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flowers. But if more material is added near the edge than near the center, as in case (c), then the sheet will become wavy at the edge, much like some leaves. And if the amount of material increases sufficiently rapidly from the center to the edge, as in case (d), then the disk will be forced to become highly corrugated, somewhat like a lettuce leaf.

So what about animals? To what extent are their mechanisms of growth the same as plants? If one looks at air passages or small blood vessels in higher animals then the patterns of branching one sees look similar to those in plants. But in most of their obvious structural features animals do not typically look much like plants at all. And in fact their mechanisms of growth mostly turn out to be rather different.

As a first example, consider a horn. One might have thought that, like a stem in a plant, a horn would grow by adding material at its tip. But in fact, like nails and hair, a horn instead grows by adding material at its base. And an immediate consequence of this is that the kind of branching that one sees in plants does not normally occur in horns.

But on the other hand coiling is common. For in order to get a structure that is perfectly straight, the rate at which material is added must be exactly the same on each side of the base. And if there is any difference, one edge of the structure that is produced will always end up being longer than the other, so that coiling will inevitably result, as in the pictures below.

And as has been thought for several centuries, it turns out that a three-dimensional version of this phenomenon is essentially what leads to the elaborate coiled structures that one sees in mollusc shells. For in a typical case, the animal which lives at the open end of the shell

Captions on this page:

Idealized horns generated by progressively adding new material, with the amount of material on the upper edge of the base always being the specified percentage larger than the amount on the lower edge. These pictures can be viewed as one-dimensional analogs of those on the facing page.

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