Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 3-space. (English) Zbl 1446.52016
In hyperbolic \(3\)-space \(\mathbb{H}^3\), packings of hyperballs (sets of points whose distance from a given plane does not exceed a given distance \(h>0\)) are related to associated tilings of \(\mathbb{H}^3\) by truncated simplices. This allows a local definition of density of hyperball packings. It is shown that the densest packing of congruent hyperballs related to doubly truncated Coxeter orthoschemes is based on the tiling with Schläfli symbol \(\{7,3,7\}\) and attains a density of \(\approx 0.81335\). The author conjectures that this cannot be improved by dropping the condition of congruence of the hyperballs.
Reviewer: Christian Richter (Jena)
MSC:
| 52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
| 51M16 | Inequalities and extremum problems in real or complex geometry |
| 52C22 | Tilings in \(n\) dimensions (aspects of discrete geometry) |
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