Transformations

by D'Arcy Thompson , published 1917

Excerpted from On Growth and Form, Chap. IX

Summary

Growth can be viewed as a series of transformations or deformations of the form of an organism.  D’Arcy Thompson, the great naturalist and polymath, explores these ideas in the most influential chapter of his classic text On Growth and Form.  Imagine taking a picture of a particular creature, say a species of fish, and drawing an x-y axis that encompasses the picture.  By putting the picture of the fish into a co-ordinate system, you can now begin to transform its dimensions by altering the x-y axis in the plane.  If you inclined the x-y axis by 70º without changing the representation of the fish, you might end up with a picture of a related species of fish.  By continuing to transform the fish in this manner, you might begin to discover new, undiscovered species of fish that might have once been an important evolutionary step in fish development.  These deformations might also yield clues to the evolutionary forces that shaped the growth and development of various fish species.

The process of comparing two forms, and of recognizing in one the deformation of the other, can teach us a lot about the development of new forms and the forces that shaped their growth.  In this way we can reconstruct a simple continuity underlying complex development or project new dimensions of transformation.

Comparison of Related Forms

We are apt to think of mathematical definitions as too strict and rigid for common use, but their rigour is combined with all but endless freedom.  The precise definition of an ellipse introduces us to all the ellipses in the world; the definition of a ‘conic section’ enlarges our concept, and a ‘curve of higher order’ all the more extends our range of freedom.  By means of these large limitations, by this controlled and regulated freedom, we reach through mathematical analysis to mathematical synthesis.  We discover homologies or identities which were not obvious before, and which our descriptions obscured rather than revealed: as for instance, when we learn that, however we hold our chain, or however we fire our bullet, the contour of the one or the path of the other is always mathematically homologous.

Once more, and this is the greatest gain of all, we pass quickly and easily from the mathematical concept of form in its statical aspect to form in its dynamical relations: we rise from the conception of form to an understanding of the forces which gave rise to it; and in the representation of form and in the comparison of kindred forms, we see in the one case a diagram of forces in equilibrium, and in the other case we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other…

Every natural phenomenon, however simple, is really composite, and every visible action and effect is a summation of countless subordinate actions.  Here mathematics shows her peculiar power, to combine and to generalize… Growth and Form are throughout of this composite nature; therefore the laws of mathematics are bound to underlie them, and her methods to be peculiarly fitted to interpret them… We must learn from the mathematician to eliminate and to discard; to keep the type in mind and leave the single case, with all its accidents, alone; and to find in this sacrifice of what matters little and conservation of what matters much one of the peculiar excellences of the method of mathematics.

In a very large part of morphology, 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 unanalyzed and undefined.  This process of comparison, of recognizing in one form a definite permutation or deformation of another, apart altogether from a precise and adequate understanding of the original ‘type’ or standard of comparison, lies within the immediate province of mathematics, and finds its solution in the elementary use of a certain method of the mathematician.  This method is the Method of Co-ordinates, on which is based the Theory of Transformations…

A system of transformation

Let us inscribe in a system of Cartesian co-ordinates the outline of an organism, however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull.  We may now treat this complicated figure, in general terms, as a function of x, y. 

If we submit our rectangular system to deformation on simple and recognized lines, altering, for instance, the direction of the axes, the ratio of x/y, or substituting for x and y some more complicated expressions, then we obtain a new system of co-ordinates, whose deformation from the original type the inscribed figure will precisely follow.  In other words, we obtain a new figure which represents the old figure under a more or less homologous strain, and is a function of the new co-ordinates in precisely the same way as the old figure was of the original co-ordinates x and y.

The problem is closely akin to that of the cartographer who transfers identical data to one projection or another; and whose object is to secure (if it be possible) a complete correspondence, in each small unit of area, between the one representation and the other.  The morphologist will not only seek to draw his organic forms in a new and artificial projection; but, in the converse aspect of the problem, he will enquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shown to be a transformed representation of the other.  This once demonstrated, it will be a comparatively easy task (in all probability) to postulate the direction and magnitude of the force capable of effecting the required transformation… For it is a maxim in physics that an effect ought not to be ascribed to the joint operation of many causes if few are adequate to the production of it…

It is clear, I think, that we may account for many ordinary biological processes of development or transformation of form by the existence of trammels or lines of constraint, which limit and determine the action of the expansive forces of growth that would otherwise be uniform and symmetrical.  This case has a close parallel in the operations of the glass-blower, to which we have already, more than once, referred in passing.  The glass-blower starts his operation with a tube, which he first closes at one end so as to form a hollow vesicle, within which his blast of air exercises a uniform pressure on all sides; but the spherical conformation which this uniform expansive force would naturally tend to produce is modified into all kinds of forms by the trammels or resistances set up as the workman lets one part or another of his bubble be unequally heated or cooled.  It was Oliver Wendell Holmes who first showed this curious parallel between the operations of the glass-blower and those of Nature, when she starts, as she so often does, with a simple tube…

Deforming fish

Among the fishes we discover a great variety of deformations, some of them of a very simple kind, while others are more striking and more unexpected.  A comparatively simple case, involving a simple shear, is illustrated [below].

The one represents, within Cartesian co-ordinates, a certain little oceanic fish known as Argyropelecus olfersi.  The other represents precisely the same outline, transferred to a system of oblique co-ordinates whose axes are inclined at an angle of 70º; but this is now (as far as can be seen on the scale of the drawing) a very good figure of an allied fish, assigned to a different genus, under the name of Sternoptyx diaphana.  The deformation illustrated by this case of Argyropelecus is precisely analogous to the simplest and commonest kind of deformation to which fossils are subject as the result of shearing-stresses in the solid rock…

[The first figure below] is a common, typical Diodon or porcupine-fish, and in [the accompanying figure] I have deformed its vertical co-ordinates into a system of concentric circles, and its horizontal co-ordinates into a system of curves which, approximately and provisionally, are made to resemble a system of hyperbolas.  The old outline, transferred in its integrity to the new network, appears as a manifest representation of the closely allied, but very different looking, sunfish, Orthagoriscus mola.  This is a particularly instructive case of deformation or transformation… It accounts, by one single integral transformation, for all the apparently separate and distinct external differences between the two fishes.  It leaves the parts near to the origin of the system, the whole region of the head, the opercular orifice and the pectoral fin, practically unchanged in form, size and position; and it shows a greater and greater apparent modification of size and form as we pass from the origin towards the periphery of the system.

In a word, it is sufficient to account for the new and striking contour in all its essential details, of rounded body, exaggerated dorsal and ventral fins, and truncated tail.  In like manner, and using precisely the same co-ordinate networks, it appears to me possible to show the relations, almost bone for bone, of the skeletons of the two fishes; in other words, to reconstruct the skeleton of the one from our knowledge of the skeleton of the other, under the guidance of the same correspondence as is indicated in their external configuration…

It may also be employed for drawing hypothetical structures, on the assumption that they have varied from a known form in some definite way.  And this process may be especially useful, and will be most obviously legitimate, when we apply it to the particular case of representing intermediate stages between two forms which are actually known to exist, in other words, of reconstructing the transitional stages through which the course of evolution must have successively traveled if it has brought about the change from some ancestral type to its presumed descendant…

Our simple, or simplified, illustrations carry us but a little way, and only half prepare us for much harder things.  But interesting as the whole subject is we must meanwhile leave it alone; recognizing, however, that if the difficulties of description and representation could be overcome, it is by means of such co-ordinates in space that we should at last obtain an adequate and satisfying picture of the processes of deformation and the directions of growth.

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1,530 words

Keywords:
Shape, form, deformation, transformation, force, transition, evolution, coordinates, mathematics, simulation, representation, morphology, constraints