Chapter 8: Implications for Everyday Systems

Section 6: Growth of Plants and Animals

Leaf shapes

Leaves are usually put into categories like the ones below, with names mostly derived from Latin words for similar-looking objects.

Some classification of leaf shapes was done by Theophrastus as early as 300 BC, and classifications similar to those above were in use by the early Renaissance period. (They appear for example in the first edition of the Encyclopedia Britannica from 1768.) Leaf shapes have been widely used since antiquity as a way of identifying plants—initially particularly for medicinal purposes. But there has been very little general scientific investigation of leaf shapes, and most of what has been done has concentrated on the expansion of leaves once they are out of their buds. Already in 1724 Stephen Hales looked at the motion of grids of marks on fig leaves, and noted that growth seemed to occur more or less uniformly throughout the leaf. Similar but increasingly quantitative studies have been made ever since, and have reported a variety of non-uniformities in growth. For a long time it was believed that after leaves came out of their buds growth was due mainly to cell expansion, but in the 1980s it became clear that many cell divisions in fact occur, both on the boundary and the interior. At the earliest stages, buds that will turn into leaves start as bumps on a plant stem, with a structure that is essentially impossible to discern. Surgically modifying such buds when they are as small as 0.1 mm can have dramatic effects on final leaf shape, suggesting that at least some aspects of the shape are already determined at that point. On a single plant different leaves can have somewhat different shapes—sometimes for example those lower on a tree are smoother, while those higher are pointier. It may nevertheless be that leaves on a single plant initially have a discrete set of possible shapes, with variations in final shape arising from differences in environmental conditions during expansion. My model for leaf shapes is presumably most relevant for initial shapes.

Traditional evolutionary explanations have not had much to say about detailed questions of leaf shape; one minor claim is that the pointed tips at the ends of many tropical leaves exist to allow moisture to drip off the leaves. The fossil record suggests that leaves first arose roughly 400 million years ago, probably when collections of branches which lay in a plane became joined by webbing. Early plants such as ferns have compound leaves in which explicit branching structure is still seen. Extremely few models for shapes of individual leaves appear to have ever been proposed. In 1917 D'Arcy Thompson mentioned that leaves might have growth rates that are simple functions of angle, and drew the first of the pictures shown below.

With new tip positions as on page 400 given by {p Exp[I θ], p Exp[-I θ], q}, rough {p, q, θ} for at least some versions of some common plants include: wild carrot (Queen Anne's lace) {0.4, 0.7, 30°}, cypress {0.4, 0.7, 45°}, coralbells {.5, .4, 0°}, ivy {.5, .6, 0°}, grape {.5, .6, 15°}, sycamore {.5, .6, 15°}, mallow {.5, .6, 30°}, goosefoot {.55, .8, 30°}, willow {.55, .8, 80 °}, morning glory {.7, .8, 0°}, cucumber {.7, .8, 15 °}, ginger {.65, .6, 15 °}.

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