Mathematics of Transformations

Mathematics of Transformations


D'Arcy Thompson'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).

Scarus
Scarus in a rectangular grid, On Growth and Form

He then performed a simple transformation of the grid structure by, for example, bending the lines in a systematic way. Then by re-drawing 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).

Pomacanthus
Pomacanthus transformed from Scarus, On Growth and Form

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 co-ordinate network which is suggested by our comparison of the two entire pelves". This is illustrated below.

Pelvis
Pelvis of Stegosaurus transformed into pelvis of Camptosaurus, On Growth and Form

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!

Click and drag to draw lines on the left grid! Download as Line Drawing

Alternatively, choose a test image:

Or, upload a pre-made line drawing:

Show grid: x-range: to y-range: to
F(x,y) = (p(x,y), q(x,y)) where:
p(x,y) = x2 + xy + y2 + x + y
q(x,y) = x2 + xy + y2 + x + y

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HD/BC October 2017
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School of Mathematics and Statistics
University of St Andrews, Scotland

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Darcy/transformation.html