A polytopal generalization of Apollonian packings and Descartes' theorem

TL;DR

This work generalizes Descartes' circle theorem to polytopal sphere packings arising from uniform polytopes by expressing a polytopal curvature relation in terms of canonical polytope data and edge-scribable realizations, culminating in a unified Descartes-type equation $$(\kappa_{f_0}-\kappa_{f_1})^2+\cdots=\ell_{\mathcal{P}}^2\kappa_{\mathcal{P}}^2$$. It develops the Lorentzian and affine models for the space of spheres, defines polytopal packings via edge-scribable polytopes, and introduces the Möbius spectrum as a new invariant that is preserved under Möbius maps (with a rigidity-based justification). The paper then specializes to Platonic solids to derive Platonic Descartes relations, construct explicit integral Platonic crystallographic packings, and provide linear representations of full symmetry groups along with integrality criteria, enabling parametrizations via Schläfli symbols. Overall, it advances the arithmetic and geometric theory of polytopal packings, linking hyperbolic reflection groups, Möbius geometry, and polytope invariants to obtain new integral packings and a robust framework for higher-dimensional generalizations. The results have potential implications for crystallographic packings, diophantine problems in sphere packings, and the study of polytope-based Apollonian-type structures across dimensions.

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.

Figures (22)

• Figure 1: An Apollonian packing.
• Figure 2: Two integral Apollonian packings. The labels are the bends, and the bends of the outer circles are $-3$ (left) and $-6$ (right).
• Figure 3: Two polyhedral crystallographic packings based on the cube (left) and the truncated tetrahedron (right).
• Figure 4: (Left) An outer point of $\mathbb{R}^{d+1}$ and its stereographic sphere in $\widehat{\mathbb{R}^d}$; (right) same setting in the affine model of the space of spheres, together with the corresponding Lorentzian vector.
• Figure 5: Three edge-scribed realisations of a $4$-pyramid. The barycentre of the second one is the origin. In the third realisation, the barycentre of the contact points of the edges with the sphere is the origin, so it is canonical.

Theorems & Definitions (22)

• Theorem 1: Descartes
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 2
• proof