On Growth and Form

by D'Arcy Wentworth Thompson

Fig. 320. Head of Ovis Ammon, shewing St Venant’s curves.

But there is yet another and a very remarkable phenomenon which we may discern in the growth of a horn, when it takes the form of a curve of double curvature, namely, an effect of torsional strain; and this it is which gives rise to the sinuous “lines of growth,” or sinuous boundaries of the separate horny rings, of which we have already spoken. It is not at first sight obvious that a mechanical strain of torsion is necessarily involved in the growth of the horn. In our experimental illustration (p. [618]), we built up a twisted coil of separate elements, and no torsional strain attended the development of the system. So would it be if the horny sheath grew by successive annular increments, free save for their relation to one another, and having no attachment to the solid core within. But as a matter of fact there is {622} such an attachment, by subcutaneous connective tissue, to the bony core; and accordingly a torsional strain will be set up in the growing horny sheath, again provided that the forces of growth therein be directed more or less obliquely to the axis of the core; for a “couple” is thus introduced, giving rise to a strain which the sheath would not experience were it free (so to speak) to slip along, impelled only by the pressure of its own growth from below. And furthermore, the successive small increments of the growing horn (that is to say, of the horny sheath) are not instantaneously converted from living to solid and rigid substance; but there is an intermediate stage, probably long-continued, during which the new-formed horny substance in the neighbourhood of the zone of active growth is still plastic and capable of deformation.

Now we know, from the celebrated experiments of St Venant[563], that in the torsion of an elastic body, other than a cylinder of circular section, a very remarkable state of strain is introduced. If the body be thus cylindrical (whether solid or hollow), then a twist leaves each circular section unchanged, in dimensions and in figure. But in all other cases, such as an elliptic rod or a prism of any particular sectional form, forces are introduced which act parallel to the axis of the structure, and which warp each section into a complex anticlastic surface. Thus in the case of a triangular and equilateral prism, such as is shewn in section in Fig. [321], if the part of the rod represented in the section be twisted by a force acting in the direction of the arrow, then the originally plane section will be warped as indicated in the diagram:—where the full contour-lines represent elevation above, and the dotted lines represent depression below, the original level. On the external surface of the prism, then, contour-lines which were originally parallel and horizontal, will be found warped into sinuous curves, such that, on each of the three faces, the curve will be convex upwards on one half, and concave upwards on the other half of the face. The ram’s horn, and still better that of Ovis Ammon, is comparable to such a prism, save that in section it is not quite equilateral, and that its three faces are not plane. The warping is therefore not precisely identical on the three faces {623} of the horn; but, in the general distribution of the curves, it is in complete accordance with theory. Similar anticlastic curves are well seen in many antelopes; but they are conspicuous by their absence in the cylindrical horns of oxen.

The better to illustrate this phenomenon, the nature of which is indeed obvious enough from a superficial examination of the horn, I made a plaster cast of one of the horny rings in a horn of Ovis Ammon, so as to get an accurate pattern of its sinuous edge: and then, filling the mould up with wet clay, I modelled an anticlastic surface, such as to correspond as nearly as possible with the sinuous outline[564]. Finally, after making a plaster cast of this sectional surface, I drew its contour-lines (as shewn in Fig. [322]), with the help of a simple form of spherometer. It will be seen that in great part this diagram is precisely

Fig. 321.

Fig. 322.

similar to St Venant’s diagram of the cross-section of a twisted triangular prism; and this is especially the case in the neighbourhood of the sharp angle of our prismatic section. That in parts the diagram is somewhat asymmetrical is not to be wondered at: and (apart from inaccuracies due to the somewhat rough means by which it was made) this asymmetry can be sufficiently accounted for by anisotropy of the material, by inequalities in thickness of different parts of the horny sheath, and especially (I think) by unequal distributions of rigidity due to the presence of the smaller corrugations of the {624} horn. It is apparently on account of these minor corrugations that, in such horns as the Highland ram’s, where they are strongly marked, the main St Venant effect is not nearly so well shewn as in the smoother horns such as those of O. Ammon and its immediate congeners[565].

A further Note upon Torsion.

The phenomenon of torsion, to which we have been thus introduced, opens up many wide questions in connection with form. Some of the associated phenomena are admirably illustrated in the case of climbing plants; but we can only deal with these still more briefly and parenthetically.