The Local-Global Principle for Integral Generalized Apollonian Sphere Packings

Abstract

Four mutually tangent spheres form two gaps. In each of these, one can inscribe in a unique way four mutually tangent spheres such that each one of these spheres is tangent to exactly three of the original spheres. Repeating the process gives rise to a generalized Apollonian sphere packing. These packings have remarkable properties. One of them is the local to global principle and will be proven in this paper.