General associative [cellular automaton] rules With a cellular automaton rule in which the new color of a cell is given by f[a 1 , a 2 ] (compare page 886 ) it turns out that the pattern generated by evolution from a single non-white cell is always nested if the function f has the property of being associative or Flat . In fact, for a system involving k colors the pattern produced will always be essentially just one of the patterns obtained from an additive rule with k or less colors. … And from the discussion on page 952 this means that any rule that shows generalized additivity must always yield a nested pattern.

Nested initial conditions [in cellular automata] The pictures below show patterns generated by rule 90 starting from the nested sequences on page 83 .

Emergence of reversibility Once on an attractor, any system—even if it does not have reversible underlying rules—must in some sense show approximate reversibility.

[Structures in] rule 30 For the first background shown, no initial region up to size 25 yields a truly localized structure, though for example starts off growing quite slowly.

String transformations An example of a rule that allows one to go from any string of A 's and B 's to any other is {"A"  "AA", "AA"  "A", "A"  "B", "B"  "A"} (Compare page 1038 .)

In mathematics, rather little is usually done with network substitutions, though the proof of the Four-Color Theorem in 1976 was for example based on showing that 300 or so possible replacement rules—if applied in an appropriate sequence—can transform any graph to have one of 1936 smaller subgraphs that require the same number of colors. (32 rules and 633 subgraphs are now known to be sufficient.)

But I suspect that essentially all of the various phenomena that we have observed in discrete systems in the past several chapters can in fact also be found even in continuous systems with fairly simple rules.

And in fact if one were just to see this pattern, one would probably assume that it came from a rule whose typical behavior is vastly simpler than code 1329.

Pictures (h) and (i) demonstrate that with the particular underlying rule used here, a highly regular network is produced.

The underlying rules for the cellular automaton are exactly the same in each case, and involve nearest neighbors and five possible colors for each cell.