# Apollonian Ball Packings and Stacked Polytopes

Discrete & Computational Geometry, Jun 2016

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 there is no six 4-cliques sharing a 3-clique. For higher dimension, we have some partial results.

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Apollonian Ball Packings and Stacked Polytopes, Discrete & Computational Geometry, 2016, DOI: 10.1007/s00454-016-9777-3