There are various deviations from perfect randomness. … (The probability for s randomly chosen integers to be relatively prime is 1/Zeta[s] .)

My papers The primary papers that I published about cellular automata and other issues related to this book were (the dates indicate when I finished my work on each paper; the papers were actually published 6-12 months later): • "Statistical mechanics of cellular automata" (June 1982) (introducing 1D cellular automata and studying many of their properties) • "Algebraic properties of cellular automata" (with Olivier Martin and Andrew Odlyzko ) (February 1983) (analyzing additive cellular automata such as rule 90) • "Universality and complexity in cellular automata" (April 1983) (classifying cellular automaton behavior) • "Computation theory of cellular automata" (November 1983) (characterizing behavior using formal language theory) • "Two-dimensional cellular automata" (with Norman Packard ) (October 1984) (extending results to two dimensions) • "Undecidability and intractability in theoretical physics" (October 1984) (introducing computational irreducibility) • "Origins of randomness in physical systems" (February 1985) (introducing intrinsic randomness generation) • "Random sequence generation by cellular automata" (July 1985) (a detailed study of rule 30) • "Thermodynamics and hydrodynamics of cellular automata" (with James Salem ) (November 1985) (continuum behavior from cellular automata) • "Approaches to complexity engineering" (December 1985) (finding systems that achieve specified goals) • "Cellular automaton fluids: Basic theory" (March 1986) (deriving the Navier–Stokes equations from cellular automata) The ideas in the first five and the very last of these papers have been reasonably well absorbed over the past fifteen or so years.

Note that even though the underlying rule involves randomness definite geometrical shapes can be produced. … The question of what ultimate forms of behavior can occur with any sequence of random choices, starting from a given configuration with a given rule, is presumably in general undecidable. … But most generalized aggregation models do not have this property: instead, the form of their internal patterns depends on the sequence of random choices made.

With completely random input, the output will on average be longer by a factor Sum[2 -(n + 1) r[n], {n, 1, ∞ }] where r[n] is the length of the representation for n .

Each repeating block of digits typically seems quite random, and has properties such as all possible subblocks of digits up to a certain length appearing (see page 1084 ).

Note that in a random network of n nodes, about n/8 such clusters typically occur.

In rule 154R, each diagonal stripe is followed by at least one 0; otherwise, the positions of the stripes appear to be quite random, with a density around 0.44.

The basic origin of this phenomenon is the averaging effect of randomness discussed in Chapter 7 (technically, it is the survival only of leading operators at renormalization group fixed points).

But the experiments I have carried out do suggest that, just as with simple register machines, searching through many millions of short programs typically yields at least a few that exhibit complex and seemingly random behavior.

In the third picture, however, one no longer sees such regularity, and instead there is behavior that seems in many respects random.