Phyllotaxis: a framework for foam topological evolution

TL;DR

This paper presents a geometric framework for understanding foam topological evolution in phyllotaxis, revealing invariance properties and stress response mechanisms through Voronoi tilings of spiral lattices.

Contribution

It introduces a novel approach linking phyllotactic arrangements to foam topology, demonstrating invariance under growth and topological transformations.

Findings

Sequence and organization are invariant under initial point position.

Foam responds to shear stress via grain boundaries with specific cell shapes.

Topological transformations are characterized by elementary $T1$ moves.

Abstract

Phyllotaxis describes the arrangement of florets, scales or leaves in composite flowers or plants (daisy, aster, sunflower, pinecone, pineapple). As a structure, it is a geometrical foam, the most homogeneous and densest covering of a large disk by Voronoi cells (the florets), constructed by a simple algorithm: Points placed regularly on a generative spiral constitute a spiral lattice, and phyllotaxis is the tiling by the Voronoi cells of the spiral lattice. Locally, neighboring cells are organized as three whorls or parastichies, labeled with successive Fibonacci numbers. The structure is encoded as the sequence of the shapes (number of sides) of the successive Voronoi cells on the generative spiral. We show that sequence and organization are independent of the position of the initial point on the generative spiral, that is invariant under disappearance ($T2$) of the first Voronoi cell or, conversely, under creation of a first cell, that is under growth. This independence shows how a foam is able to respond to a shear stress, notably through grain boundaries that are layers of square cells slightly truncated into heptagons, pentagons and hexagons, meeting at four-corner vertices, critical points of $T1$ elementary topological transformations.