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 $d\ge 3$. 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.

Corrected some types and revised some proofs. Use fancy pictures rendered by PovRay. 18 pages

References in corpus (3)

• Optimality and uniqueness of the (4,10,1/6) spherical code
• Lorentzian Coxeter systems and Boyd-Maxwell ball packings
• Uniqueness of the (22,891,1/4) spherical code

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