Some attempts toward 3-dimensional Phyllotaxy

TL;DR

This paper explores various methods to extend 2D phyllotactic set constructions into higher dimensions, including Euclidean, spherical, hyperbolic, and hyperspherical spaces, introducing new approaches and examples.

Contribution

It proposes novel approaches for higher-dimensional phyllotaxy, including stacking, radial methods, Hopf fibration, and cut-and-project techniques, expanding the understanding of these structures.

Findings

Constructed 3D phyllotactic sets via stacking Euclidean sets

Analyzed radially triggered 3D phyllotactic solutions

Generated a 4D example using product sets and cut-and-project

Abstract

This paper investigates several distinct attempts to generalize in higher dimension the standard 2-dimensional phyllotaxy set construction. We first recall known contructions for these sets on $2D$ manifolds of constant curvature (the Euclidean plane $\mathbb{R}^2$, the sphere $\mathbb{S}^2$ and the hyperbolic plane $\mathbb{H}^2$). We then propose a first attempt to get a $3D$ phyllotactic set by piling up suitably shifted Euclidean $2D$ phyllotactic sets. A different, radially triggered, solution is then analyzed. An interesting phyllotactic set on the hypersphere $\mathbb{S}^3$ is then generated using a Hopf fibration approach. Finally,a simple 4-dimensional example is presented, generated as a simple product of two 2-dimensional planar sets. A $3D$ phyllotaxy candidate is then derived by applying a "Cut and Project" algorithm.