Some attempts toward 3-dimensional Phyllotaxy

TL;DR

The work tackles extending deterministic 2D phyllotaxy to 3D by exploring several distinct constructions, including vertical stacking of 2D layers, radial reordering from spherical templates, Hopf-fibration-based arrangements on $\\mathbb{S}^3$, and a Cut-and-Project approach from 4D. It provides explicit formulae for generating 3D phyllotactic sets, discusses their geometric properties, and demonstrates both deterministic and numerical methods to assemble homogeneous 3D patterns. The study reveals shell-like structures and potential dislocations that arise in 3D, and proposes methods to recover 3D patterns from 4D lattices, offering a rich framework for further exploration of higher-dimensional phyllotaxy and its packing characteristics. Overall, it lays foundational pathways for deterministic 3D phyllotaxy with implications for sphere packings, quasicrystal-inspired arrangements, and geometric organization in higher dimensions.

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.

Figures (10)

• Figure 1: Phyllotaxy on $\mathbb{R}^2$, with disks centred on the generative spiral.
• Figure 2: Phyllotaxy $P(\mathbb{S}^2,s,F_{11})$ on the sphere $\mathbb{S}^2$, with $F_{11}=89$ sites represented by black small spheres, located on the underlying generative spiral, and surrounded by larger spheres whose colour goes from red (north pole) to blue (south pole) as $s$ increases.
• Figure 3: Phyllotaxy on the hyperbolic plane $\mathbb{H}^2$ in the Poincaré disk representation. This set contains 1000 sites with a factor $a=1/50$. The portion of the Poincar/'e unit disk represented here is a small disk (of radius much smaller than unity).
• Figure 4: Five sets $P(\mathbb{R}^2,s+j r)$ which nicely fill in the interstices created by each other. The set $j=0$ is represented by larger red disks, and $j=1,2,3,4$ are represented by smaller disks of respective colours green, blue, magenta and orange.
• Figure 5: A 3-dimensional phyllotaxy set $P(\mathbb{R}^3,s+ j r)$ We use the same colour choices as in Fig \ref{['fig:phylloR2R3']}, with repeated colours corresponding to almost (but not exactly)identical sets vertically shifted by $5h$.