For one might have assumed that any apparent randomness in the final shape of something like a crystal must always be a consequence of randomness in its original seed, or in the environment in which it grew. But in fact, as the pictures below show—and as we have seen many times in this book—it is also possible for randomness to arise intrinsically just through the application of simple underlying rules. And contrary to what has always been assumed, I suspect that this is actually how the apparent randomness that one sometimes sees in shapes formed by crystalline materials often comes about.

As an example, the second picture below shows the rule 110 cellular automaton evolving from random initial conditions. The picture The generation of a nested pattern by rule 184 starting from random initial conditions. … A highly compressed representation of the evolution of rule 110 from random initial conditions in which only the first cell in every 14×7 block is sampled.

Most often, the key ingredient is randomness. … The next page shows a classic example of this phenomenon, based on so-called random walks. Each random walk is made by taking a discrete particle, and then at each step randomly moving the particle one position to the left or right.

The Breaking of Materials In everyday life one of the most familiar ways to generate randomness is to break a solid object. … So what is the origin of this randomness? At first one might think that it must be a reflection of random small-scale irregularities within the material.

Note (a) for Defining the Notion of Randomness

Note (a) for The Intrinsic Generation of Randomness

The picture below shows an example where a larger amount of apparent randomness is seen. … A multiway system with behavior that shows some signs of apparent randomness. … At least with the initial condition used here, despite considerable early apparent randomness, the differences in number of elements do repeat (shifted by 1) every 1071 steps.

There is no doubt that they do, and as one example I will briefly discuss here what is probably the most obvious feature of essentially all financial markets: the apparent randomness with which prices tend to fluctuate. … So what is the origin of this randomness? … And it is then assumed that these estimates are ultimately affected by all sorts of events that go on in the world, making random movements