In class 2, there are many different possible final states, but all of them consist just of a certain set of simple structures that either remain the same forever or repeat every few steps. … And typically all reasonable definitions will then assign any particular system to the same class.

Indeed, as I will discuss in this chapter and the chapters that follow, I believe that this mechanism is in fact ultimately responsible for a large fraction, if not essentially all, of the randomness that we see in the natural world. … But since one realistically cannot keep track of all these things, the ocean will inevitably seem in many respects unpredictable and random.

In all these examples, however, the randomness that is involved comes from the same basic mechanism: it is explicitly inserted from outside at each step in the evolution of the system. But it turns out that all that really seems to matter is that randomness is present: the mechanism through which it arises appears to be largely irrelevant.

For it is not necessary that no overlaps exist at all in the replacements—only that no overlaps occur in whatever sequences of elements can actually be generated by the evolution of the substitution systems. And in the end there are then all sorts of substitution systems which have the property that the causal networks they generate are always independent of the order in which their rules are applied.

But as of now I do not know of any fundamental reason why this might be so, and following my arguments in Chapter 8 I would not be at all surprised if the process of biological evolution had simply missed even methods of perception that are, in some sense, fairly obvious. … Free from any effects of terrestrial biological evolution might it have developed all sorts of higher forms of perception and analysis?

All the networks shown are truncated at the leaves of each tree.

Collisions [in rule 110] A fundamental result is that the sum of the widths of all persistent structures involved in an interaction must be conserved modulo 14.

It does not repeat with any short period or show any obvious structure for almost all keys. … (Most but not all of the results from my original paper are included in this book, together with various new results.) … From the beginning, computations of spacetime entropies for rule 30 (see page 960 ) gave indications that for strong cryptography one should not sample all cells in a column, and in 1991 Willi Meier and Othmar Staffelbach described essentially the explicit cryptanalysis approach shown on page 601 .

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 . … Exceptions are known to include so-called Pisot numbers such as GoldenRatio , √ 2 + 1 and Root[# 3 - # - 1 &, 1] (the numerically smallest of all Pisot numbers) for which Mod[h n , 1] becomes 0 or 1 for large n .

With any such shift rule, all states lie on cycles, and the lengths of these cycles are the divisors of the size n . … For prime k , each cycle (except all 0's) corresponds to a term in the product Factor[x k n - 1 - 1, Modulus  k] .