On Growth and Form in Nature and Art: The Projective Geometry of Plant Buds and Greek Vases

Stephen Eberhart
Bridges: Mathematical Connections in Art, Music, and Science (2000)
Pages 267–278


D' Arcy Thompson's pioneering book On Growth and Form showed how a square grid could be laid over the profile of one species of animal and then made to fit that of another related animal by a suitable deformation of space, thus allowing e.g. the shapes of missing bones to be estimated when reconstructing the fossil skeleton of an unknown species. Where Thompson had resorted to quite arbitrary spatial distortions for his examples, George Adams realized that the kind of conformal maps first discussed by Felix Klein (collineations of plane and space with invariant "path curves") would fit at least certain parts of plants and animals by conservative means, being in effect linear transformations in a non-Euclidean setting. This was then extended by Lawrence Edwards to quadratic models, showing how certain pairs of parts of a given plant or animal can be formally related in a species-true manner. I have applied these approaches to Greek amphora:, showing how their body and base and/or neck shapes are similarly related.