🧬 Game of Life (2D)

The rules

Life lives on a grid of cells, each alive or dead. All cells update at the same moment. Count the 8 neighbours around a cell (the Moore neighbourhood). In Conway's rule, written B3/S23:

• Birth: a dead cell with exactly 3 live neighbours becomes alive.
• Survival: a live cell with 2 or 3 live neighbours stays alive; otherwise it dies (lonely with fewer than 2, crowded with more than 3).

Every other Life-like rule uses the same idea with different digit lists. B36/S23 is HighLife, B2/S is Seeds. You can tick any digits above and watch a new universe appear.

What the colours and heights show

Each cell remembers its age, the number of generations it has been alive. In Age colour mode a newborn is cyan, then green, yellow, orange, pink and violet for the oldest. A cell that has just died glows red for a moment.

Afterglow and the space-time slice

Cells that have just died leave a red afterglow that fades in about half a second, so you can see where life was lost. The fourth graph is a space-time diagram of the yellow row across the middle of the board: every line is that row one generation later, newest at the top. A still life is a straight vertical line, a blinker a flashing dot, a glider a slanted streak. The 3D page stacks the whole board this way as glass layers.

Statistics

Density is p = live cells / all cells. A colony is a group of live cells connected through any of their 8 neighbours. The page also remembers a fingerprint (hash) of recent boards; if the same board comes back, it reports a cycle: period 1 is a still life, period 2 a blinker-like oscillator.

The mean-field map

If live cells were scattered at random with density p, the number of live neighbours of a cell would follow the binomial law P(k) = C(8,k) pk (1-p)8-k. The expected density after one step is then

f(p) = p · Σk in S P(k) + (1 - p) · Σk in B P(k)

The yellow curve in the third graph is f(p) for the current rule; the dashed line is f(p) = p. Where they cross, a well-mixed soup would stay at the same density. This predicts the first step of a random soup very well (check 1 below tests it). After that the live cells clump together into islands, neighbours are no longer independent, and real Life settles at a much lower density (roughly 2 to 4 per cent for Conway) than the formula suggests. The pink dots show the real path.

Famous facts

• John Conway introduced Life in 1970 (Martin Gardner described it in Scientific American).
• The glider gun was found by Bill Gosper's group in 1970 and showed that a finite pattern can grow without bound.
• Life is Turing complete: gliders can carry signals and build logic gates, so a big enough Life board can in principle run any computer program (this is shown with constructions, not by this page).
• The R-pentomino stabilises after 1103 generations, the acorn after 5206, and diehard disappears after 130 on an unlimited plane.

Honest simplifications

The board here is finite: with wrap it is a torus (what leaves on the right returns on the left), without it the outside is dead, so gliders crash into the walls and the acorn or R-pentomino end differently from the infinite plane. The cycle detector only looks back about 400 generations, so a glider that needs a whole lap of a big torus to return is not reported. Age and the statistics are exact for the board you see; only the mean-field curve is an approximation.

• Press Glider gun and watch the population graph climb in steps of 5 every 30 generations.
• Press R-pentomino with the board set to 128 and look at the third graph: the pink path wanders far from the yellow mean-field curve.
• Untick Born digit 3 and tick 2 (rule B2/S23): what happens to the soup?
• Use Stamp with the Pulsar, then draw a single extra cell next to it and see whether the oscillator survives.
• Run Seeds and look at the space-time slice: every cell lives exactly one generation.
• Turn off Wrap and fire a glider at the wall: it crashes and leaves debris, often a block.