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In retrospect: On Growth and Form

Philip Ball celebrates a classic work on the mathematics that shape living structures, from antlers to cells.

On Growth and Form

  • D'Arcy Wentworth Thompson
Cambridge University Press: 1917.
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Like Newton's Principia, D'Arcy Wentworth Thompson's On Growth and Form is a book more often name-checked than read. Both are hefty — Thompson's revised edition in 1942 weighed in at more than 1,000 pages, to the alarm of Cambridge University Press.

And both books stand apart from their age. Each contains ideas ahead of its time, yet seems rooted in earlier traditions. First published in 1917, with the modern synthesis of neo-Darwinian biology two or three decades away and genes still a nascent concept, On Growth and Form looked in some ways archaic by the time the second edition appeared — yet it continues to inspire.


Thompson's agenda is captured in the book's epigraph from statistician Karl Pearson (first published in this journal in 1901): “I believe the day must come when the biologist will — without being a mathematician — not hesitate to use mathematical analysis when he requires it.” Thompson presents mathematical principles as a shaping agency that may supersede natural selection, showing how the structures of the living world often echo those in inorganic nature.

Thompson's route to this view was, like the man himself, idiosyncratic. The son of a Cambridge classicist, he went to Edinburgh to study medicine but switched to zoology at Cambridge — the same trajectory as Darwin. There he supplemented his income by teaching Greek. He returned to Scotland as (in effect) a marine biologist at the University of Dundee before moving a few miles down the North Sea coast to the University of St Andrews, where he occupied the Chair of Natural History.

His frustration at the 'Just So' explanations of morphology offered by Darwinians burst out in a 1894 paper presented at the British Association meeting; he argued that physical forces, not heredity, may govern biological form.

On Growth and Form elaborates at length on this theme. “In general no organic forms exist save such as are in conformity with physical and mathematical laws,” he wrote. Thompson's demonstration of this claim takes him through a formidable range of topics. To name a few: the mathematical laws that relate growth, flight and locomotion to mass and size (a topic currently experiencing a renaissance); the shapes of cells, bubbles and soap films; geometrical compartmentalization and honeycombs; corals; banded minerals; the intricate shells of molluscs and of the minuscule protozoan radiolarians; antlers and horns; plant shapes; bone microstructure; skeletal mechanics; and the morphological comparison of species.

The book's central motif is the logarithmic spiral, which appears on the plaque commemorating Thompson's former residence in St Andrews. He saw it first in foraminifera, and again in seashells, horns and claws, insect flight paths and the arrangement of leaves in some plants. This, to Thompson, was evidence of the universality of form and the reduction of diverse phenomena to a few mathematical governing principles.

How much influence did On Growth and Form have? Evolutionary and developmental biologists often genuflect to Thompson's breadth and imagination while remaining sceptical that he told us much of lasting value.

Thompson was reacting against the Darwinism of his age, whereby, in its first flush of enthusiasm, it seemed adequate to account for every feature with a plea to adaptation. Thompson's insistence that biological form had to make sense in engineering terms was a necessary reminder. But it did not challenge the idea that natural selection was evolution's scalpel — it merely imposed constraints on the forms that might emerge. When he sent the manuscript of On Growth and Form to his publisher, Thompson wrote: “where it undoubtedly runs counter to conventional Darwinism, I do not rub this in, but leave the reader to draw the obvious moral for himself.”

The logarithmic spiral was, to Thompson, evidence of the universality of form.

Thompson believed that evolution could sometimes advance in a leap rather than a shuffle — still a hotly discussed issue. And the debate that Thompson tried to initiate about contingency versus necessity in biological form has not yet really been engaged. There are still biologists who believe that almost every feature of an organism must be adaptive. There are still open questions about how deterministic the course of evolution is.

This is one reason to keep On Growth and Form in the canon. Another is the modern appreciation of self-organization as a means of developing complex form and pattern from simple physical rules — evident, for example, in fractal patterns, animal swarming and perhaps even the subtle regularities of human society. Here Thompson is not as easy to enlist as a prophet as one might expect.

Many of the systems he looked at, such as the striped markings of animals, and the formation of polygonal crack networks, are now recognized as paradigmatic examples of spontaneous self-organization in complex systems. But Thompson often gives such things only a glancing mention while either confessing that he has no real explanation or assuming that it must be a simple one. He says of the chemical precipitation patterns called Liesegang rings, “For a discussion of the raison d'être of this phenomenon, the student will consult the textbooks of physical and colloid chemistry.” The student would have found little there in 1917, and some aspects of this chemistry are still being clarified.

The tradition from which Thompson's great work emerged was rather different from the early interest in complex systems by the likes of Henri Poincaré: it was indebted to the biophysics and biomechanics of anatomists such as Wilhelm His and Wilhelm Roux. It is probably this strand that ties Thompson most securely to the present, for much of cell biology now centres on how the mechanics of cell structures determine the fates, forms and functions of tissues. This undervalued aspect of biophysics is becoming more integrated into the rest of molecular biology, as we come to realize how much mesoscale mechanics modulates gene and protein behaviour.

But much of the admiration for On Growth and Form expressed by fans such as Peter Medawar and Stephen Jay Gould stems from a more general consideration: Thompson's breadth of scholarship, coupled to the elegance of his writing. He was a classicist as well as a scientist (the many Greek and Latin quotes in On Growth and Form pass untranslated), and there is something of the antiquarian in his persona. At a time when science was succumbing to the specialization that has now become something of a liability, Thompson showed the value of synoptic thinkers who are prepared to risk being quite wrong here and there for the sake of an inspirational vision. Like the modern mavericks James Lovelock, Benoit Mandelbrot, Gould and Stephen Wolfram, he presented his ideas in an extended, almost incontinent, gush, rather than with a conventional succession of closely argued papers.

Such figures excite strong responses. They are sometimes exasperating. But we must make sure that they do not — in an age of Big Science, citation-counting, tenure battles and funding crises — become extinct.

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Correspondence to Philip Ball.

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The D'Arcy Thompson Zoology Museum at the University of Dundee, UK, is currently displaying the first works acquired through a grant from the Art Fund to build a collection of art inspired by Thompson's work. A special issue of Interdisciplinary Science Reviews on D'Arcy Thompson and his legacy will be published in March 2013.

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Ball, P. In retrospect: On Growth and Form. Nature 494, 32–33 (2013).

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