关于叶序的测地线

摘要：

向日葵的种子通常由映射$n\longmapsto φ_θ(n)=\sqrt{n}e^{2iπnθ}$建模，导致大致均匀的再分配，其中两个连续的种子由发散角$2πθ$分离，其中$θ$是黄金分割率。我们关联到一个任意的真实发散角$2πθ$的测地线路径$γ_θ: \mathbb R_{>0}\longrightarrow \mathrm{PSL}_2(\mathbb Z)\backslash \mathbb H$的模曲线，并使用它的局部描述的图像$φ_θ(\mathbb N)$的叶序图$φ_θ$。

英文摘要：

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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