On geodesics of phyllotaxis

Bacher, Roland

Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 3-30.

Seeds of sunflowers are often modelled by $n⟼{\varphi }_{\theta }\left(n\right)=\sqrt{n}{e}^{2i\pi n\theta }$ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance $2\pi \theta$ for $\theta$ the golden ratio. We associate to such a map ${\varphi }_{\theta }$ a geodesic path ${\gamma }_{\theta }:{ℝ}_{>0}⟶{\mathrm{PSL}}_2\left(\mathbb{Z}\right)\setminus ℍ$ of the modular curve and use it for local descriptions of the image ${\varphi }_{\theta }\left(\mathbb{N}\right)$ of the phyllotactic map ${\varphi }_{\theta }$.

Classification : 92B99, 11H31, 52C15
Mots-clés : Lattice, hyperbolic geometry, phyllotaxis, sunflower-map

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