In the first case, however, the black walls that are present seem to prevent any long-range transmission of information at all.

Implementation [of conserved quantity test] Whether a k -color cellular automaton with range r conserves total cell value can be determined from Catch[Do[ (If[Apply[Plus, CAStep[rule, #] - #] ≠ 0, Throw[False]] &)[ IntegerDigits[i, k, m]], {m, w}, {i, 0, k m - 1}]; True] where w can be taken to be k 2r , and perhaps smaller.

And over the course of time a wide range of mathematical and aesthetic principles were suggested.

For longer-range rules, the network must satisfy constraints of the kind discussed on page 483 .

Pollen The grains of pollen produced by different species of plants have a remarkable range of different forms.

But probabilistic models based on other underlying systems can yield sequences with long-range correlations.

General powers [of numbers] It has been known in principle since the 1930s that Mod[h n , 1] is uniformly distributed in the range 0 to 1 for almost all values of h .

Emulating Cellular Automata with Other Systems In the previous section we discovered the rather remarkable fact that cellular automata can be set up to emulate an extremely wide range of other types of systems.

For n = 3 this polytope comes close to filling the region of all possible colors, but for no n can it completely fill it—which is why practical displays and printing processes can produce only limited ranges of colors. An important observation, related to the fact that limitations in color ranges are usually not too troublesome, is that the perceived colors of objects stay more or less constant even when viewed in very different lighting, corresponding to very different wavelength distributions.

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] .