In detail, some of the patterns are definitely more complicated than those seen in elementary rules. But at the level of overall behavior, there are no fundamental differences. And in the case of nested patterns even the specific structures seen are usually the same as for elementary rules. Thus, for example, the structure in codes 237 and 948 is the most common, followed by the one in code 1749. The only new structure not already seen in elementary rules is the one in code 420—but this occurs only quite rarely.

About 85% of all three-color totalistic cellular automata produce behavior that is ultimately quite regular. But just as in elementary cellular automata, there are some rules that yield behavior that seems in many respects random. A few examples of this are given on the facing page.

Beyond fairly uniform random behavior, there are also cases similar to elementary rule 110 in which definite structures are produced that interact in complicated ways. The next page gives a few examples. In the first case shown, the pattern becomes repetitive after about 150 steps. In the other two cases, however, it is much less clear what will ultimately happen. The following pages [67, 68] continue these patterns for 3000 steps. But even after this many steps it is still quite unclear what the final behavior will be.

Looking at pictures like these, it is at first difficult to believe that they can be generated just by following very simple underlying cellular automaton rules. And indeed, even if one accepts this, there is still a tendency to assume that somehow what one sees must be a consequence of some very special feature of cellular automata.

As it turns out, complexity is particularly widespread in cellular automata, and for this reason it is fortunate that cellular automata were the very first systems that I originally decided to study.

But as we will see in the remainder of this chapter, the fundamental phenomena that we discovered in the previous chapter are in no way restricted to cellular automata. And although cellular automata remain some of the very best examples, we will see that a vast range of utterly different systems all in the end turn out to exhibit extremely similar types of behavior.

Exportable Images for This Page:

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