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In class 2, there are many different possible final states, but all of them consist just of a certain set of simple structures that either remain the same forever or repeat every few steps.

In class 3, the behavior is more complicated, and seems in many respects random, although triangles and other small-scale structures are essentially always at some level seen.

And finally, as illustrated on the next few pages [236, 237, 238, 239], class 4 involves a mixture of order and randomness: localized structures are produced which on their own are fairly simple, but these structures move around and interact with each other in very complicated ways.

I originally discovered these four classes of behavior some nineteen years ago by looking at thousands of pictures similar to those on the last few pages [232, 233, 234]. And at first, much as I have done here, I based my classification purely on the general visual appearance of the patterns I saw.

But when I studied more detailed properties of cellular automata, what I found was that most of these properties were closely correlated with the classes that I had already identified. Indeed, in trying to predict detailed properties of a particular cellular automaton, it was often enough just to know what class the cellular automaton was in.

And in a sense the situation was similar to what is seen, say, with the classification of materials into solids, liquids and gases, or of living organisms into plants and animals. At first, a classification is made purely on the basis of general appearance. But later, when more detailed properties become known, these properties turn out to be correlated with the classes that have already been identified.

Often it is possible to use such detailed properties to make more precise definitions of the original classes. And typically all reasonable definitions will then assign any particular system to the same class.

Captions on this page:

Examples of class 4 cellular automata with totalistic rules involving nearest neighbors and three possible colors for each cell. Each picture shows 1500 steps of evolution from random initial conditions.

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