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The pictures on the facing page show typical examples of pigmentation patterns in animals, and demonstrate that even across a vast range of different types of animals just a few kinds of patterns occur over and over again. So how are these patterns produced? Even though some of them seem quite complex, it turns out that once again there is a rather simple kind of rule that can account for them.

The idea is that when a pattern forms, the color of each element will tend to be the same as the average color of nearby elements, and opposite to the average color of elements further away. Such an effect could have its origin in the production and diffusion of activator and inhibitor chemicals, or, for example, in actual motion of different types of cells. But regardless of its origin, the effect itself can readily be captured just by setting up a two-dimensional cellular automaton with appropriate rules.

The pictures below show what happens with two slightly different choices for the relative importance of elements that are further away. In both cases, starting from a random distribution of black and white elements there quickly emerge definite patterns—in the first case a collection of spots, and in the second case a maze-like or labyrinthine structure.

The next page shows the final patterns obtained with a whole array of different choices of weightings for elements at different distances. A certain range of patterns emerges—almost all of which turn out to be quite similar to patterns that one sees on actual animals.

Evolution of simple two-dimensional cellular automata in which the color of each cell at each step is determined by looking at a weighted sum of the average colors of cells up to distance 3 away. In both rules shown the cell itself and its nearest neighbors enter with weight 1. Cells at distances 2 and 3 enter with negative weights— -0.4 per cell for the first rule, and -0.2 for the second. A cell becomes black if the weighted sum is positive, and white otherwise. Starting from random initial conditions, both rules quickly evolve to stationary states that look very much like pigmentation patterns seen in animals.

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