The Study of Shape Transformation after D’Arcy Thompson

Abstract

The boundary represents the shape, but it is not the boundary which grows. A rubber band released from tension happens to return to its elongate resting shape. In the course of release, the band is active, having its own preferred curvatures. There may exist biological situations in which an analogous determination occurs, but in the vast majority of shape changes the boundary is passive. It is pushed out or pulled in, pressured hither and thither by the changes everywhere inside of the shape. The inside, in growing, carries the boundary along with it, embedded in whatever distortions are wreaked about it. Intuitively we know what the distortions have to be like: smooth correspondences with no tears or folds, no suddenly disappearing or appearing tissues.