Four Rules

On April 11, 2020, John Horton Conway died of COVID in a hospital in New Brunswick. He was eighty-two. The newspapers wrote: “The mathematician who invented the Game of Life has died.” His colleagues wrote: “One of the greatest minds in combinatorics has died” — the science of how many ways you can rearrange, connect, and count things that, seemingly, don’t need counting. Both were right — and both missed the point. Because Conway didn’t “invent a game.” Conway showed that from four rules you can build a universe — and that the universe won’t notice it’s built from four rules. From the inside, it will look like the real thing.

---

The rules — here they are. Imagine an infinite sheet of graph paper. Each cell is either black (alive) or white (dead). At each step, every cell looks at its eight neighbors and decides its fate. Fewer than two live neighbors — dies of loneliness (try living in a building where you’re the only tenant). Two or three — lives, fine, tolerable. More than three — dies of overcrowding (now try living in a building where every apartment is a communal flat). And one more: a dead cell with exactly three live neighbors — comes alive. Three neighbors — and an empty space becomes a living thing.

Four rules. The cells don’t know each other, don’t remember the past, don’t plan the future. They have no brain, no desires, nothing except one question: how many live neighbors?

Set up the cells however you like — and watch.

Some configurations die immediately, like a bad joke. Some freeze into “still lifes” — shapes that never change again, like a retiree who’s found his park bench. Some start pulsing — “oscillators” that toggle between states forever, like a person who can’t decide whether to put on a jacket or not. And some of them fly. A small pattern of five cells, called a “glider,” shifts one cell diagonally every four steps and keeps that motion going forever, with neither engine nor any other reason than the rules of the game themselves. Five dots on a sheet of graph paper — and they have a direction. They — crawl. Like a small stubborn creature that nobody created.

Conway was looking for this: the simplest rules from which anything can grow. He found them in 1970. Called it the “Game of Life,” though in fact what he had launched was an experiment: he set the rules, after which the system played by itself, against itself, and accounted to itself. And the experiment went further than anyone expected.

First they found “guns” — things that shoot gliders, like a printer nobody gave an assignment: producing new objects out of nothing. Then — patterns that grow forever, self-complicating with each step. Then — logic gates. A gate is a simple component that takes two signals and outputs one. If both signals are “yes” — the AND gate says “yes.” If at least one is “yes” — the OR gate says “yes.” If the signal is “yes” — the NOT gate says “no.” Three components. Like three kinds of bricks. From them you can build any computer — any computer at all, from a calculator to a supercomputer. And people built one — inside the Game of Life. From streams of gliders crashing into each other. A full-fledged computer capable of computing anything your laptop can. From four rules and black-and-white squares on a sheet of graph paper.

In 2006, programmer Brice Due built another Game of Life inside the Game of Life — the OTCA metapixel. A world that simulates itself. From four rules — a universe capable of containing universes. Like a matryoshka, except each next doll is the same size as the previous one.

Conway, for the record, hated his “Game of Life.” His whole life. It annoyed him that his other work — serious mathematics he’d devoted decades to — drowned in the shadow of a cellular automaton he’d come up with over tea. As if Pushkin had become famous not for Eugene Onegin but for a doodle in the margins of a notebook. But that “doodle” turned out to be a door. Through it, mathematics saw what was previously thought impossible: from simple rules — without design, without instructions, without anyone’s “mind” — complexity, structure, motion emerge. An entire world.

After which a question hung in the air that hasn’t been put to bed since 1970: what if our world is a cellular automaton too, just with more complex rules and a finer grid?