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The solution is Fold[Mod[#1 + k, #2, 1]&, 0, Range[n]] , or FromDigits[RotateLeft[IntegerDigits[n, 2]], 2] for k = 2 .
It shows class 2 behavior in which information propagates only over limited distances, so that except when the total size of the system is comparable to the range of the rule, boundary conditions are not crucial.
And indeed, it seems that the basic themes of repetition, nesting, randomness and localized structures that we already saw in specific cellular automata in the previous chapter are actually very general, and in fact represent the dominant themes in the behavior of a vast range of different systems.
In the three cases shown in the center a whole range of mixtures of different repetitive patterns are possible.
And as various examples in Chapter 8 demonstrate, across a whole range of physical, biological and other systems there can indeed be remarkable similarities.
The answer, as we have seen many times in this book, is that across a very wide range of programs there is great universality in the behavior that occurs.
But in trying to understand the range of behavior that can occur in reversible systems it is often convenient to consider classes of cellular automata with rules that are specifically constructed to be reversible.
And within the piece of the network corresponding to the particle, the effective structure of space may be very different—with for example more long-range connections added to reduce the effective overall distance.
As I discussed in the last two sections [ 14 , 15 ], causal invariance of the underlying rules implies that such structures should be able to move at a range of uniform speeds through the background.
For one knows that given a single fixed underlying language, it is possible to describe an almost arbitrarily wide range of things.