What you're looking at
An elementary cellular automaton is the simplest interesting computational
universe: an infinite row of cells, each black or white, updated in discrete time steps.
A cell's next state depends only on itself and its two neighbours — three bits in,
one bit out. There are 23 = 8 possible neighbourhoods and
28 = 256 possible rules. That's the whole zoo.
Stephen Wolfram catalogued all 256 in the 1980s and, in 2002, published A New Kind of Science. The provocative claim: from rules so trivial you can write them on a napkin, you get nested fractals (rule 90), pure randomness that passes statistical tests (rule 30), and full Turing-complete computation (rule 110, proved by Matthew Cook). The argument is that nature might be doing the same thing — simple local rules, complex global behaviour — and that this changes how we should think about modelling, complexity and physics.
The numbering convention is Wolfram's: write the rule as an 8-bit binary number where bit
i is the output for neighbourhood i (with i read as
left·centre·right in binary). Rule 30 is 00011110; rule
110 is 01101110. Try them. The diagram above the canvas shows the eight
transitions for whichever rule is loaded.
A few rules worth trying
- Rule 30 — from a single black cell, produces a stream of bits so statistically random that Wolfram used it as the random number generator in Mathematica for years.
- Rule 90 — the Sierpinski triangle, derived purely from XOR of two neighbours.
- Rule 110 — the famous one. Cook proved it is Turing-complete: you can in principle compute anything with it.
- Rule 184 — a minimal traffic-flow model. Try it with a random initial condition.
- Rule 54 — complex localised structures (gliders) on a periodic background. Still not fully understood.
Where this comes from
- A New Kind of Science — the full book, free online.
- MathWorld: Elementary Cellular Automaton — the canonical reference.
- The Wolfram Atlas — deep dive into each of the 256 rules.
- Wikipedia: Elementary cellular automaton — classification, history, references.
- Rule 110 — Turing completeness — Cook's proof.
- Wolfram's 15-year retrospective — what he thinks held up.