For some purposes a more direct analog of messages is not programs or rules for systems like cellular automata but instead initial conditions. … But if one was just given a collection of initial conditions without any underlying rules one would then need to find out what underlying rules one was supposed to use in order to determine their meaning. Yet the system will always do something, whatever rules one uses.

All the rules shown start with f[1]=f[2]=1 .

(The rule is the same as (g) on the previous page .)

Self-similarity [of rule 90] The pattern generated by rule 90 after a given number of steps has the property that it is identical to what one would get by going twice as many steps, and then keeping only every other row and column. … Nesting occurs in all cellular automata with additive rules (see page 955 ).

3D class 4 [cellular automaton] rules With a cubic lattice of the type shown on page 183 , and with updating rules of the form LifeStep3D[{p_, q_, r_}, a_List] := MapThread[If[ #1  1 && p ≤ #2 ≤ q || #2  r, 1, 0]&, {a, Sum[RotateLeft[ a, {i, j, k}], {i, -1, 1}, {j, -1, 1}, {k, -1, 1}] - a}, 3] Carter Bays discovered between 1986 and 1990 the three examples {5, 7, 6} , {4, 5, 5} , and {5, 6, 5} . The pictures below show successive steps in the evolution of a moving structure in the second of these rules.

Other models [of mutation] Sequential substitution systems are probably more realistic than cellular automata as models of genetic programs, since elements can explicitly be added to their rules at will. As a rather different approach, one can consider a fixed underlying rule—say a class 4 cellular automaton—with modifications in initial conditions. The notion of universality in Chapter 11 implies that under suitable conditions this should be equivalent to modifications in rules.

[Examples of] reducible systems The color of a cell at step t and position x can be found by starting with initial condition Flatten[With[{w = Max[Ceiling[Log[2, {t, x}]]]}, {2 Reverse[IntegerDigits[t, 2, w]] + 1, 5, 2 IntegerDigits[x, 2, w] + 2}]] then for rule 188 running the cellular automaton with rule {{a : (1 | 3), 1 | 3, _}  a, {_, 2 | 4, a : (2 | 4)}  a, {3, 5 | 10, 2}  6, {1, 5 | 7, 4}  0, {3, 5, 4}  7, {1, 6, 2}  10, {1, 6 | 11, 4}  8, {3, 6 | 8 | 10 | 11, 4}  9, {3, 7 | 9, 2}  11, {1, 8 | 11, 2}  9, {3, 11, 2}  8, {1, 9 | 10, 4}  11, {_, a_ /; a > 4, _}  a, {_, _, _}  0} and for rule 60 running the cellular automaton with rule {{a : (1 | 3), 1 | 3, _}  a, {_, 2 | 4, a : (2 | 4)}  a, {1, 5, 4}  0, {_, 5, _}  5, {_, _, _}  0}

[No text on this page] An example of unbounded growth in rule 110.

[No text on this page] Fifteen hundred steps of rule 30 evolution.

The rules for the three cellular automata involve only nearest neighbors, and allow 12 possible colors for each cell.