Elementary cellular automata
One row of cells. Each looks at itself and its two neighbours. Eight possible neighbourhoods, one output bit each - so exactly 256 rules exist, and every one of them is on this page.
How to read it
The top row is the starting configuration; each row below it is the next generation. Time runs downwards, so the whole history of the automaton is a single image. The eight boxes in the panel are the rule itself - each shows one possible neighbourhood and the cell it produces. Click any of them to flip that output and you will land on a different rule number.
The rule number is simply those eight output bits read as a binary number, with the leftmost neighbourhood (111) as the most significant bit. That is Wolfram's numbering, and it is why Rule 30 and Rule 110 have the names they do.
The four classes
Wolfram sorted all 256 rules into four behavioural classes, and the classification turns out to apply to cellular automata generally - including Conway's Life, which is class IV.
| Class | Behaviour | Try |
|---|---|---|
| I | Everything collapses to a uniform state | 0, 32, 160, 255 |
| II | Settles into stable or repeating structures | 4, 108, 94, 250 |
| III | Chaotic and statistically random | 30, 45, 90, 150 |
| IV | Localised structures that interact - the class that can compute | 110, 54, 137 |
The ones worth knowing
- Rule 30
- From a single live cell, produces a triangle whose left side is regular and whose right side is indistinguishable from noise. Wolfram used its centre column as Mathematica's random number generator for years. The shell of the sea snail Conus textile carries a strikingly similar pattern.
- Rule 90
- Exactly the Sierpiński triangle. Each cell is the XOR of its two neighbours, which makes the whole thing equivalent to Pascal's triangle modulo 2.
- Rule 110
- The one that matters. It supports a periodic background, several distinct gliders travelling at different speeds, and controllable collisions between them. Matthew Cook proved in 2004 that it is capable of universal computation - the simplest system known to be Turing complete.
- Rule 184
- A genuine traffic model. Cells are cars, and the rule reproduces the transition from free flow to a jam, including phantom jams that propagate backwards through the traffic.
- Rule 150
- XOR of all three cells in the neighbourhood - a nested, self-similar fractal that is easier to analyse algebraically than most.
Try Rule 110 from a random starting row rather than a single cell: the gliders only become obvious once there is enough going on for them to collide.