Four rules. Universes follow.
Martin Gardner introduced Conway's rules in his October 1970 Scientific American column. Two pages of magazine, four lines of rule, and a community of mathematicians has spent fifty years discovering what was already inside them. The Sandbox lets you draw and run any pattern — but first, the four rules in action.
Take a grid. Each square is alive or dead. Apply four small rules to every square at once, then again, and again. Out of those four lines emerge spaceships that travel, factories that fire, oscillators that breathe, and — given enough room — calculators that add. Nobody puts the spaceships there. They just appear.
Born if exactly three live neighbours, survives with two or three, else die. That's the whole rule, written B3/S23. Drop a 2×2 Block and it never changes. Drop a straight line of three cells, a Blinker, and it flips between horizontal and vertical forever — one rule pulse per heartbeat.
John Conway invented this in 1970 as a parlour game. By 1982 hobbyists had built logic gates from glider streams; by 2000 a Turing machine; by 2010 a CPU; by 2014 a Game of Life simulator running inside a Game of Life. Universal computation, discovered hiding inside a toy.
Every cell counts its eight neighbours. A dead cell with exactly three live neighbours wakes up — birth. A live cell with two or three neighbours stays alive — survival. Anything else dies, from loneliness with fewer than two or from suffocation with more than three. That's it. The whole physics of this world fits in fourteen words. Everything else — the spaceships, the guns, the computers — is just consequence.
. . . . C . . . .
Some shapes never move: the 2×2 Block, the Beehive, the Loaf — still lifes. Others pulse: the Blinker flips period 2, the Toad period 2, the Beacon period 2, the Pulsar period 3. And some shapes move across the grid by repeating themselves shifted: spaceships. The Glider, five cells, slides one square diagonally every four generations. The Lightweight Spaceship cruises straight. Catalogues now list tens of thousands of named patterns.
Inline sandbox · click cells, press play, try a preset.
Click or drag to toggle cells. Then press play, or step one tick at a time.
Conway had conjectured that no Life pattern could grow forever. A bounded universe seemed reasonable. He offered fifty dollars to anyone who broke the conjecture. In November 1970, Bill Gosper's group at MIT found a pattern of 36 cells that fires a glider every 30 generations, forever — the first glider gun. The population grows without bound. Conway paid up. Suddenly Life was unbounded, and the door to building things inside it was wide open.
Treat a stream of gliders as a wire carrying bits — glider present is 1, absent is 0. Collisions between glider streams implement AND, OR, NOT. Loops of glider streams store bits — memory. Glider guns provide the clock. With those primitives you can build any Boolean circuit, and from any Boolean circuit, any computation. Paul Rendell built a working Turing machine entirely from Life patterns in 2000; Adam Goucher's metafier turned full CPUs into Life mosaics. Conway's four-line rule is Turing-complete.
Same starting soup · different B/S rules · watch them diverge.
Flip a digit and the universe changes. Same seed, different physics.
In 2006 Brice Due designed the OTCA metapixel — a 2048×2048 Life pattern whose collective behaviour, viewed from far enough away, is itself a single Life cell. Tile a whole grid with OTCA metapixels and you get Game of Life running inside Game of Life, indistinguishable in the limit from the original. Self-replicating patterns followed: in 2010 Andrew Wade built Gemini, a pattern of 846,278 live cells spanning a roughly 4.2-million-cell-wide bounding box, copying itself every 33.7 million generations. The toy contains itself.
Conway's B3/S23 is one point in a vast space. HighLife (B36/S23) adds period-12 replicators. Day & Night is symmetric under colour inversion. Seeds (B2/S) is purely generative — nothing ever survives. Margolus neighbourhoods partition the grid into 2×2 blocks and update them as units, giving reversible automata. Bill Gosper's Hashlife algorithm memoises space-time regions to simulate 10^20 generations in seconds. There is a whole biology to discover, one rule string at a time.
How and where this technique lives in the world today.
Topics in the same vein.
The Sandbox holds a 120×70 toroidal grid, seven classic patterns and a draw-by-dragging tool. Open it, click around, see what survives.
→ Open the Sandbox