On geodesics of phyllotaxis

Abstract

Seeds of sunflowers are often modelled by the map $n⟼{\phi }_{\theta }(n)=\sqrt{n}{e}^{2i\pi n\theta }$ 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 ${\gamma }_{\theta }:{\mathbb{R}}_{>0}⟶{\mathrm{PSL}}_2(\mathbb{Z})\backslash \mathbb{H}$ of the modular curve and use it for local descriptions of the image ${\phi }_{\theta }(\mathbb{N})$ of the phyllotactic map ${\phi }_{\theta }$.