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Hyperball packings related to octahedron and cube tilings in hyperbolic space. (English) Zbl 1452.52014

Author’s abstract: 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 Böröczky-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 \(0.84931\).

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
52B15 Symmetry properties of polytopes
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