Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Other types of [2D] constraints

Constraints based on smaller templates simply require smaller numbers of repetitive patterns: :4; :7; :17; :11; :12. To extend the class of systems considered in the main text, one can increase the size of the templates, or increase the number of possible colors for each cell. For 3×3 templates with two colors extensive randomized searches have failed to discover examples where non-repetitive patterns are forced to occur. Another extension of the constraints in the main text is to require that not just a single template, but every template in the set, must occur somewhere in the pattern. Searches of such systems have also failed to discover examples of forced non-repetitive patterns beyond the one shown in the text.

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