Seeds of sunflowers are often modelled by n⟼ϕ θ (n)=ne 2iπnθ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance 2πθ for θ the golden ratio. We associate to such a map ϕ θ a geodesic path γ θ :ℝ >0 ⟶PSL 2 (ℤ)∖ℍ of the modular curve and use it for local descriptions of the image ϕ θ (ℕ) of the phyllotactic map ϕ θ .