Geometric Similarity in Allometric Growth: A Contribution to the Problem of Scaling in the Evolution of Size

Abstract

The coefficient b of the power function y = bxa has long been misinterpreted as a measure of size-independent differences between regressions. Just the opposite is true; b is a scale factor that expresses differences in size between comparable animals of the same shape on two or more regressions of constant α. When α is invariant for two regressions, a similarity criterion s can be extracted from the two b-values (s = [b1/b2]1/(1-α)); s measures the relative difference in size at which animals on the two curves have the same shape. If this calculated difference equals the observed difference in size, then the transposition (shift of regression line without change of slope) occurred in order to maintain geometric similarity in a new size range. I present examples of geometric similarity via transposition for body shape in gulls, brain weight in felids and primates, tooth shape in canids, skull form in bovids, the evolution of Gryphaea, the growth of horses, and differences between local races of lobsters and molluses. The literature on dwarfism in humans and animals shows that proportioned change in size can have a simple genetic and developmental basis. As a mode of size change, geometric similarity may be important in macroevolution because: (1) it allows size to change rapidly by uncoupling the usual correlation of growth and development and (2) it can produce a phylogenetic increase in effective organ size when the expected correlation of that organ with body size is negatively allometric.