Game of Life

Drag on the grid to draw cells (drag from a live cell to erase). Scroll to zoom, middle-drag or two fingers to pan. Keys: space play/pause, S step, C clear, R random, F fit.

Population against generation (last 240 steps).

What's the maths?

Where's the maths? This is a cellular automaton — a corner of mathematics that sits between number patterns, geometry on a grid and logic. Every square on the board is either alive or dead, and one very short rule decides the whole next picture: count your eight neighbours, then apply the rule. Nothing is random and nothing is drawn by hand — every generation you see is forced by the one before it, exactly like finding the next term of a number sequence. The mathematics is in spotting the pattern: which shapes stay the same forever, which repeat after a fixed number of steps, and which crawl across the board. Those three behaviours are the grid version of a constant sequence, a periodic sequence, and a sequence with a constant difference. The surprise is that a rule this simple can produce gliders, guns and structures that never settle down — order and chaos from three lines of arithmetic.

The rule. Let n be the number of live cells among a cell's eight neighbours (including diagonals). Then:

Mathematicians write this as B3/S23. That is the entire definition of the game — everything else on the screen is a consequence of it.

Still lifes are shapes that map to themselves: they have period 1. In a block, each of the four cells sees exactly 3 live neighbours, so all four survive; every cell just outside sees at most 2, so nothing is born. Nothing can change.

Oscillators come back to their starting shape after a fixed number of steps. The blinker has period 2; the pulsar has period 3 — step it three times and every one of its 48 cells is back where it began. Written as a sequence of pictures P₀, P₁, P₂, P₃, … we have P_k+3 = P_k for every k.

Spaceships are oscillators that come back shifted. The glider satisfies P_k+4 = P_k translated by the vector (1, 1): same shape, one square right and one square down, every 4 generations. So it travels 1 diagonal square per 4 steps — a speed of c/4, where c is one cell per generation, the fastest anything can move in this universe. The lightweight spaceship repeats after 4 steps shifted by (2, 0), twice as fast.

Why it matters. The Gosper glider gun emits a new glider every 30 generations, so its population grows without limit — proof that a finite starting pattern need not settle down. With guns you can build AND, OR and NOT gates out of colliding gliders, which is why the Game of Life can, in principle, compute anything a computer can.