Phyllotaxis on surfaces of constant Gaussian curvature

TL;DR

This paper explores how the distribution patterns of discs arranged by phyllotaxis change on surfaces with constant Gaussian curvature, revealing effects on homogeneity, isotropy, and self-similarity.

Contribution

It extends the study of phyllotactic arrangements from flat planes to curved surfaces, analyzing the impact of curvature on structural invariants and grain boundary properties.

Findings

Quasicrystalline sequences of grain boundaries are invariant to curvature.

Self-similarity is lost on curved surfaces, especially hyperbolic and spherical geometries.

Local order within grains remains consistent except near the sphere's equator.

Abstract

A close packed organization with circular symmetry of a large number of small discs on a plane is obtained when the centres of the discs are distributed according to the algorithm of phyllotaxis. We study here the distributions obtained on surfaces of constant Gaussian curvatures, positive for the sphere or negative for the hyperbolic plane. We examine how the properties of homogeneity, isotropy and self-similarity typical of the distribution on the plane, and resulting from the presence of circular grain boundaries with quasicrystalline sequences, are affected by the curvature of the bearing surface. The quasicrystalline sequences of the grain boundaries appear indeed to be structural invariants, but the widths of the grains they separate vary differently with the curvature of the surface. The self similarity of the whole organization observed on the plane is therefore lost on the hyperbolic plane and the sphere. The evolutions of the local order within the grains show no differences except on the equatorial belt of the sphere where the isotropy is decreased owing to the symmetry of this finite surface around its equator.