Plant patterns

Abstract

The hexagons on a pineapple contrast with the ribs observed, for example, on pumpkins or saguaro cacti. This dissertation demonstrates how these various configurations, and also the related patterns of phyllotaxis (the arrangement of leaves into whorls or spirals) can be understood as the energy-minimizing buckling pattern of a compressed shell (the plant's tunica) on an elastic foundation. The key new idea is that the elastic energy is minimized by special triads or sequences of triads of periodic deformations whose local wavevectors add to zero. Although triad configurations arise from a variety of microscopic mechanisms in natural and laboratory systems, we show that the particular choices of wavevectors that are observed on plants arise in a nontrivial way from properties specific to a mechanical model. Furthermore, the theory predicts correlations between types of phyllotaxis and shapes of plant surface configurations and suggests experiments that can further test the mechanical theory of plant pattern formation. The dissertation concludes with a derivation of Cross-Newell equations governing pattern formation far from onset in nonisotropic systems and in systems with hexagonal planforms.