Conway's Game of Life

The Game of Life is a cellular automaton invented by the mathematician John Conway in 1970. There are no players and no moves. You set a board of square cells alive or dead, press play, and four rules decide what the next generation looks like. Everything that happens afterwards was already decided by the picture you drew.

The four rules

Every cell has eight neighbours — the squares touching it edgewise and cornerwise. On each generation, every cell on the board is judged at once, against the board as it was before any of them changed:

• A live cell with fewer than two live neighbours dies.
• A live cell with two or three live neighbours survives.
• A live cell with more than three live neighbours dies.
• A dead cell with exactly three live neighbours comes alive.

That is the whole of it. Life is usually written B3/S23 — born on three, survives on two or three — and this tool runs that rule and no other. The simultaneity matters more than it sounds: if cells were updated one at a time, a cell changed early in the sweep would change the count its neighbour is judged against, and the result would depend on the order you happened to walk the board in.

What the four rules turn out to do

Almost every pattern falls into one of a few fates. Most collapse into still lifes — blocks, beehives and loaves that never change again — or into small oscillators that flip between two or three shapes forever. A few travel: the glider is five cells that reproduce themselves one square down and one square right every four generations, walking diagonally across the board until they hit something.

The interesting ones do neither for a very long time. Press Random and you get a board about a third full. On the medium board that is roughly 2,180 live cells, and it will usually spend somewhere between five hundred and twelve hundred generations churning before it settles down to about two hundred — a tenth of what it started with, mostly blocks and blinkers with the occasional glider still crossing the debris.

Three patterns worth loading

Diehard is seven cells that vanish completely — but take exactly 130 generations to do it, and grow to forty cells on the way. It is the cleanest demonstration that you cannot tell by looking whether a pattern is about to die.

The R-pentomino is five cells. On the large board here it thrashes for 1,105 generations, peaking at 314 live cells and throwing gliders into all four corners, before it finally settles into a two-step cycle of 110 cells. Conway and his students traced its first few hundred generations by hand on a Go board in 1970.

The Gosper glider gun is the one that changed what people thought the game was. It returns to its own shape every thirty generations and spits out a glider each time, so the population grows without limit. Bill Gosper found it in 1970 and it settled a question Conway had put a $50 prize on: whether any pattern could grow forever. On a finite board like this one the gliders eventually reach the wall and die, so what you see is a steady state rather than unlimited growth.

Edges, and why this board has a setting for them

Conway's game is played on an infinite plane. A real board has to end somewhere, and the choice of how changes what patterns do. With Wrap edges off, the world outside the board counts as dead, so a glider reaching the edge is destroyed. With it on, the board becomes a torus: the left edge is joined to the right and the top to the bottom, so a glider leaving one side arrives on the other.

Neither is more correct, and both give answers that differ from the textbook. The acorn settles after about 2,008 generations on the large bounded board here, against the 5,206 usually quoted — because the quoted figure is for a plane with no walls to crash into. It is worth flipping the switch on a pattern you know and watching the answer change.

Drawing, pasting and sharing

Click or drag on the board to draw cells; dragging from a live cell erases instead. On a touchscreen, tap a cell to toggle it — dragging pans the board, which is what you want once you have zoomed in past the edge of the screen.

The pattern code box speaks RLE, the run-length format Life patterns have been published in for decades. Paste one in from any pattern collection and press Load pattern; press Copy RLE to get what is currently on the board back out in the same format, ready to paste anywhere else. Copy link puts the whole board in the URL, so a board you drew can be sent to somebody exactly as it stands. Nothing is uploaded — the pattern travels in the part of the address after the #, which browsers never send to a server.

When the run stops by itself

A board that has stopped changing keeps burning battery to redraw the same picture, so the run halts on the two endings where nothing moves at all: every cell dead, or a pattern identical to the generation before it.

An oscillating board is not one of them and keeps going. When a board becomes identical to the generation two before it — which is what a blinker, a toad or a beacon settles into — the tool says so and carries on running, because blinkers flipping and beacons pulsing is most of what a soup leaves behind and most of what there is to watch. Knowing the board will now repeat forever is worth telling you; it is not a reason to take it away.

Longer cycles are not detected at all. A pulsar repeats every three generations and a pentadecathlon every fifteen, and neither is ever reported as settled.