And what I discover is that whatever kind of underlying rules one uses, the behavior that emerges turns out to be remarkably similar to the basic examples that we have already seen in cellular automata. … Four basic examples from the previous chapter of behavior produced by cellular automata with simple underlying rules.

Another initial condition [for rule 90] Inserting a single in a background of blocks in rule 90 yields the pattern below in which both the white and striped regions have fractal dimension 2.

Rule orderings [for cellular automata] The fact that successive rules often show very different behavior does not appear to be affected by using alternative orderings such as Gray code (see page 901 .)

Despite the simplicity of their underlying rules, the final patterns produced show immense complexity.

Despite the simplicity of their underlying rules, the final patterns produced show immense complexity.

Given a string p which gets transformed either to q or r by the original rules, one can always imagine adding a new rule q  r or r  q that makes the paths from p immediately converge. To do this explicitly for all possible p that can occur would however entail having infinitely many new rules. But as noted by Donald Knuth and Peter Bendix in 1970 it turns out often to be sufficient just iteratively to add new rules only for each so-called critical pair q , r that is obtained from strings p that represent minimal overlaps in the left-hand sides of the rules one has.

Neighbor-dependent [2D] substitution systems Given a list of individual replacement rules such as {{_, 1}, {0, 1}}  {{1, 0}, {1, 1}} , each step in the evolution shown corresponds to Flatten2D[Partition[list, {2, 2}, 1, -1] /. rule] One can consider rules in which some replacements lead to subdivision of elements but others do not.

Nearby cellular automaton rules In a range r cellular automaton the new color of a particular cell depends only on cells at most a distance r away. One can make an equivalent cellular automaton of larger range by having a rule in which cells at distance more than r have no effect. One can then define nearby cellular automata to be those where the differences in the rule involve only cells close to the edge of the range.

Properties [of second-order cellular automata] The pattern from rule 67R with simple initial conditions grows irregularly, at an average rate of about 1 cell every 5 steps. The right-hand side of the pattern from rule 173R consists three triangles that repeat progressively larger at steps of the form 2 (9 s -1) . Rule 90R has the property that of the diamond of cells at relative positions {{-n,0},{0,-n},{n,0},{0,n}} it is always true for any n that an even number are black.

In each case, all cells are initially white, and one of the rules given on the left is applied for the specified number of steps.