Thought Toys · Emergence · Exhibit 29

The simplest rule that makes complexity

Take a row of cells, each black or white. To get the next row, look at each cell and its two neighbours and consult a tiny rulebook — eight lines long. That's the whole machine. Yet one rulebook draws a perfect fractal, another spins endless chaos, and a third builds a working computer. Turn the dial and see which universe you land in.

A space-time diagram. The top strip shows the current rule as eight before-and-after pictures: each three-cell neighbourhood and the single cell it produces. Below, the row of cells evolves downward step by step, starting from a single black cell in the middle, painting a pattern that depends entirely on the rule.

the law, in full: 00011110 · from one seed, a triangle that never repeats

↳ your turn — drag the rule, or jump to 90, 110, 184

What you're seeing

The strip across the top is the rule. There are only eight situations a cell can face — its left neighbour, itself, and its right neighbour, each black or white — and the rulebook simply says, for each, what the cell becomes next. Eight yes/no answers, so eight bits, so a number from 0 to 255. That number is the entire law of this universe; the dial is just choosing which of the 256 possible laws to obey.

Below the strip, a single lit cell in the middle is rewritten over and over, each new row drawn under the last, so time runs downward. Watch what eight little bits can do. Rule 90 folds a flawless Sierpinski triangle out of nothing — the same nested pattern at every scale. Rule 30 pours out a stream so irregular that it was used for decades as a source of random numbers. Rule 110 grows drifting stripes that collide and scatter like particles — and those collisions, it turns out, can carry out any computation at all. And Rule 0? It just dies.

That spread is the surprise. The machinery couldn't be simpler — a one-line row, a peek at two neighbours — yet sliding the dial walks you through four kinds of world that Stephen Wolfram sorted into classes — dies out, settles into repetition, dissolves into chaos, or makes lasting moving structures. Complexity here isn't poured in from outside; the rule is trivial and the start is one dot. It emerges, purely from the rule being iterated. Whatever richness you see was always latent in those eight bits.

The rule, exactly. Number the eight neighbourhoods 111, 110, … , 000 as the binary 7 … 0. The rule's code in binary lists the output cell for each, high bit first: cellnext(i) = bit(4·left + 2·self + right) of R So Rule 30 = 00011110 means the neighbourhoods 001, 010, 011, 100 turn the cell on and the rest off. Verified in node (improve/verify/29-eca.js): Rule 90 from a single seed matches an independent Pascal's-triangle-mod-2 oracle (Lucas' theorem) cell-for-cell over 64 rows — it is exactly the Sierpinski fractal — and Rule 30's centre column has no period up to 64 over 400 steps, with balanced bits: genuine aperiodic chaos. Counter-examples (the low-complexity classes): Rule 0 erases everything after a single step (no structure can survive), and Rule 51 — "flip the centre, ignore the neighbours" — blinks with a strict period of 2 forever. Same machine, opposite ends of the spectrum: the eight bits, and nothing else, decide which world you get. (Rule 184, the "traffic" rule, is the same idea behind Exhibit 1 — cars as cells, jams as the moving pattern.)

Also in Emergence: Fireflies that sync →

All 11 in Emergence
  1. 01Phantom Traffic Jams
  2. 02Schelling's Segregation
  3. 07Conway's Game of Life
  4. 11Percolation
  5. 15Reaction & diffusion
  6. 18Sandpiles & self-organized criticality
  7. 22Diffusion-limited aggregation
  8. 29The simplest rule that makes complexity — you are here
  9. 32Fireflies that sync
  10. 41Small-world networks
  11. 43The Ising model

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 29.