On geodesics of phyllotaxis

1: Seeds of sunflowers are often modelled by the map n 7−→ φθ(n) = √ ne2iπnθ 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 γθ : R>0 −→ PSL2(Z)\H of the modular curve and use it for local descriptions of the image φθ(N) of the phyllotactic map φθ. Given a real parameter θ, we call the map φθ : N −→ C defined by φθ(n) = √ ne the phyllotactic map of divergence angle 2πθ (measured in radians). The image φθ(N) of a phyllotactic map is the phyllotactic set (of parameter θ or divergence angle 2πθ). A phyllotactic set φθ(N) is uniformly discrete (i.e. two distinct elements of φθ(N) are at distance at least ǫ for some strictly positive ǫ) with uniform density if θ = [a0; a1, a2, . . . ] = a0 + 1 a1 + 1 a2+... is irrational with bounded coefficients a0, a1, a2, . . . in its continued fraction expansion. Among all possible parameters, the value given by the golden ratio 1+ √ 5 2 = [1; 1, 1, 1, . . . ] (or closely related numbers) stands out and gives a particularly nice configuration. Figure 1 displays a few hundred small points of φ(1+ √ 5)/2(N). Finite approximations of φ(1+ √ 5)/2(N) can be observed in capitula (heads) of sunflowers or daisies (the map φ(1+ √ 5)/2, sometimes also called the sunflower-map, has been proposed in [14] as a model for heads of sunflowers). Joining close points of φ(1+ √ 5)/2(N) we get parastichy spirals appearing in pairs of crisscrossing families enumerated by two consecutive elements of the Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, . . . . Explaining the occurence of the