Class 4 [cellular automaton] rules Other examples of class 4 totalistic rules with k = 3 colors include 357 (page 282 ), 438, 600, 792, 924, 1038, 1041, 1086, 1329 (page 282 ), 1572, 1599 (see page 70 ), 1635 (see page 67 ), 1662, 1815 (page 236 ), 2007 (page 237 ) and 2049 (see page 68 ).

Simulating mobile automata Given a mobile automaton like the one from page 73 with rules in the form used on page 887 —and behavior of any complexity—the following will yield a causal-invariant substitution system that emulates it: Map[StringJoin, Map[{"AAABB", "ABABB", "ABAABB"} 〚 # + 1 〛 &, Map[Insert[# 〚 1 〛 , 2, 2]  Insert[# 〚 2, 1 〛 , 2, 2 + # 〚 2, 2 〛 ] &, rule], {2}], {2}]

Density in rule 90 From the superposition principle above and the number of black cells at step t in a pattern starting from a single black cell (see page 870 ) one can compute the density after t steps in the evolution of rule 90 with initial conditions of density p to be (see also page 602 ) 1/2 (1 - (1 - 2 p)^(2^DigitCount[t,2,1]))

Reversibility [of networks and universe] By including both forward and backward versions of every transformation it is straightforward to set up reversible rules for network evolution. It is not clear, however, whether the basic rules for the universe are really reversible.

Rule 37R Complicated structures are fairly easy to get with this rule.

Rule 60 Turing machines One can emulate rule 60 using the 8-case s = 3 , k = 3 Turing machine (with initial condition Append[list + 1, 1] surrounded by 0 's) {{1, 2}  {2, 2, 1}, {1, 1}  {1, 1, 1}, {1, 0}  {3, 1, -1}, {2, 2}  {2,1, 1}, {2, 1}  {1, 2, 1}, {3, 2}  {3, 2, -1}, {3, 1}  {3, 1, -1}, {3, 0}  {1, 0, 1}} or by using the 6-case s = 2 , k = 4 Turing machine (with initial condition Append[3list, 0] with 0 's on the left and 1 's on the right) {{1, 3}  {2, 2, 1}, {1, 2}  {1, 3, -1}, {1, 1}  {1, 0, -1}, {1, 0}  {1, 1, 1}, {2, 3}  {2, 1, 1}, {2, 0}  {1, 2, 1}} This second Turing machine is directly analogous to the one for rule 110 on page 707 . Random searches suggest that among s = 3 , k = 3 Turing machines roughly one in 25 million reproduce rule 60 in the same way as the machines discussed here.

And of the 196 possible rules involving two colors and blocks of length at most three, 112 have this property. But there are also an additional 20 rules which allow some overlap but which nevertheless yield convergence after one step. … In these rules some of the elements essentially just supply context, but are not affected by the replacement.

[No text on this page] Examples of generalized mobile automata with various rules.

If a more general network is allowed then rules based on analogs of network substitution systems from page 508 can be used. (One can also construct an infinite tree from a general network by following all its possible paths, as on page 277 , but in most cases there will be no simple way to apply symbolic system rules to such a tree.)

The rule 30 cellular automaton, for example, can be described as follows. … Now each successive person in each successive row determines the color of the card they hold up by looking at the person directly above them, and above them immediately to their left and right, and then applying the simple rule on page 27 . A photograph of the stadium will then show the pattern produced by rule 30.