Voronoi spiral tilings

Abstract

The parameter set of Voronoi spiral tilings gives a dual of van Iterson's bifurcation diagram for phyllotactic spirals. We study the Voronoi tilings for the Bernoulli spiral site sets, as the simplest spirals in the centric representation with similarity symmetry. Their parameter set is composed of a family of real algebraic curves in the complex plane, with the Farey sequence structure. This naturally extends to the parameter set for multiple tilings, i.e., the tilings of the covering spaces of the punctured plane. We show the denseness of the parameters z = reiθ for quadrilateral Voronoi spiral multiple tilings.The techniques of dynamical systems are applied to the group of similarity symmetry. The parastichy numbers and the distortion of the Voronoi regions depend on the rational approximations of θ/2π. We consider the limit set of the shapes of the quadrilateral tiles by taking the limit as r → 1, with θ fixed. If θ/2π is a quadratic irrational number, then the limit set is a finite set of rectangles. In particular, if θ/2π is linearly equivalent to the golden section, then the limit is the square.