Rhombic Tilings and Primordia Fronts of Phyllotaxis

TL;DR

This paper introduces rhombic tilings and primordia fronts on cylinders to model plant pattern formation, analyzing their properties and dynamics through a discrete system that generalizes cylindrical lattices and explains Fibonacci petal numbers.

Contribution

It develops a mathematical framework for rhombic tilings and primordia fronts, providing partial proofs of attractor properties and linking geometry to phyllotactic patterns.

Findings

Rhombic tilings form periodic orbits of the dynamical system S.

Primordia fronts determine the dynamics and transitions in phyllotaxis.

The model explains the occurrence of Fibonacci numbers in petal arrangements.

Abstract

We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate config- urations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically in nearby systems. A key tool in understanding the geometry of tilings and the dynamics of S is the concept of pri- mordia front, which is a closed ring of tangent disks around the cylinder. We show how fronts determine the dynamics, including transitions of parastichy numbers, and might explain the Fibonacci number of petals often encountered in compositae.