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cellular automata. And indeed the previous page shows that if one looks at the evolution of a one-dimensional slice through each two-dimensional pattern the results one gets are strikingly similar to what we have seen in ordinary one-dimensional cellular automata.

But looking at such slices cannot reveal much about the overall shapes of the two-dimensional patterns. And in fact it turns out that for all the two-dimensional cellular automata shown on the last few pages [173, 174, 175], these shapes are always very regular.

But it is nevertheless possible to find two-dimensional cellular automata that yield less regular shapes. And as a first example, the picture on the facing page shows a rule that produces a pattern whose surface has seemingly random irregularities, at least on a small scale.

In this particular case, however, it turns out that on a larger scale the surface follows a rather smooth curve. And indeed, as the picture on page 178 shows, it is even possible to find cellular automata that yield overall shapes that closely approximate perfect circles.

But it is certainly not the case that all two-dimensional cellular automata produce only simple overall shapes. The pictures on pages 179-181 show one rule, for example, that does not. The rule is actually rather simple: it just states that a particular cell should become black whenever exactly three of its eight neighbors—including diagonals—are black, and otherwise it should stay the same color as it was before.

In order to get any kind of growth with this rule one must start with at least three black cells. The picture at the top of page 179 shows what happens with various numbers of black cells. In some cases the patterns produced are fairly simple—and typically stop growing after just a few steps. But in other cases, much more complicated patterns are produced, which often apparently go on growing forever.

The pictures on page 181 show the behavior produced by starting from a row of eleven black cells, and then evolving for several hundred steps. The shapes obtained seem continually to go on changing, with no simple overall form ever being produced.

And so it seems that there can be great complexity not only in the detailed arrangement of black and white cells in a two-dimensional cellular automaton pattern, but also in the overall shape of the pattern.

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