D’Arcy’s method of ‘deformation’ worked as follows. Firstly he drew the form he was interested in over a rectangular grid. The form could be the outline of a whole animal or a section of one, such as a leg bone or skull. In the illustrative example below the form used is that of a Parrotfish (Scarus).
He then performed a simple transformation of the grid structure by, for example, bending the lines in a systematic way. Then by redrawing the form point by point onto this new grid the shape of a different known structure could be found. For example, if we deform the rectangular grid from before by curving the lines outwards from the centre we recognise the new form as that of an Angelfish (Pomacanthus).
The case of the Angelfish is particularly interesting, as noted D’Arcy, “upon the body of the Pomacanthus there are striking color bands, which correspond in direction very closely to the lines of our new coordinates.”
D’Arcy believed that in reality these transformations were due to physical forces exerted on the species. He placed less emphasis on physical matter and more on the forces that shape it.
Another way in which D’Arcy used his method was to predict what bones of dinosaurs may have looked like. Thanks to the collection of fossils, the shape of the pelvis bone of a Stegosaurus is known. However part of the fossil pelvis of a Camptosaurus is missing. D’Arcy suggested that taking the known structure of the Stegosaurus pelvis and transforming it to known part of the Camptosaurus pelvis is sufficient to know what this missing part of bone looked like. In his own words, he “completed this missing part of the bone in harmony with the general coordinate network which is suggested by our comparison of the two entire pelves”. This is illustrated below.
D’Arcy believed that this method had real physical implications and that the forces which were responsible for shaping an animal could be easily found from the transformation. He believed that these “external forces” were at least as important as natural selection in the process of evolution.
Try D’Arcy’s method for yourself below. Thanks to Rhiannon Michelmore for developing it!
Show grid:  xrange: to yrange: to  
F(x,y) = (p(x,y), q(x,y)) where:  
p(x,y) = x^{2} + xy + y^{2} + x + y  
q(x,y) = x^{2} + xy + y^{2} + x + y 
Introduction

Overview & D'Arcy's Life

On Growth and Form

Heilmann & Shufeldt

Maths of Transformations

Correspondence

D'Arcy and Mathematics

Coordinate Transformations

Logarithmic Spirals

Forms of Cells

Forms and Mechanical Efficiency

Shrinkage

Wartime and D'Arcy

The Leg as a Pendulum

Recreational Maths

Fibonacci Sequence

CellAggregates

Claxton Fidler

Eric Harold Neville

John Marshall

Alfred North Whitehead

Charles Robert Darling

Peter Guthrie Tait

William Peddie

Geoffrey Thomas Bennett

Dorothy Wrinch



The support of The Strathmartine Trust towards this website is gratefully acknowledged
HD & BC © July 2017 Copyright information  School of Mathematics and Statistics
University of St Andrews, Scotland 
The URL of this page is:
http://www.mcs.stand.ac.uk/~dat/