Geometric Properties of the Icosahedral-Hexagonal Grid on the Two-Sphere

An icosahedral-hexagonal grid on the two-sphere is created by dividing the faces of an icosahedron and projecting the vertices onto the sphere. This grid and its Voronoi tessellation have several desirable features for numerical simulations of physical processes on the sphere. While several methods to construct the icosahedral grid mesh have been proposed over the past decades, and empirical data have been collected to understand and help improve the grid, rarely have analytical analyses been done to investigate the basic geometric properties of the grid. In this paper, we present an analytical analysis of several geometric properties of the icosahedral grids based on two basic constructions: recursive and nonrecursive construction. We point out that these geometric properties can be improved with modified construction procedures. We demonstrate how these improvements impact the numerical integration of PDEs over the sphere.