Note (c) for Chaos Theory and Randomness from Initial Conditions…But vastly more common in practice is instability only at specific critical points—say bifurcation points—combined with either intrinsic randomness generation or randomness from the environment.

Note (b) for Chaos Theory and Randomness from Initial Conditions…Small pieces of paper spin in a repetitive way; but larger pieces of paper tend to flutter in a seemingly random way (as discussed, among others, by James Clerk Maxwell in 1853). … I suspect that in these cases the randomness that occurs has an intrinsic origin, rather than being the result of sensitive dependence on initial conditions.

Note (b) for The Intrinsic Generation of Randomness…Random number generators A fairly small number of different types of random number generators have been used in practice, so it is possible to describe all the major ones here. … Practical stream cipher systems can thus be used as random number generators. … I invented the rule 30 cellular automaton random number generator in 1985.

In the pictures at the top of the facing page , the first network in each case represents random initial conditions in which any possible sequence of black and white cells can occur.

Note (d) for The Intrinsic Generation of Randomness…Perturbations of size δ are then added using v + Sign[v - 1/2] Random[] δ .

In each case the initial conditions that are used are completely random.

The pictures on the next two pages [ 246 , 247 ] show various steps in the evolution of some simple two-dimensional cellular automata starting from random initial conditions.

The facing page shows a typical example with random initial conditions.

Nothing as simple is true for the left-hand edge, and indeed this seems to execute an essentially random walk—with an average motion of about 0.2428 cells per step. Note that in the approximation that the colors of all cells in the pattern are assumed completely independent and random there should be motion by 0.25 cells per step.

History [of emergence of order] The fact that despite initial randomness processes like friction can make systems settle down into definite configurations has been the basis for all sorts of engineering throughout history. … When the study of feedback and stability became popular in the 1940s, there were many results about how specific simple fixed or repetitive behaviors in time could emerge despite random input. … And when I first generated pictures of the behavior of arbitrary cellular automaton rules, what struck me most was the order that emerged even from random initial conditions.