Domains of Convergence for Polyhedral Packings

TL;DR

This paper extends the theory of circle packings to polyhedral packings, analyzing their convergence domains using algebraic and geometric tools, revealing a connection to Tits cones and root systems.

Contribution

It generalizes the Apollonian packing framework to all polyhedral packings and characterizes their convergence domains via the Tits cone.

Findings

The domain of absolute convergence is the Tits cone for an infinite root system.

Develops the theory of the Apollonian group and Descartes quadratic form for polyhedral packings.

Establishes a link between convergence domains and algebraic structures like root systems.

Abstract

Polyhedral circle packings are generalizations of the Apollonian packing. We develop the theory of the Apollonian group, Descartes quadratic form, and related objects for all polyhedral packings. We use these tools to determine the domain of absolute convergence of a generating function that can be associated to any polyhedral packing. This domain of convergence is the Tits cone for an infinite root system.