color will occur at any specific step, one still knows for example that black and white will on average always occur equally often.
But it turns out that there are cellular automata whose behavior is in effect still more complex—and in which even such averages become very difficult to predict. The pictures on the next several pages [32, 33, 34, 35, 36, 37, 38] give a rather dramatic example. The basic form of the rule is just the same as before. But now the specific rule used—that I call rule 110—takes the new color of a cell to be black in every case except when the previous colors of the cell and its two neighbors were all the same, or when the left neighbor was black and the cell and its right neighbor were both white.
The pattern obtained with this rule shows a remarkable mixture of regularity and irregularity. More or less throughout, there is a very regular background texture that consists of an array of small white triangles repeating every 7 steps. And beginning near the left-hand edge, there are diagonal stripes that occur at intervals of exactly 80 steps.
But on the right-hand side, the pattern is much less regular. Indeed, for the first few hundred steps there is a region that seems essentially random. But by the bottom of the first page all that remains of this region is three copies of a rather simple repetitive structure.
Yet at the top of the next page the arrival of a diagonal stripe from the left sets off more complicated behavior again. And as the system progresses, a variety of definite localized structures are produced.
Some of these structures remain stationary, like those at the bottom of the first page while others move steadily to the right or left at various speeds. And on their own, each of these structures works in a fairly simple way. But as the pictures illustrate, their various interactions can have very complicated effects.
And as a result it becomes almost impossible to predict—even approximately—what the cellular automaton will do.
Will all the structures that are produced eventually annihilate each other, leaving only a very regular pattern? Or will more and more structures appear until the whole pattern becomes quite random?
The only sure way to answer these questions, it seems, is just to run the cellular automaton for as many steps as are needed, and to