Combinatorics and Geometry to Arithmetic of Circle Packings

Combinatorics and Geometry to Arithmetic of Circle Packings Abstract: The Koebe-Andreev-Thurston/Schramm theorem assigns a conformally rigid fi- nite circle packing to a convex polyhedron, and then successive inversions yield a conformally rigid infinite circle packing. For example, starting with the tetrahedron, we take a configuration of four pairwise tangent circles and invert successively to obtain the classical Apollonian Circle Packing. The latter, an object of much recent study, is ”arithmetic”, in that there are realizations for which all circles have curvatures in the rational integers. Our aim, in joint work with Alex Kontorovich, is to classify polyhedra with this property and study the integral curvatures of the resulting circle packings. We will start from scratch and report on work toward this goal, with some emphasis on Archimedean and Catalan polyhedra.