With rules (a) and (b), each different region effectively remains separated forever. But with rules (c) and (d) the regions gradually mix. … In rules (a) and (b) the regions remain separated forever, but in rules (c) and (d) they gradually diffuse into each other.

Patterns generated by rule 90 with various initial conditions. … Unlike rule 30 or rule 22 therefore, rule 90 cannot intrinsically generate randomness starting from simple initial conditions. … Note that the pictures above show only half as many steps of evolution as the corresponding pictures of rule 22 on the previous page .

But in this book we have discovered that even by following very simple rules it is possible to obtain forms of great complexity. … In both cases the rules are set up so that every stem in effect just branches into exactly three new stems at each step. … At each step every growing stem is replaced by a collection of three new stems according to the rules shown.

This is a consequence of the fact that the cellular automaton rule allows only certain blocks to appear in the pattern, as illustrated in the picture below. (e) is generated by a two-dimensional cellular automaton; (f) is the sequence that appears on the center column of rule 30. Cellular automaton rule 30, and the 3×2 blocks which appear in large patterns generated by it.

[No text on this page] Further examples of three-dimensional cellular automata, but now with rules that depend on all 26 neighbors that share either a face or a corner with a particular cell. In the top pictures, the rule specifies that a cell should become black when exactly one of its 26 neighbors was black on the step before. In the bottom pictures, the rule specifies that a cell should become black only when exactly two of its 26 neighbors were black on the step before.

[No text on this page] Rule 22 with various different simple initial conditions. … But in the bottom case, it is instead in many respects random, much like rule 30.

The details of the geometrical rules used are different in each case. … But with geometrical replacement rules of the kind shown on the facing page there is a problem with this. … The pattern obtained by repeatedly applying the simple geometrical rule shown on the right.

As the pictures on the next page demonstrate, rules like 254 and 90 that have fairly simple behavior lead to formulas that stay fairly simple. But for rule 30 the formulas rapidly get much more complicated. … And given this, one can imagine finding for any particular rule the formula that involves the smallest number of Nand functions.

The pictures on the left show patterns produced by the ordinary evolution of cellular automata with elementary rules 188 and 60. … For rule 188 the cellular automaton that does this involves 12 colors; for rule 60 it involves 6. In general, to find the color of a cell after t steps of rule 188 or rule 60 evolution takes about Log[2, t] steps.

The rule used here includes diagonal neighbors, and so involves a total of 8 neighbors for each cell, as indicated in the icon on the bottom left. The rule specifies that the center cell should become black if either 3 or 5 of its 8 neighbors were black on the step before, and should otherwise stay the same color as it was before. … In an extension to 8 neighbors of the scheme used in the pictures a few pages back , the rule has code number 175850.