The third element of the rule is at first used only on some steps—but after step 50 it appears to be used somewhere in every step. … Without this additional block of black, only the first case in the rule can ever apply.

It shows class 2 behavior in which information propagates only over limited distances, so that except when the total size of the system is comparable to the range of the rule, boundary conditions are not crucial. … The rule on page 336 gives dappled patterns with progressively larger spots.

Endpapers The goldenrod pages inside the front cover show the center 900 or so cells of the first 500 or so steps in the evolution of the rule 30 cellular automaton of page 29 from a single black cell.

But in recent years, with more extensive measurement methods, there has been increasing evidence for precise deterministic underlying rules.

I termed two-color nearest-neighbor cellular automata "elementary" to reflect the idea that their rules are as simple as possible.

Examples where this works include the surjective rules 30 and 90.

Properties [of cyclic tag systems] Assuming that black and white elements occur in an uncorrelated way, then the sequences in a cyclic tag system with n blocks should grow by an average of Count[Flatten[rules], 1]/n - 1 elements at each step. … Rules such as {{1, 0}, {0, 1}} and {{1, 1}, {0}} therefore yield repetitive behavior with sequences of limited length. … The rules for the relevant substitution system may however depend on the initial conditions for the cyclic tag system.

[Networks generated by] random replacements As indicated in the note above, applying the second rule (T1, shown as (b) on page 511 ) at an appropriate sequence of positions can transform one planar network into any other with the same number of nodes. The pictures below show what happens if this rule is repeatedly applied at random positions in a network. Each time it is applied, the rule adds two edges to one face, and removes them from another.

But this extra structure can be important in the application of other rules—and can for example emulate something like having multiple colors of connections.

Possible purposes [for systems] As part of asking whether the rules for a system are somehow minimal for a given purpose, one can ask what properties the system has that could reasonably be considered a purpose at all. … And so similarly, one would be unlikely to think that generating the center column from rule 30 could represent any sort of meaningful purpose—unless one was operating within the framework that I have developed in this book.