Mod 3 [cellular automaton] rule Code 420 is an example of an additive rule, and yields a pattern corresponding to Pascal's triangle modulo 3, as discussed on page 870 .

Block occurrences [in rule 30] The pictures below show at which step each successive block of length up to 8 first appears in evolution according to various cellular automaton rules starting from a single black cell. For rule 30, the numbers of steps needed for each block of lengths 1 through 10 to appear at least once is {1, 2, 4, 12, 22, 24, 33, 59, 69, 113} .

Representing the strings by lists, one can write rules in the form {{1, 1, s___}  {s, 1, 0}, {1, s___}  {s, 1, 0, 1}} so that the evolution is given by MWTSEvolve[rule_, list_, t_] := Nest[Flatten[Map[ReplaceList[#, rule] &, #], 1] &, list, t]

Projections from 3D [cellular automata] Looking from above, with closer cells shown darker, the following show patterns generated after 30 steps, by (a) the rule at the top of page 183 , (b) the rule at the bottom of page 183 , (c) the rule where a cell becomes black if exactly 3 out of 26 neighbors were black and (d) the same as (c), but with a 3×3×1 rather than a 3×1×1 initial block of black cells:

Implementation [of 2D substitution systems] With the rule on page 187 given for example by {1  {{1, 0}, {1, 1}}, 0  {{0, 0}, {0, 0}}} the result of t steps in the evolution of a 2D substitution system from a initial condition such as {{1}} is given by SS2DEvolve[rule_, init_, t_] := Nest[Flatten2D[# /. rule] &, init, t] Flatten2D[list_] := Apply[Join, Map[MapThread[Join, #] &, list]]

Densities in other [cellular automaton] rules The pictures below show how the densities on successive steps depend on the initial density. … Rule 236 is class 2, and the density retains a memory of its initial value. But in the class 3 rules 126 and 30, the densities converge quickly to a fixed value.

[Subset of] elementary rules The examples shown have rule numbers n for which IntegerDigits[n, 2, 8] matches {_, i_, _, j_, i_, _, j_, 0} .

For rule 254, for example, it is 8t + 2 , while for rule 90 it is 4t + 2 . … Thus for rule 30 it is {7, 14, 29, 60, 129} and for rule 110 {7, 15, 27, 52, 88} . The size of the minimal BDD can depend on the order in which variables are specified; thus for example, just reflecting rule 30 to give rule 86 yields {6, 11, 20, 36, 63} .

But one can also consider two-dimensional cellular automata that involve a whole grid of cells, with the color of each cell being updated according to a rule that depends on its neighbors in all four directions on the grid, as in the picture below. The form of the rule for a typical two-dimensional cellular automaton. … Usually I consider so-called totalistic rules in which the new color of the center cell depends only on the average of the previous colors of its four neighbors, as well as on its own previous color.

The reversibility of the underlying rules has some obvious consequences, such as the presence of triangles pointing sideways but not down. But despite their reversibility, the rules still manage to produce the kinds of complex behavior that we have seen in cellular automata and many other systems throughout this book. … There are some constraints on the details of the kinds of collisions that are possible, but reversible rules typically tend to work very much like ordinary ones.