Cellular Automata and Fractal Evolution

Cellular Automaton Number of States

The meaning of the number of states is clear. We denote this number by S.

We shall consider mostly binary (S = 2) CA, with states alive (black) or dead (white).

(Von Neumann designed a self-replicating CA with S = 29 states per cell.)

Note the number of states per cell is S and the number of cells making up a neighborhood is N, so there are S^N neighborhood configurations, arrangements of cells in the neighborhood.

This is easy to see:

There are S choices of cell states for the first cell in the neighborhood.

Independently of this, there are S choices of cell states for the second cell, hence there are S² combinations of states for the first two cells of the neighborhood.

Continuing, there are S³ combinations of states for the first three cells of the neighborhood,

...,

and there are S^N combinations of cells for the neighborhood of N cells.

We illustrate these configurations for

One-dimensional, N=3 nbhds

One-dimensional, N=5 nbhds

Two-dimensional, von Neumann nbhds

Two-dimensional, Moore nbhds

Return to Cellular Automaton Basics.