A polytopal generalization of Apollonian packings and Descartes' theorem

TL;DR

This paper extends Descartes' theorem to polytopal sphere packings from uniform polytopes, enabling the construction of integral Apollonian packings based on Platonic solids and introducing a new spectral invariant.

Contribution

It generalizes Descartes' theorem for polytopal sphere packings, linking geometric invariants of uniform polytopes to Apollonian packings and introducing a spectral invariant.

Findings

Derived a quadratic equation for polytopal packings using geometric invariants.

Constructed integral Apollonian packings from Platonic solids.

Introduced a new spectral invariant for edge-scribable polytopes.

Abstract

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.