# Hyperball packings related to truncated cube and octahedron tilings in hyperbolic space

### Abstract

In this paper, we study congruent and noncongruent hyperball (hypersphere) packings to the truncated regular cube and octahedron tilings. These are derived from the Coxeter truncated orthoscheme tilings $\{4,3,p\}$ $(6< p \in \mathbb{N})$ and $\{3,4,p\}$ $(4< p \in \mathbb{N})$, respectively, by their Coxeter reflection groups in hyperbolic space $\mathbb{H}^{3}$. We determine the densest hyperball packing arrangement and its density with congruent and noncongruent hyperballs.

We prove that the locally densest (noncongruent half) hyperball configuration belongs to the truncated cube with a density of approximately $0.86145$ if we allow $6< p \in \mathbb{R}$ for the dihedral angle $2\pi/p$. This local density is larger than the B\"or\"oczky--Florian density upper bound for balls and horoballs. But our locally optimal noncongruent hyperball packing configuration cannot be extended to the entire hyperbolic space $\mathbb{H}^3$. We determine the extendable densest noncongruent hyperball packing arrangement to the truncated cube tiling $\{4,3,p=7\}$ with a density of approximately $0.84931$.

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