On Growth and Form


On Growth and Form is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson. The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
The book covers many topics including the effects of scale on the shape of animals and plants, large ones necessarily being relatively thick in shape; the effects of surface tension in shaping soap films and similar structures such as cells; the logarithmic spiral as seen in mollusc shells and ruminant horns; the arrangement of leaves and other plant parts ; and Thompson's own method of transformations, showing the changes in shape of animal skulls and other structures on a Cartesian grid.
The work is widely admired by biologists, anthropologists and architects among others, but less often read than cited. Peter Medawar explains this as being because it clearly pioneered the use of mathematics in biology, and helped to defeat mystical ideas of vitalism; but that the book is weakened by Thompson's failure to understand the role of evolution and evolutionary history in shaping living structures. Philip Ball and Michael Ruse, on the other hand, suspect that while Thompson argued for physical mechanisms, his rejection of natural selection bordered on vitalism.

Overview

's most famous work, On Growth and Form was written in Dundee, mostly in 1915, but publication was put off until 1917 because of the delays of wartime and Thompson's many late alterations to the text. The central theme of the book is that biologists of its author's day overemphasized evolution as the fundamental determinant of the form and structure of living organisms, and underemphasized the roles of physical laws and mechanics. At a time when vitalism was still being considered as a biological theory, he advocated structuralism as an alternative to natural selection in governing the form of species, with the smallest hint of vitalism as the unseen driving force.
Thompson had previously criticized Darwinism in his paper Some Difficulties of Darwinism. On Growth and Form explained in detail why he believed Darwinism to be an inadequate explanation for the origin of new species. He did not reject natural selection, but regarded it as secondary to physical influences on biological form.
Using a mass of examples, Thompson pointed out correlations between biological forms and mechanical phenomena. He showed the similarity in the forms of jellyfish and the forms of drops of liquid falling into viscous fluid, and between the internal supporting structures in the hollow bones of birds and well-known engineering truss designs. He described phyllotaxis and its relationship to the Fibonacci sequence.
Perhaps the most famous part of the book is Chapter 17, "The Comparison of Related Forms," where Thompson explored the degree to which differences in the forms of related animals could be described, in work inspired by the German engraver Albrecht Dürer, by mathematical transformations.
The book is descriptive rather than experimental science: Thompson did not articulate his insights in the form of hypotheses that can be tested. He was aware of this, saying that "This book of mine has little need of preface, for indeed it is 'all preface' from beginning to end."

Editions

The first edition appeared in 1917 in a single volume of 793 pages published by Cambridge University Press. A second edition, enlarged to 1116 pages, was published in two volumes in 1942. Thompson wrote in the preface to the 1942 edition that he had written "this book in wartime, and its revision has employed me during another war. It gave me solace and occupation, when service was debarred me by my years. Few are left of the friends who helped me write it." The unabridged edition is no longer in print in English, but an edition of 346 pages was abridged by John Tyler Bonner, and is widely published under the same title. The book, often in the abridged edition, has been reprinted more than 40 times, and has been translated into Chinese, French, German, Greek, Italian, and Spanish.

Contents

The contents of the chapters in the first edition are summarized below. All but Chapter 11 have the same titles in the second edition, but many are longer, as indicated by the page numbering of the start of each chapter. Bonner's abridgment shortened all the chapters, and removed some completely, again as indicated at the start of each chapter's entry below.

1. Introductory

2. On Magnitude

3. The Rate of Growth

4. On the Internal Form and Structure of the Cell

5. The Forms of Cells

6. A Note on Adsorption

7. The Forms of Tissues, or Cell-aggregates

8. The same (continued)

9. On Concretions, Spicules, and Spicular Skeletons

10. A Parenthetic Note on Geodetics

11. The Logarithmic Spiral 'The Equiangular Spiral' in 2nd Ed.

12. The Spiral Shells of the Foraminifera

13. The Shapes of Horns, and of Teeth or Tusks: with A Note on Torsion

14. On Leaf-arrangement, or Phyllotaxis

The numbers that result from such spiral arrangements are the Fibonacci sequence of ratios 1/2, 2/3, 3.5... converging on 0.61803..., the golden ratio which is

15. On the Shapes of Eggs, and of certain other Hollow Structures

16. On Form and Mechanical Efficiency

17. On the Theory of Transformations, or the Comparison of Related Forms

.

Epilogue

Reception

Modern

"J. P. McM", reviewing the book in Science in 1917, wrote that "the book is one of the strongest documents in support of the mechanistic view of life that has yet been put forth", contrasting this with "vitalism". The reviewer was interested in the "discussion of the physical factors determining the size of organisms, especially interesting being the consideration of the conditions which may determine the minimum size".
J. W. Buchanan, reviewing the second edition in Physiological Zoology in 1943, described it as "an imposing extension of his earlier attempt to formulate a geometry of Growth and Form" and "beautifully written", but warned that "the reading will not be easy" and that "A vast store of literature has here been assembled and assimilated". Buchanan summarizes the book, and notes that Chapter 17 "seems to the reviewer to contain the essence of the long and more or less leisurely thesis... The chapter is devoted to comparison of related forms, largely by the method of co-ordinates. Fundamental differences in these forms are thus revealed", and Buchanan concludes that the large "gaps" indicate that Darwin's endless series of continuous variations is not substantiated. But he does have some criticisms: Thompson should have referenced the effects of hormones on growth; and the relation of molecular configuration and form; genetics is barely mentioned, and experimental embryology and regeneration are overlooked. The mathematics used consists of statistics and geometry, while thermodynamics is "largely absent".
Edmund Mayer, reviewing the second edition in The Anatomical Record in 1943, noted that the "scope of the book and the general approach to the problems dealt with have remained unchanged, but considerable additions have been made and large parts have been recast". He was impressed at the extent to which Thompson had kept up with developments in many sciences, though he thought the mentions of quantum theory and Heisenberg uncertainty unwise.
George C. Williams, reviewing the 1942 edition and Bonner's abridged edition for the Quarterly Review of Biology, writes that the book is "a work widely praised, but seldom used. It contains neither original insights
that have formed a basis for later advances nor instructive fallacies that have stimulated fruitful attack. This seeming paradox is brilliantly discussed by P. B. Medawar Pluto's Republic." Williams then attempts a "gross simplification" of Medawar's evaluation:

Contemporary

The architects Philip Beesley and Sarah Bonnemaison write that Thompson's book at once became a classic "for its exploration of natural geometries in the dynamics of growth and physical processes." They note the "extraordinary optimism" in the book, its vision of the world as "a symphony of harmonious forces", and its huge range, including:
Beesley and Bonnemaison observe that Thompson saw form "as a product of dynamic forces.. shaped by flows of energy and stages of growth." They praise his "eloquent writing and exquisite illustrations" which have provided inspiration for artists and architects as well as scientists.
Computer-scientist and physicist, Stephen Wolfram, writes:
The statistician Cosma Shalizi writes that the book "has haunted all discussion of these matters ever since."
Shalizi states that Thompson's goal is to show that biology follows inevitably from physics, and to a degree also from chemistry. He argues that when Thompson says "the form of an object is a 'diagram of forces,'", Thompson means that we can infer from an object the physical forces that act upon it. Shalizi calls Thompson's account of the physics of morphogenesis
Shalizi notes Thompson's simplicity, explaining the processes of life "using little that a second-year physics undergrad wouldn't know. ". He notes that Thompson deliberately avoided invoking natural selection as an explanation, and left history, whether of species or of an individual's life, out of his account. He quotes Thompson's "A snow-crystal is the same today as when the first snows fell": adding "so, too, the basic forces acting upon organisms", and comments that we have forgotten other early twentieth century scientists who scorned evolution. In contrast, he argues,
The anthropologist Barry Bogin writes that Thompson's book
Bogin observes that Thompson originated the use of transformational grids to measure growth in two dimensions, but that without modern computers the method was tedious to apply and was not often used. Even so, the book stimulated and lent intellectual validity to the new field of growth and development research.
Peter Coates recalls that
Coates argues however that the book goes far beyond expressing knowledge elegantly and influentially, in a form "that can be read for pleasure by scientists and nonscientists"; it is in his view
The science writer Philip Ball observes that
Ball quotes the 2nd Edition's epigraph by the statistician Karl Pearson: "I believe the day must come when the biologist will—without being a mathematician—not hesitate to use mathematical analysis when he requires it." Ball argues that 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", and notes his "frustration at the 'Just So' explanations of morphology offered by Darwinians." Instead, Ball argues, Thompson elaborates on how not heredity but physical forces govern biological form. Ball suggests that "The book's central motif is the logarithmic spiral", evidence in Thompson's eyes of the universality of form and the reduction of many phenomena to a few principles of mathematics.
The philosopher of biology Michael Ruse wrote that Thompson "had little time for natural selection." Instead, Thompson emphasised "the formal aspects of organisms", trying to make a case for self-organization through normal physical and chemical processes. Ruse notes that, following Aristotle, Thompson used as an example the morphology of jellyfish, which he explained entirely mechanically with the physics of a heavy liquid falling through a lighter liquid, avoiding natural selection as an explanation. Ruse is not sure whether Thompson believed he was actually breaking with "mechanism", in other words adopting a vitalist view of the world. In Ruse's opinion, Thompson can be interpreted as arguing that "we can have completely mechanical explanations of the living world" – with the important proviso that Thompson apparently felt there was no need for natural selection. Ruse at once adds that "people like Darwin and Dawkins undoubtedly would disagree"; they would insist that

Influence

For his revised On Growth and Form, Thompson was awarded the Daniel Giraud Elliot Medal from the United States National Academy of Sciences in 1942.
On Growth and Form has inspired thinkers including the biologists Julian Huxley and Conrad Hal Waddington, the mathematician Alan Turing and the anthropologist Claude Lévi-Strauss. The book has powerfully influenced architecture and has long been a set text on architecture courses.
On Growth and Form has inspired artists including Richard Hamilton, Eduardo Paolozzi, and Ben Nicholson. In 2011 the University of Dundee was awarded a £100,000 grant by The Art Fund to build a collection of art inspired by his ideas and collections, much of which is displayed in the D'Arcy Thompson Zoology Museum in Dundee.
To celebrate the centenary of On Growth and Form numerous events are being staged around the world, including New York, Amsterdam, Singapore, London, Edinburgh, St Andrews and in Dundee where the book was written. The On Growth and Form 100 website was set up in late 2016 to map all of this activity.