D'Arcy Thompson was a Scottish naturalist and polymath. Thompson held a professorial chair at St. Andrews and Dundee, Scotland, for the amazing period of 64 years, a record for tenure unlikely ever to be broken. Although he wrote more than 300 scientific articles and books, his reputation is based primarily on his efforts to reduce biological phenomena to mathematics in his magnum opus On Growth and Form (1917).¹ In this book, full of marvelous sketches of such things as Nautilus shells and honeycombs, Thompson claimed that much about animals and plants could be understood by the laws of physics, as mirrored in the structures and patterns of mathematics. His most novel idea was to show how mathematical functions could be applied to the shape of one organism to continuously transform it into other, physically similar organisms. One memorable example is the squeezing and stretching of a rectangular Cartesian grid that transforms the fish species Scarus sp. to the species Pomacanthus. Thompson used the same principle to transform skulls of baboons into those of other primates, and to show how corresponding bones like the shoulder blade are related in different species. Thompson acquired a local reputation as a mild eccentric; indeed, older inhabitants of St. Andrews still recall seeing him strolling about town with a parrot on his shoulder.

 

 

D'Arcy Thompson used mostly linear and quadratic maps to compare different life forms. Linear maps between two shapes require four coefficient variables, while quadratic maps use 10 variables. Thompson wrote in On Growth and Form, "I know that in the study of material things number, order, and position are the threefold clue to exact knowledge: and that these three, in the mathematician's hands, furnish the first outlines for a sketch of the Universe." Doubtless, if he were alive today, he would be heavily into "morphing" – the digital technique that allows a computer to do exactly the same kind of transformation of one object to another, but vastly more efficiently.

 

Reference

1. Thompson, d'Arcy W. On Growth and Form. Cambridge: Cambridge University Press, 1917. (Dover reprint edition, 1992.)