On geodesics of phyllotaxis

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 $\mathrm{P}\mathrm{S}\mathrm{L}$_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 ${\phi }_{\theta }$.