vast majority of cases the occurrence of complexity—say in the shapes of leaves—is in essence just a side effect of the particular rules of growth that happen to result from the underlying properties of the plant.

The pictures on the next page show the array of possible forms that can be produced by rules in which each stem splits into exactly two new stems at each step. The vertical black line on the left-hand side of the page represents in effect the original stem at each step, and the pictures are arranged so that the one which appears at a given position on the page shows the pattern that is generated when the tip of the right-hand new stem goes to that position relative to the original stem shown on the left.

In some cases the patterns obtained are fairly simple. But even in these cases the pictures show that comparatively small changes in underlying rules can lead to much more complex patterns. And so if in the course of biological evolution gradual changes occur in the rules, it is almost inevitable that complex patterns will sometimes be seen.

But just how suddenly can the patterns change? To get some idea of this one can construct a kind of limit of the array on the next page in which the total number of pictures is in effect infinite, but only a specific infinitesimal region of each picture is shown. Page 407 gives results for four choices of the position of this region relative to the original stem. And instead of just displaying black or white depending on whether any part of the pattern lies in the region, the picture uses gray levels to indicate how close it comes.

The areas of solid black thus correspond to ranges of parameters in the underlying rule for which the patterns obtained always reach a particular position. But what we see is that at the edges of these areas there are often intricate structures with an essentially nested form. And the presence of such structures implies that at least with some ranges of parameters, even very small changes in underlying rules can lead to large changes in certain aspects of the patterns that are produced.

So what this suggests is that it is almost inevitable that features such as the shapes of leaves can sometimes change greatly even when the underlying properties of plants change only slightly. And I suspect

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