in underlying rules, the overall shapes of the clusters produced remain very much the same.

In all these examples, however, the randomness that is involved comes from the same basic mechanism: it is explicitly inserted from outside at each step in the evolution of the system.

But it turns out that all that really seems to matter is that randomness is present: the mechanism through which it arises appears to be largely irrelevant. And in particular what this means is that randomness which comes from the mechanism of intrinsic randomness generation discussed in the previous section is able to make systems with discrete components behave in seemingly continuous ways.

The picture on the next page shows a two-dimensional cellular automaton where this happens. There is no randomness in the rules or the initial conditions for this system. But through the mechanism of intrinsic randomness generation, the behavior of the system exhibits considerable randomness. And this randomness turns out to lead to an overall pattern of growth that yields the same basic kind of smooth roughly circular form as in the aggregation model.

Having seen this, one might then wonder whether in fact any system that involves randomness will ultimately produce smooth overall patterns of growth. The answer is definitely no. In discussing two-dimensional cellular automata in Chapter 5, for example, we saw many examples where randomness occurs, but where the overall forms of growth that are produced have a complicated structure with no particular smoothness or continuity.

As a rough guide, it seems that continuous patterns of growth are possible only when the rate at which small-scale random changes occur is substantially greater than the overall rate of growth. For in a sense it is only then that there is enough time for randomness to average out the effects of the underlying discrete structure.

And indeed this same issue also exists for processes other than growth. In general the point is that continuous behavior can arise in systems with discrete components only when there are features that evolve slowly relative to the rate of small-scale random changes.

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From Stephen Wolfram: A New Kind of Science [citation]