Some attempts toward 3-dimensional Phyllotaxy

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.