Examples of patterns produced by two-dimensional cellular automata set up to mimic the growth of crystals. The rules in each case take a cell to become black if the specified number of its neighbors (including diagonals) on a square grid are black on the step before. These rules are such that once a cell has become black, corresponding to solid, it never reverts to white again. In each case a row of initial black cells of the specified length was used.

The occurrence of these last forms is at first especially surprising. 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 above 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.