  We present a generalization of Descartes' theorem for the family of polytopal
sphere packings arising from uniform polytopes. The corresponding quadratic
equation is expressed in terms of geometric invariants of uniform polytopes
which are closely connected to canonical realizations of edge-scribable
polytopes. We use our generalization to construct integral Apollonian packings
based on the Platonic solids. Additionally, we also introduce and discuss a new
spectral invariant for edge-scribable polytopes.
