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Rule 110: Elementary Cellular Automaton

An elementary cellular automaton is a 1D line of binary cells that evolves in discrete steps. Each cell's next state depends on its current state and its two neighbors: just three bits, giving 2³ = 8 possible neighborhoods.

Wolfram's Numbering

Stephen Wolfram numbered each rule by treating the eight output bits as a binary number. Rule 110 (binary 01101110) maps:

Neighborhood	111	110	101	100	011	010	001	000
Output	0	1	1	0	1	1	1	0

Why Rule 110 Matters

In 2004, Matthew Cook proved that Rule 110 is Turing-complete: it can simulate any computation, making it one of the simplest known universal systems.

This Notebook

We start with a single active cell on the right edge and run 150 generations on a 201-cell tape. Each row in the output image represents one generation, producing the characteristic nested triangular pattern.

(output too large to embed; run the cell to regenerate it)

Patterns & Turing Completeness

The triangular structures you see are nested, reminiscent of Sierpinski triangles. Rule 110 produces a rich mixture of:

• Regular triangular regions that tile the space.
• Glider-like structures that propagate and interact.
• Complex boundaries where different regions collide.

It is precisely these glider interactions that enable Turing completeness: information can be encoded in the gliders, and their collisions can implement logic gates.

Try Modifying

• Change the initial conditions: try multiple active cells, or a random start.
• Implement other rules: Rule 30 produces pseudorandom output, Rule 90 generates a perfect Sierpinski triangle.
• Vary the tape width and number of generations to see larger-scale structure.