Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Relation to 1D cellular automata

A picture that shows the evolution of a 1D cellular automaton can be thought of as a 2D array of cells in which the color of each cell satisfies a constraint that relates it to the cells above according to the cellular automaton rule. This constraint can then be represented in terms of a set of allowed templates; the set for rule 30 is as follows:

To reproduce an ordinary picture of cellular automaton evolution, one would have to specify in advance a whole line of black and white cells. Below this line there would then be a unique pattern corresponding to the application of the cellular automaton rule. But above the line, except for reversible rules, there is no guarantee that any pattern satisfying the constraints can exist.

If one specifies no cells in advance, or at most a few cells, as in the systems discussed in the main text, then the issue is different, however. And now it is always possible to construct a repetitive pattern which satisfies the constraints simply by finding repetitive behavior in the evolution of the cellular automaton from a spatially repetitive initial condition.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]