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Beyond randomness, the last example in the previous chapter was rule 110: a cellular automaton whose behavior becomes partitioned into a complex mixture of regular and irregular parts. This particular cellular automaton is essentially unique among the 256 rules considered here: of the four cases in which such behavior is seen, all are equivalent if one just interchanges the roles of left and right or black and white.

So what about more complicated cellular automaton rules?

The 256 "elementary" rules that we have discussed so far are by most measures the simplest possible—and were the first ones I studied. But one can for example also look at rules that involve three colors, rather than two, so that cells can not only be black and white, but also gray. The total number of possible rules of this kind turns out to be immense—7,625,597,484,987 in all—but by considering only so-called "totalistic" ones, the number becomes much more manageable.

The idea of a totalistic rule is to take the new color of each cell to depend only on the average color of neighboring cells, and not on their individual colors. The picture below shows one example of how this works. And with three possible colors for each cell, there are 2187 possible totalistic rules, each of which can conveniently be identified by a code number as illustrated in the picture. The facing page shows a representative sequence of such rules.

We might have expected that by allowing three colors rather than two we would immediately get noticeably more complicated behavior.

Example of a totalistic cellular automaton with three possible colors for each cell. The rule is set up so that the new color of every cell is determined by the average of the previous colors of the cell and its immediate neighbors. With 0 representing white, 1 gray and 2 black, the rightmost element of the rule gives the result for average color 0, while the element immediately to its left gives the result for average color 1/3—and so on. Interpreting the sequence of new colors as a sequence of base 3 digits, one can assign a code number to each totalistic rule.

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