So then I went back and started looking by eye at mobile automata with large numbers of randomly chosen rules.

Indeed, even after a million steps, when the Results of applying the rule n  If[EvenQ[n], 5n/2, (n + 1)/2] , starting with different initial choices of n .

Often it is very difficult to predict whether this will be so just by looking at the underlying rules.

But in fact what we have seen here is that once again the fundamental mechanisms responsible already occur in a much more minimal way in programs that have some remarkably simple underlying rules.

Any consistent choice of such slices will correspond to a possible evolution history—with the same underlying rules, but potentially a different scheme for determining the order in which to apply replacements.

There seems to be no simple criterion for deciding from the rule what type of spectrum will be obtained. … With k colors each giving a string of the same length s the recurrence relation is Thread[Map[ ϕ [#][t + 1, ω ] &, Range[k] - 1]  Apply[Plus, MapIndexed[Exp[  ω (Last[#2] - 1) s t ] ϕ [#1][t, ω ] &, Range[k] - 1 /. rules, {-1}], {1}]/ √ s ] Some specific properties of the examples shown include: (a) (Thue–Morse sequence) The spectrum is essentially Nest[Range[2 Length[#]] Join[#, Reverse[#]] &, {1}, t] . … (Z transform or generating function methods can be applied directly only for substitution systems with rules such as {1  list, 0  1 - list} .)

These tests mostly seem simpler than those shown on page 597 obtained by running a cellular automaton rule on the data. Over the years, essentially every proposed statistical test of randomness has been applied to the center column of rule 30. … So as of now, the center column of rule 30 appears to pass every single proposed statistical test of randomness.

With the state of a 2-color tag system encoded as an integer according to FromDigits[Reverse[list] + 1, 3] the following takes the rule for any such tag system (in the first form from page 894 ) and yields a primitive recursive function that emulates a single step in its evolution: TSToPR[{n_, rule_}] := Fold[Apply[c, Flatten[{#1, Array[p, # 2], c[r[z, c[r[p[1], s], c[r[z, p[2]], c[r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[2]]], p[1]]], p[#2]]}]] & , c[c[r[p[1], s], p[1], c[r[p[1], r[z, c[s, c[s, s]]]], c[c[r[z, c[r[p[1], s], c[r[z, c[s, z]], c[r[p[1], r[z, c[r[p[1], s], c[r[z, p[2]], c[ r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[2]]], p[1]]]], p[2], p[3]]], p[1]]], p[1], p[1]], p[1]], p[2]]], p[n + 1], MapIndexed[c[r[z, c[r[p[1], p[4]], p[2], p[3], p[4]]], c[r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[Length[#2] + 1]], # 1 〚 1 〛 , #1 〚 2 〛 ] & , Nest[Partition[#1, 2] & , Table[Nest[c[s, #] & z, FromDigits[Reverse[IntegerDigits[i, 2, n] /. rule] + 1, 3]], {i, 0, 2 n - 1}], n - 1], {0, n - 1}]], Range[n, 1, -1]] (For tag system (a) from page 94 this yields a primitive recursive function of size 325.)

Isotropy [in lattice systems] Any pattern grown from a single cell according to rules that do not distinguish different directions on a lattice must show the same symmetry as the lattice. But we have seen that in fact many rules actually yield almost circular patterns with much higher symmetry. … Even though it is not inevitable from lattice symmetry, one might think that if there is some kind of effective randomness in the underlying rules then sufficiently large patterns would still often show some sort of average isotropy.

Cover image The image on the cover of this book is derived from the first 440 or so steps (with perhaps 10 at each end cut off by trimming) of the pattern generated by evolution according to the rule 110 cellular automaton discussed on page 32 , with an initial condition consisting of repeats of followed by repeats of .