it turns out that class 4 behavior is typically seen—as illustrated on the facing page. And what is particularly remarkable is that this behavior involves the same kinds of localized structures and other features that we saw in ordinary discrete class 4 cellular automata.

So what about two-dimensional cellular automata? Do these also exhibit the same four classes of behavior that we have seen in one dimension? The pictures on the next two pages [246, 247] show various steps in the evolution of some simple two-dimensional cellular automata starting from random initial conditions. And just as in one dimension a few distinct classes of behavior can immediately be seen.

But the correspondence with one dimension becomes much more obvious if one looks not at the complete state of a two-dimensional cellular automaton at a few specific steps, but rather at a one-dimensional slice through the system for a whole sequence of steps.

The pictures on page 248 show examples of such slices. And what we see is that the patterns in these slices look remarkably similar to the patterns we already saw in ordinary one-dimensional cellular automata. Indeed, by looking at such slices one can readily identify the very same four classes of behavior as in one-dimensional cellular automata.

So in particular one sees class 4 behavior. In the examples on page 248, however, such behavior always seems to occur superimposed on some kind of repetitive background—much as in the case of the rule 110 one-dimensional cellular automaton on page 229.

So can one get class 4 behavior with a simple white background? Much as in one dimension this does not seem to happen with the very simplest possible kinds of rules. But as soon as one goes to slightly more complicated rules—though still very simple—one can find examples.

And so as one example page 249 shows a two-dimensional cellular automaton often called the Game of Life in which all sorts of localized structures occur even on a white background. If one watches a movie of the behavior of this cellular automaton its correspondence to a one-dimensional class 4 system is not particularly obvious. But as soon as one looks at a one-dimensional slice—as on page 249—what one sees is immediately strikingly similar to what we have seen in many one-dimensional class 4 cellular automata.

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