means that the network of connections between stems necessarily has a very simple nested form. But if one looks at the actual geometrical arrangement of stems there is no longer such simplicity; indeed, despite the great simplicity of the underlying rules, considerable complexity is immediately evident even in the pictures at the bottom of the facing page.
The pictures on the next page show patterns obtained with various sequences of choices for the lengths and angles of new stems. In a few cases the patterns are quite simple; but in most cases they turn out to be highly complex—and remarkably diverse.
The pictures immediately remind one of the overall branching patterns of all sorts of plants—from algae to ferns to trees to many kinds of flowering plants. And no doubt it is from such simple rules of growth that most such overall branching patterns come.
But what about more detailed features of plants? Can they also be thought of as consequences of simple underlying rules of growth?
For many years I wondered in particular about the shapes of leaves. For among different plants there is tremendous diversity in such shapes—as illustrated in the pictures on page 403. Some plants have leaves with simple smooth boundaries that one might imagine could be described by traditional mathematical functions. Others have leaves with various configurations of sharp points. And still others have leaves with complex and seemingly somewhat random boundaries.
So given this diversity one might at first suppose that no single kind of underlying rule could be responsible for what is seen. But looking at arrays of pictures like the ones on the next page one makes a remarkable discovery: among the patterns that can be generated by simple substitution systems are ones whose outlines look extremely similar to those of a wide variety of types of leaves.
There are patterns with smooth edges that look like lily pads. There are patterns with sharp points that look like prickly leaves of various kinds. And there are patterns with intricate and seemingly somewhat random shapes that look like sycamore or grape leaves.
It has never in the past been at all clear how leaves get the shapes they do. Presumably most of the processes that are important take place while leaves are still folded up inside buds, and are not yet very solid.