Chapter 8: Implications for Everyday Systems

Section 2: The Growth of Crystals

Other models [of crystal growth]

There are many ways to extend the simple cellular automata shown here. One possibility is to allow dependence on next-nearest as well as nearest neighbors. In this case it turns out that non-convex as well as convex faceted shapes can be obtained. Another possibility is to allow cells that have become black to turn white again. In this case all the various kinds of patterns that we saw in Chapter 5 can occur. A general feature of cellular automaton rules is that they are fundamentally local. Some models of crystal growth, however, call for long-range effects such as a temperature field which changes throughout the crystal in an effectively instantaneous way. It turns out, however, that many seemingly long-range effects can actually be captured quite easily in cellular automata. In a typical case, this can be done by introducing a third possible color for each cell, and then having rapidly changing arrangements of this color.

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