On geodesics of phyllotaxis

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

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