Apollonian Ball Packings and Stacked Polytopes

Abstract

We investigate in this paper the relation between Apollonian d-ball packings and stacked (d+ 1) -polytopes for dimension d≥ 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.