Phyllotaxis: a non conventional crystalline solution to packing efficiency in situations with radial symmetry

TL;DR

This paper explores the geometric and topological properties of phyllotactic arrangements of disks within circular domains, highlighting their unique symmetry and defect structures compared to traditional crystallography.

Contribution

It introduces a novel analysis of phyllotactic patterns using Voronoi and Delaunay structures, emphasizing their unique inflation-deflation symmetry due to circular constraints.

Findings

Phyllotactic arrangements exhibit a distinct inflation-deflation symmetry.

The organization of disks can be characterized by Voronoi cells and Delaunay triangulation.

Defects in the pattern relate to concepts from condensed matter physics.

Abstract

Phyllotaxis, the search for the most homogeneous and dense organizations of small disks inside a large circular domain, was first developed to analyze arrangements of leaves or florets in plants. Then it has become an object of study not only in botany, but also in mathematics, computer simulations and physics. Although the mathematical solution is now well known, an algorithm setting out the centers of the small disks on a Fermat spiral, the very nature of this organization and its properties of symmetry remain to be examined. The purpose of this paper is to describe a phyllotactic organization of points through its Voronoi cells and Delaunay triangulation and to refer to the concept of defects developed in condensed matter physics. The topological constraint of circular symmetry introduces an original inflation-deflation symmetry taking the place of the translational and rotational symmetries of classical crystallography.