Apollonian Ball Packings and Stacked Polytopes

TL;DR

This paper explores the relationship between Apollonian ball packings and stacked polytopes across dimensions, providing a complete characterization in 3D and partial results in higher dimensions.

Contribution

It establishes a full characterization of when the tangency graph of an Apollonian 3-ball packing corresponds to a stacked 4-polytope, and offers partial insights for higher dimensions.

Findings

In 3D, the tangency graph of an Apollonian 3-ball packing matches the 1-skeleton of a stacked 4-polytope under a specific clique-sharing condition.

The paper provides a necessary and sufficient condition for the 3D case.

Partial results are presented for dimensions higher than 3.

Abstract

We investigate in this paper the relation between Apollonian $d$-ball packings and stacked $(d+1)$-polytopes for dimension $d\ge 3$. For $d=3$, the relation is fully described: we prove that the $1$-skeleton of a stacked $4$-polytope is the tangency graph of an Apollonian $3$-ball packing if and only if no six $4$-cliques share a $3$-clique. For higher dimension, we have some partial results.