Rule

Presets:

How It Works

An elementary cellular automaton is a 1D grid of cells, each either on or off. At each step, every cell's next state is determined by its current state and its two neighbors — a 3-cell neighborhood with 8 possible patterns.

A rule number (0-255) encodes which patterns produce an "on" cell. Since there are 8 possible 3-cell patterns and each can map to on or off, there are 28 = 256 possible rules.

The diagram above shows each rule's 8 cases. The top row is the input pattern (left neighbor, cell, right neighbor), and the bottom cell is the output.

Some notable rules:

  • Rule 30 — chaotic, used in Mathematica's random number generator
  • Rule 90 — produces a Sierpinski triangle
  • Rule 110 — proven Turing-complete
  • Rule 184 — models traffic flow

Connection to Conway's Life

These 1D automata are the simplest members of the same family as Conway's Game of Life. Life is a 2D cellular automaton where each cell has 8 neighbors instead of 2, and the rule considers the count of live neighbors rather than the exact pattern. But the core idea is identical: simple local rules produce complex global behavior. Rule 110's Turing-completeness shows that even the 1D case is capable of arbitrary computation.

Wolfram's Four Classes

Stephen Wolfram classified all 256 elementary rules into four classes based on their long-term behavior. This classification shows that even the simplest possible rules — one dimension, two states, nearest neighbors — can produce the full spectrum from trivial uniformity to universal computation.

Class I — Uniform

Evolution leads to a homogeneous state. All cells become the same regardless of initial conditions.

Class II — Periodic

Evolution leads to stable or periodically repeating structures.

Class III — Chaotic

Evolution produces pseudo-random, aperiodic patterns.

Class IV — Complex

Localized structures emerge and interact in complicated ways. Capable of universal computation.

The progression from Class I to Class IV mirrors a phase transition: too little activity produces uniformity, too much produces chaos, and the boundary between order and chaos — Class IV — is where complex, computation-capable behavior appears. Rule 110, a Class IV rule, has been proven Turing-complete, meaning it can simulate any computation given sufficient time and space.