Recursive Semi-Regular Circle Packings on the Sphere

Gregory S. Chirikjian

Associate Professor
Dept. of Mechanical Engineering
Johns Hopkins University

Abstract

It has been know for thousands of years that only five three-dimensional polyhedra exist which have polygonal faces that are both congruent and regular. These so-called perfect solids are the tetrahedron (four triangular faces), cube (six square faces), octahedron (eight triangular faces), dodecahedron (twelve pentagonal faces), and icosahedron (twenty triangular faces). By projecting faces of any of these polyhedra onto a sphere from the common center of the sphere and polyhedron, the sphere is divided into regions of equal area and shape. Regular circle packings on the sphere are generated by inscribing circles in each of these regions (which is the same as first inscribing circles in each polygonal face of a perfect solid and then projecting onto the sphere).

The focus of this talk is the generation of "semi-regular" circle packings on the sphere. These are generated by observing the duality of the perfect solids (i.e., connecting the centers of all adjacent faces of any perfect solid results in another perfect solid). In particular, the tetrahedron is dual to itself, the cube is dual to the octahedron, the dodecahedron is dual to the icosahedron. A group of rotational symmetries exists for each pair of dual perfect solids. By subdividing the sphere into units formed by overlaying the projections of dual perfect solids onto the sphere, and observing the symmetries of these units, it is reviewed how sixteen different packings of congruent circles on the sphere are generated in addition to the regular packings. It is then shown how one can pack an arbitrary number of "almost congruent" circles on the sphere by recursively subdividing these units and inscribing circles in each resulting subdivision. In either case, these semi-regular packings of circles inherit the symmetry group of their parent polyhedra.

This work was motivated by the need to approximate functions on the sphere with axi-symmetric basis functions to a finer resolution than allowed by the regular spherical circle packings. To this end, it is described how a basis for the space of square integrable functions on the sphere is generated using these recursive circle packings.