Discrete and continuous invariance in phyllotactic tilings

Phyllotaxis refers to the arrangement of primordia (the first stage in the development of a structure such as a leaf) on plants and phyllotactic planforms refer to the shapes of the primordia in a phyllotactic arrangement. This paper focuses on invariances in phyllotactic planforms as the van Iterson parameter Gamma-a measurement of the ratio of the size of the annular generative region at the plant tip where the patterns form to primordium area-varies. We demonstrate discrete invariance in phyllotactic planforms, by which we mean a similarity in the planform under a scaling Gamma --> Gamma phi(n), where phi is the golden number and n is an integer. Continuous invariance in planforms is then motivated by examples in which the shapes of primordia are homogeneous as n varies over the real numbers. We also show how continuous invariance results from classical number-theoretical theorems on the approximation of irrational numbers (such as phi) by rational numbers. We define these notions first for the underlying phyllotactic lattice and then for primordium shapes and amplitude equations resulting from partial differential equation (PDE) models.