So in the end, any features of the behavior of a system that go beyond pure repetition will tend to seem to our ears essentially random. … In different situations the reasons for using such probabilistic models have been somewhat different, but before the discoveries in this book one of the key points was that it seemed inconceivable that there could be deterministic models that would reproduce the kinds of complexity and apparent randomness that were so often seen in practice.

And indeed sequence (a) is certainly not random; in fact it is purely repetitive. … So do these fluctuations represent evidence that sequences (g) and (h) are not in fact random?

The patterns that are obtained by this procedure turn out for reasons that are still not particularly clear to have a random but on average nested form. … Rapid rearrangement of gray cells on successive steps can then have a similar effect to the random walks that occur in the usual DLA model. … But so long as there is effective randomness in the successive positions of these cells, and so long as the total number of them is conserved, then it appears that DLA-like results are usually obtained.

Non-standard diffusion To get ordinary diffusion behavior of the kind that occurs in gases—and is described by the diffusion equation—it is in effect necessary to have perfect uncorrelated randomness, with no structure that persists too long. … In rule (c) there is considerable apparent randomness, but it turns out that there are also fluctuations that last too long to yield ordinary diffusion.

And since the mid-1990s such systems (usually characterized as random trees or random context-free languages) have sometimes been used in analyzing data that is expected to have grammatical structure of some kind.

Two views of the evolution of rule 54 from typical random initial conditions.

Tumors In both plants and animals tumors seem to grow mainly by fairly random addition of cells to their surface—much as in the aggregation models shown on page 332 .

The evidence is based on applying various standard statistical tests of randomness, and remains somewhat haphazard. (Already in 1888 John Venn had noted for example that the first 707 digits of π lead to an apparently typical 2D random walk.) … It has nevertheless been known since the work of Emile Borel in 1909 that numbers picked randomly on the basis of their value are almost always normal.

It is important to realize however that this randomness has little to do with the details of the initial conditions. … And it has often been observed that the sequences generated in these simulations look quite random. But as we now see, such randomness cannot in fact be a consequence of the chaos phenomenon and of sensitive dependence on initial conditions.

At a small scale, there are some obvious triangular and other structures, but beyond these the pattern looks essentially random. So just as in simple programs like cellular automata, it seems that simple systems based on numbers can also yield behavior that is highly complex and apparently random.