivalent about the Game of Life's fame. He once said he hated it because it overshadowed his serious mathematical work. People who knew nothing about the Conway groups knew about the glider.
With Simon Kochen (2006), Conway proved the "Free Will Theorem": if experimenters have free will to choose measurement settings, then so do elementary particles. This controversial result in quantum foundations generated heated debate among physicists.
Conway died on April 11, 2020, at age 82, from COVID-19 during the early days of the pandemic. The mathematical community mourned the loss of one of its most brilliant and beloved figures.
The Game of Life launched the field of artificial life and inspired research in self-organization, emergence, and complex adaptive systems from biology to economics.
The Leech lattice and Conway groups have applications in error-correcting codes. The Leech lattice provides the densest lattice packing in 24 dimensions, relevant to communication systems.
Monstrous Moonshine connects the Monster group to conformal field theory and string theory. Understanding "why" Moonshine works remains a driving question in mathematical physics.
Combinatorial game theory provides algorithms for analyzing positions in Go, Chess endgames, and other games. Conway's framework is used in computer game-playing programs.
The Conway polynomial for knots is used in knot classification, with applications to DNA topology (how enzymes unknot DNA strands) and materials science (polymer entanglement).
Conway's work on the Leech lattice contributed to the proof of optimal sphere packing in dimensions 8 and 24 (Viazovska et al., Fields Medal 2022).
Siobhan Roberts (2015). The definitive biography, capturing Conway's personality, mathematics, and the joy he brought to everyone around him.
John H. Conway (1976, 2nd ed. 2001). His masterpiece on surreal numbers and combinatorial game theory. Playful, deep, and utterly original.
Berlekamp, Conway & Guy (1982, 4 vols). An encyclopedic treatment of combinatorial games, full of wit and mathematical beauty.
Conway et al. (1985). The definitive reference for finite simple groups and their properties. An indispensable tool for group theorists.
"You know, people think mathematics is complicated. Mathematics is the simple bit. It's the stuff we can understand. It's cats that are complicated."
— John Horton Conway1937 – 2020