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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.
With random initial conditions, the period is typically up to about 3 n/2 .
So then I went back and started looking by eye at mobile automata with large numbers of randomly chosen rules.
But for π , as well as for cube roots, fourth roots, and so on, the continued fraction representations one gets seem essentially random.
With such rules, the sequence of networks obtained no longer needs to form any kind of simple progression, and indeed one finds that even the total number of nodes at each step can vary in a way that seems in many respects completely random.
But explicit construction, based on correspondence with one-dimensional cellular automata, leads to the example shown at the top of the facing page : a system with 56 allowed templates in which the only pattern satisfying the constraint is a complex and largely random one, derived from the rule 30 cellular automaton.
A cellular automaton with a simple rule that generates a pattern which seems in many respects random.
As a first example, consider a procedure that at each step picks a square at random, then reverses its color whenever doing so reduces the total number of squares that violate the constraint.
And much as in physical systems like fluids, what ultimately leads to this is the presence of small-scale apparent randomness that washes out details of individual cells or molecules—as well as of conserved quantities that force certain overall features not to change too quickly.
It takes fewer steps for networks to be built up, but the results are qualitatively similar to those on the previous page : rule (a) yields a nested structure, rule (b) gives repetitive behavior, while rule (c) produces behavior that seems complicated and in some respects random.