Growth, Form and Structuralism

Figure: Skull transformations

D’Arcy Wentworth Thompson, Scotland 1917

D’Arcy W. Thompson, a Scottish mathematical biologist, combined his expertise on natural history with mathematics in the early 1900’s drawing new perspectives into the evolutionary thoughts of his time. He promoted structuralism to dictate species form, reacting against the Darwinistic ideas taking hold of the scientific field. In1917 Thompson published his most influential work, On Growth and Form (Figure 1), where he explained the transformation between species and gave a scientific explanation on the non-evolutionary structures of life17. He presented mathematical principles that may succeed natural selection, showing that even life structures are found in inorganic nature18 . Thompson believed that reducing the organic structures into mathematical principles would reveal that evolution goes through contingency instead of necessity in biology. This new new approach questioned the emphasis on the deterministic force of natural selection in the evolutionary processes.

Thompson’s approach to comparing and analyzing the growth of organisms through physics and mathematics differed from zoology. Zoology analyzed organic forms by comparing each organisms’ anatomy, evolution, and phylogenetics. He instead developed theories for transformation from one species turning into another fully, not progressively 19. His book contained a chapter known to be the most influential, “On the Theory of Transformations, or the Comparison of Related Forms,” which shows how the species’ differences in form are geometrically represented20. He also explored ways in which the differences can be explained in simple mathematical transformations. Instead of analyzing the structure as a whole, he isolated a couple of factors and compared them on a logarithmic scale, discovering the ratio of the growth rates in different structures. Thompson’s approach to comparing and analyzing the growth of organisms through physics and mathematics differed from zoology. Zoology analyzed organic forms by comparing each organisms’ anatomy, evolution, and phylogenetics. He instead devel- oped theories for transformation from one species turning into another fully, not progressively 21.. His book contained a chapter known to be the most influential, “On the Theory of Transformations, or the Comparison of Related Forms,” which shows how the species’ differences in form are geometrically represented 22. He also explored ways in which the differences can be explained in simple mathematical transformations. Instead of analyzing the structure as a whole, he isolated a couple of factors and compared them on a logarithmic scale, discovering the ratio of the growth rates in different structures.

Another example of the relationships he investigated were between mechanical and biological forms. As an example, he explored qualitative similarities between a Jellyfish and water droplets falling into a viscous fluid. Another correlation was the internal support in a bird’s hollow bones and the truss in engineering applications. He also related forms and mathematics through the Fibonacci sequence, exemplified in the structure of a shell (Figure 3).

To demonstrate numerical relations, he created cartesian transformations that showed the variations in form between species that were related. As Thompson showed on a human skull, a chimpanzee, and a dog, the overlaid cartesian grid consistently deformed (Figure 2)23. Continuity and differential change were key to the sheet’s deformation, as one species could transform into any other species exclusively by deformation. What he touched uponwas that the key to adaptation lay in the transformation. If the sheets were continued into a pattern, a new species would be created 24 . Because of the cartesian transformation’s unwieldiness, it hasn’t been used as often, but his method of analysis inspired other scientists. ‘Our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalysed and undefined’ 25- D’Arcy Thompson. Thompson’s theories of transformation have inspired thinkers, biologists, and mathematicians as well as architects and artists.

"And in a sense what I found strongly supports a core idea of Thompson’s: that the forms of organisms are not so much determined by evolution, as by what it’s possible for processes to produce. Thompson thought about physical processes and mathematical forms; 60-plus years later I was in a position to explore the more general space of computational processes." - Stephen Wolfram 26

Suggested readings:

Thompson, D. W., 1992. On Growth and Form. Dover reprint of 1942 2nd ed. (1st ed., 1917). ISBN 0-48667135-6

Scholtz, G., Knötel, D. & Baum, D. D’Arcy W. Thompson’s Cartesian transformations: a critical evaluation. Zoomorphology 139, 293–308, 27 June 2020

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

Arhat Abzhanov The old and new faces of morphology: the legacy of D’Arcy Thompson’s ‘theory of transformations’ and ‘laws of growth’ Development 2017 144: 4284-4297; doi: 10.1242/dev.137505

Caudwell, C and Jarron, M, 2010. D’Arcy Thompson and his Zoology Museum in Dundee. University of Dundee Museum Services.

M. Kemp, Spirals of life: D’Arcy Thompson and Theodore Cook, with Leonardo and Durer in retrospect, Physis Riv. Internaz. Storia Sci. (NS) 32 (1) (1995)

Richards, Oscar W. (1955). “D’Arcy W. Thompson’s mathematical transformation and the analysis of growth”. Annals of the New York Academy of Sciences

Smart, Steve. On growth and form 100. 31 March 2021.

Britannica, The Editors of Encyclopaedia. “Sir D’Arcy Wentworth Thompson”. Encyclopedia Britannica, 17 Jun. 2020

Figure 1 (top): Skull transformations Figure 2 (bottom): Skull transformations

Figure 3: Illustration of a sea shell.. Figure 4: Gastropoda pulmonata analysis

Figure 5 (top): An approximation of a Christaller solution applied to an area of non-uniform population distribution, Bunge 1962. Figure 6 (bottom): Illustration of a sea shell with a mathematical diagram. D’arcy Thompson, On Growth and Form.