that this is precisely why such diverse shapes of leaves are occasionally seen even in plants that otherwise appear very similar.

But while features such as the shapes of leaves typically differ greatly between different plants, there are also some seemingly quite sophisticated aspects of plants that typically remain almost exactly the same across a huge range of species.

One example is the arrangement of sequences of plant organs or other elements around a stem. In some cases successive leaves, say, will always come out on opposite sides of a stem—180° apart. But considerably more common is for leaves to come out less than 180° apart, and in most plants the angle turns out to be essentially the same, and equal to almost exactly 137.5°.

It is already remarkable that such a definite angle arises in the arrangement of leaves—or so-called phyllotaxis—of so many plants. But it turns out that this very same angle also shows up in all sorts of other features of plants, as shown in the pictures at the top of the facing page. And although the geometry is different in different cases, the presence of a fixed angle close to 137.5° always leads to remarkably regular spiral patterns.

Over the years, much has been written about such patterns, and about their mathematical properties. For it turns out that an angle between successive elements of about 137.5° is equivalent to a rotation by a number of turns equal to the so-called golden ratio (1+Sqrt[5])/2 ≃ 1.618 which arises in a wide variety of mathematical contexts—notably as the limiting ratio of Fibonacci numbers.

And no doubt in large part because of this elegant mathematical connection, it has usually come to be assumed that the 137.5° angle and the spiral patterns to which it leads must correspond to some kind of sophisticated optimization found by an elaborate process of natural selection.

But I do not believe that this is in fact the case. And instead what I strongly suspect is that the patterns are just inevitable consequences of a rather simple process of growth not unlike one that was already discussed, at least in general terms, nearly a century ago.