Three rules. Endless universes.
In 1970, John Conway put down three rules for cells on a grid. Fifty years later, people are still finding surprises hidden in them.
A cell is alive or dead. Each step every cell counts its 8 Moore neighbours and updates itself using three lines of rules:
- A live cell with 2 or 3 live neighbours survives.
- A dead cell with exactly 3 live neighbours is born.
- Any other cell dies.
That's it. That's the whole game. What comes out of these three rules is astonishing: still lifes, oscillators, spaceships that move across the grid, patterns that emit new patterns forever, and — proven in 2000 — a full universal Turing machine. This is emergence, the canonical demonstration that arbitrary complexity can hide inside arbitrarily simple local rules.
Pick a pattern · hit play · watch three rules produce a universe
🖱 Click any cell to toggle it — or drag to paint. Newly born cells glow orange; cells that have survived a while settle into blue. Toroidal edges (top wraps to bottom, left to right) mean gliders keep circling forever.
A much quieter cellular automaton
Same grid. Same 8-cell Moore neighbourhood. One tiny rule change: instead of Conway's 2/3 survival rule, each cell just copies whichever value most of its neighbours hold. That's it. Local majority wins. Let's watch it play out.
Watch consensus solidify into stable clusters
Try a fresh start at low density (say 25%) — the minority side gets absorbed almost entirely. Try 50% — the grid crystallises into a small handful of large regions. Try 75% — mirror image of the low-density outcome. Consensus is always local, never global.
Where these show up
- Conway's Life. The direct study is niche, but the *idea* — that huge state spaces can emerge from small rules — reshaped how physicists, biologists, and complexity scientists think about pattern formation. Stephen Wolfram's A New Kind of Science pushed the argument to its limits.
- Local Majority. Voter models, opinion dynamics, dialects, network coordination. Every time you notice that your neighbourhood, workplace, or friend group agrees on things you'd have no strong opinion on alone, some flavour of local-majority dynamics is at work.
- Same-grid siblings. Schelling segregation (Ch 15.3) sits alongside these — cells with preferences moving around. See our Schelling story for the direct comparison.
Related models
- 🟦🟧 Schelling segregation — same local-rule family, different update: agents move instead of flip.
- 🎲 Entropy — the four Wolfram classes of CA behaviour (equilibrium, periodic, random, complex) are the headline there.
- 🏔 NK rugged landscape — another "simple rules → surprisingly rich outcomes" model.
