Archimedean Voronoi spiral tilings

We study the transition of the number of spirals (called parastichy in the theory of phyllotaxis) within a Voronoi tiling for Archimedean spiral lattices. The transition of local parastichy numbers within a tiling is regarded as a transition at the base site point in a continuous family of tilings. This gives a natural description of the quasiperiodic structure of the grain boundaries. It is proved that the number of tiles in the grain boundaries are denominators of rational approximations of the argument (called the divergence angle) of the generator. The local parastichy numbers are non-decreasing functions of the plastochron parameter. The bifurcation diagram of local parastichy numbers has a Farey tree structure. We also prove Richards' formula of spiral phyllotaxis in the case of Archimedean Voronoi spiral tilings, and show that, if the divergence angle is a quadratic irrational number, then the shapes of tiles in the grain boundaries are close to rectangles. If the divergence angle is linearly equivalent to the golden section, then the shape of tiles in the grain boundaries is close to square.