Rule 22—like rule 90 from page 26 —gives a pattern with fractal dimension Log[2,3] ≃ 1.58 ; rule 150 gives one with fractal dimension Log[2, 1+Sqrt[5]] ≃ 1.69 .

The simplest case involves one ball bouncing around in a region of a definite shape. In a rectangular region, the position is given by Mod[a t, {w, h}] and every point will be visited if the parameters have irrational ratios. … For a system of balls in a region with cyclic boundaries, a complicated proof due to Yakov Sinai from the 1960s purports to show that every ball eventually visits every point in the region, and that certain simple statistical properties of trajectories are consistent with randomness.

But it turns out that what one perceives as happening in a system like a mobile automaton can depend greatly on whether one is looking at the system from outside, or whether one is oneself somehow part of the system. For from the outside, one can readily see each individual step in the evolution of a mobile automaton, and one can tell that there is just a single active cell that visits different parts of the system in sequence. … But if the observer itself just consists of a collection of cells inside a mobile automaton, then no such change can occur except on steps when the active cell in the mobile automaton visits this collection of cells.

Of these, {2, 4, 12, 40, 144, 544, 2128, 8544, …} are themselves fixed points. Of combinator expressions up to size 6 all evolve to fixed points, in at most {1, 1, 2, 3, 4, 7} steps respectively (compare case (a)); the largest fixed points have sizes {1, 2, 3, 4, 6, 10} (compare case (b)). … The maximum number of levels in these expressions (see page 897 ) grows roughly linearly, although Depth[expr] reaches 14 after 26 and 25 steps, then stays there.

When the material is stretched, the number is A kneading process similar to ones used to make noodles or taffy, which exhibits very sensitive dependence on initial conditions. … Two examples of what can happen when the kneading process above is applied to nearby collections of points. In both cases the points initially diverge exponentially, as implied by chaos theory.

In a typical case, one splits the deck of cards in two, then carefully riffles the cards so as to make alternate cards come from each part of the deck. … But by doing Nest[s, Range[52], 26] one ends up with a simple reversal of the original deck, as in the pictures below.

[Causal networks for] 2D mobile automata As in 2D random walks, active cells in 2D mobile automata often do not return to positions they have visited before, with the result that no causal connections end up being created.

When GCD[k, n]  1 the dot can never visit position 0. … In general, the dot will visit position m = k^IntegerExponent[n, k] every MultiplicativeOrder[k, n/m] steps.

One of these is that every node always has exactly 3 incoming and 3 outgoing connections. Another feature is that there is always a path of doubled connections (associated with the active cell) that visits every node in some order.

An argument for the Second Law from around 1900, still reproduced in many textbooks, is that if a system is ergodic then it will visit all its possible states, and the vast majority of these will look random. But only very special kinds of systems are in fact ergodic, and even in such systems, the time necessary to visit a significant fraction of all possible states is astronomically long. … The argument suffers however from the same difficulties as the ones for chaos theory discussed in Chapter 6 and does not in the end explain in any real way the origins of randomness, or the observed validity of the Second Law.