Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Searching for patterns [that satisfy constraints]

The basic approach to finding a pattern which satisfies a particular constraint on an infinite array of cells is to start with a pattern which satisfies the constraint in a small region, and then to try to extend the pattern. Often the constraint will immediately force a unique extension of the pattern, at least for some distance. But eventually there will normally be places where the pattern is not yet uniquely determined, and so a series of choices have to be made. The procedure used to find the results in this book attempts to extend patterns along a square spiral, making whatever choices are needed, and backtracking if these turn out to be inconsistent with the constraint. At every step in the procedure, regularities are tested for that would imply the possibility of an infinite repetitive pattern. In addition, whenever there is a choice, the first cases to be tried are set up to be ones that tend to extend whatever regularity has developed so far. And when backtracking is needed, the procedure always goes back to the most recent choice that actually affected whatever inconsistency was discovered. And in addition it remembers what has already been worked out, so as to avoid, for example, unnecessarily working out the pattern on the opposite side of the spiral again.

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