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We present a generalization of Descartes' theorem for the family of polytopalsphere packings arising from uniform polytopes. The corresponding quadraticequation is expressed in terms of geometric invariants of uniform polytopeswhich are closely connected to canonical realizations of edge-scribablepolytopes. We use our generalization to construct integral Apollonian packingsbased on the Platonic solids. Additionally, we also introduce and discuss a newspectral invariant for edge-scribable polytopes.

A polytopal generalization of Apollonian packings and Descartes' theorem