Geometry of Growth and Form

Commentary on D'Arcy Thompson

John Milnor commented on D’Arcy Thompson’s On Growth and Form, first published in 1917, which studied how the shape of an organism changes as it grows and the way that shape changes in the course of evolution. Milnor discussed the hypothesis that these changes are roughly conformal; but found that it did not match the data. (One obvious difficulty is that a conformal transformation cannot transform the skinny bones of a small animal into the thick bones of a large one.) Yet he did find a real tendency for cross-ratios along many straight lines to be preserved under growth or under evolution. (If all cross-ratios were preserved, then the transformation would be projective, which is also not compatible with the data.) Milnor proposed three possible explanations, but was unable to decide between them: transformations that preserve cross-ratios confer some selective advantage and tend to be chosen by natural selection; the biochemical systems that regulate growth tend to yield transformations that preserve cross-ratios, even when other patterns of growth would work just as well or better; the approximately preserved cross-ratios are just numerical accidents, with no biological meaning at all.