John Horton Conway (1937-2020)

Mathematician John Horton Conway died on April 11, 2020, at the age of 82. Brilliant but quirky, he made significant contributions to a wide range of fields, from knot theory and finite groups to number theory and combinatorial games. Fascinated by the mathematics underlying games and puzzles, he was very active in the realm of recreational mathematics.

Over the years, I encountered Conway and heard him speak many times, both at Princeton University and at a variety of conferences, notably in recent years at the Gathering4Gardner events in Atlanta. I wrote about his "Game of Life" and the invention of cellular automata in my first book, The Mathematical Tourist: Snapshots of Modern Mathematics (W.H. Freeman, 1988).

The Game of Life

Imagine an immense checkerboard grid stretching as far as the eye can see. Most of the checkerboard's squares, or cells, are empty; a few are occupied by strange beings—creatures very sensitive to their immediate neighbors. Their individual fates teeter on numbers. Too many neighbors means death by overcrowding and too few death by loneliness. A cozy trio of neighbors leads to a birth and a pair of neighbors to comfortable stability.

At each time step, this cellular universe shuffles itself. Births and deaths change old patterns into new arrangements. The patterns evolve—sometimes into a static array that simple marks time, sometimes into a sequence of shapes repeated again and again, sometimes into a chain of arrangements that propagates throughout the checkerboard universe.

The mathematical game called "Life" generates a remarkably diverse array of thought-provoking patterns and scenarios. Invented in 1970 by the British mathematician John H. Conway, it vividly demonstrates how a set of simple rules can lead to a complex world displaying a rich assortment of interesting behavior.

Conway's aim was to create a cellular pastime based on the simplest possible set of rules that would still make the game unpredictable. Moreover, he wanted the rules to be complete enough so that once started, the game could play itself. Growth and change would occur in jumps, one step inexorably leading to the next.

The result would be a little universe founded on logic, in which everything would be predestined, but there would be no obvious way for a spectator or player to determine the fate of future generations except by letting the game play itself out.

To find appropriate rules, Conway and his students at Cambridge University investigated hundreds of possibilities. They did thousands of calculations, looking at innumerable special cases to expose hidden patterns and underlying structures. They tried triangular, square, and hexagonal lattices, scribbling across acres of paper. They used large numbers of poker chips, coins, shells, and stones in their search for a viable balance between life and death.

The game they came up with is played on a infinite grid of square cells. Each cell is surrounded by eight neighbors, four along its sides, four at its corners. It is initially marked as either occupied or vacant, creating some sort of arbitrary starting configuration.

Changes occur in jumps, with each cell responding according to the rules. Any cell having two occupied cells as neighbors stays in its original state. A cell that is alive stays alive, and one that is empty stays empty. Three living neighbors adjacent to an empty cell leads to tricellular mating. A birth takes place, filling the empty cell. In such a neighborhood, a cell already alive continues to live. However, an occupied cell surrounded by four or more living cells is emptied. Unhappily, death also occurs if none or only one of an isolated living cell's neighbors is alive.

A cell's eight nearest neighbors have a strong influence on its destiny.

These simple rules engender a surprisingly complex world that displays a wide assortment of interesting events and patterns—a microcosm that captures elements of life, birth, growth, evolution, and death.

The game was first introduced to the public in October 1970, in Martin Gardner's "Mathematical Games" column in Scientific American. It aroused tremendous interest, and the game became an additive passion for many people.

Because it was relatively easy to implement as a computer program, it also quickly became a favorite computer exercise. All kinds of people—students and professors, amateurs and professionals—spent years of computer time following the evolution of countless starting patterns.

"Life" aficionados gleefully pursued elusive arrangements and searched for unusual types of behavior. Many different forms evolved on the checkerboard and were painstakingly cataloged, sporting evocative names such as pipe, horse, snake, honeycomb cell, ship, loaf, frog, danger signal, glider, beacon, powder keg, spaceship, toad, pinwheel, and gun.

Some of these arrangements vegetated in a single contented state, and others pulsated, switching back and forth between one configuration and another.

One simple pattern evolves over time into a sequence that alternates between two different forms.

The possibilities were endless, and the game presented a variety of intriguing mathematical puzzles. For example, are there patterns that can have no predecessor? Several such "Garden of Eden" arrays were eventually discovered.

Example of a "Garden of Eden" array.

Other investigations revealed that while a given pattern leads to only one sequel pattern, it can have several possible predecessors.

Different starting states can lead to identical vegetating states.

Thus, a particular configuration can have a number of different pasts but only one future. That makes it difficult for a viewer, glued to a computer screen, to backtrack if a particularly interesting pattern appears fleetingly during the course of a run. There is no guaranteed way to travel backward in time to recreate a past "Life."

The computer also brought animation to the game. A rapidly computed sequence of generations could be viewed as pulsating shapes, creeping growths, lingering dusts, fragmenting forms, and chaotically dancing figures.

Other enthusiasts adapted Conway's game for surfaces other than the infinite plane. Players can follow the game on the surface of a cylinder, a torus, or even a Möbius strip. They can also pursue their creatures of "Life" on structures in three and higher dimensions.

The attraction of Conway's original game, and the chief reason for its popularity, is that although it is completely predictable on a cell-by-cell basis, the large-scale evolution of patterns defies intuition. Will a pattern grow without limit? Will it settle into a single stable object? Will it send off a shipload of colonists?

Conway managed to balance the system's competing tendencies for growth and death so precariously that "Life" is always full of surprises.

See also "Wild Beasts around the Corner," "Computing in a Surreal Realm," "Growing Sprouts," "Powerful Sequences," and "Punctured Polyhedra,"