Growth-gradients and the Development of Animal Form

Abstract

D'ARCY THOMPSON in his “Growth and Form”, Chapter xi., deals lucidly with the properties of logarithmic spirals, and the reasons for their frequent occurrence in organisms. He points out that for them to arise, (1) parts of the growing edge must be growing at different rates, the growth-rates of any two points on the edge preserving a constant ratio of growth-rates for so long as a regular logarithmic spiral is produced; (2) the growth-rate must fall off more or less steadily from one end of the growing surface to the other; (3) the products of growth must be laid on as so much dead matter, or at least matter incapable of further growth. In his own words (p. 500) the logarithmic-spiral form of an organic structure can be explained “if we presuppose that the increments of growth take place at a constant angle to the growing surface, but more rapidly at the ‘outer edge’ than at [the ‘inner edge’], and that this difference of velocity maintains a constant ratio. Let us also assume that the whole structure is rigid, the new accretions solidifying as soon as they are laid on”.