Parametrizations of [biological] growth

The idea that different objects—say different human bodies or faces—can be related by changing a small number of geometrical parameters was used by artists such as Albrecht Dürer in the 1500s, and may have been known to architects and others in antiquity. (In modern times this idea is associated for example with the notion that just a few measurements are sufficient to specify the fitting of clothes.) D'Arcy Thompson in 1917 suggested that shapes in many different species could also be related in this way. In the case of shells and horns he gave a fairly precise analysis, as discussed above. But he also drew various pictures of fishes and skulls, and argued that they were related by deformations of coordinates. Largely from this grew the field of morphometrics, in which the relative positions of features such as eyes or tips of fins are compared in different species. And although statistical significance is reduced by considering only discrete features, some evidence has emerged that different species do indeed have shapes related by changes in fairly small numbers of geometrical parameters. Such changes could be accounted for by changes in growth rates, but it is noteworthy that my results above on branching and folding make it clear that in general changes in growth rates can have much more dramatic effects.

As emphasized by D'Arcy Thompson, even a single organism will change shape if its parts grow at different rates. And in the 1930s and 1940s it became popular to study differential growth, typically under the name of allometry. Exponential growth was usually assumed, and there was much discussion about the details and correctness of this. Practical applications were made to farm animals, and later to changes in facial bone structure during childhood. But despite some work in the past twenty years using models based on fluids, solid mechanics and networks of rigid elements, much about differential growth remains unclear. (A better approach may be one similar to general relativity.)