On geodesics of phyllotaxis

TL;DR

This paper explores the mathematical structure of sunflower seed arrangements by linking the phyllotactic map to geodesic paths on the modular curve, providing a new geometric perspective on seed distribution.

Contribution

It introduces a novel connection between divergence angles in phyllotaxis and geodesic paths on the modular curve, offering a geometric framework for analyzing seed arrangements.

Findings

Geodesic paths characterize seed distribution patterns.

The modular curve framework provides local descriptions of seed arrangements.

New insights into the mathematical structure of phyllotaxis patterns.

Abstract

Seeds of sunflowers are often modelled by the map $n\longmapsto \varphi_\theta(n)=\sqrt{n}e^{2i\pi n\theta}$ leading to a roughly uniform repartition with two consecutive seeds separated by the divergence angle $2\pi\theta$ for $\theta$ the golden ratio. We associate to an arbitrary real divergence angle $2\pi \theta$ a geodesic path $\gamma_\theta: \mathbb R_{>0}\longrightarrow \mathrm{PSL}_2(\mathbb Z)\backslash \mathbb H$ of the modular curve and use it for local descriptions of the image $\varphi_\theta(\mathbb N)$ of the phyllotactic map $\varphi_\theta$.