Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings. (English) Zbl 1402.52019
The author defines some congruent and non-congruent hyperball packings derived from the tilings of the hyperbolic space into truncated regular tetrahedrons and compute density upper bounds for obtained packings.
There are only two types of truncated regular tetrahedron tilings of \(\mathbb{H}^n\).
For \(n=3\) there is a family of tilings with Schläfli sympols \(\{p,3,3\}\). In this case the density function is maximal for \(p=7\) and corresponding upper bound is \(\approx 0.82251\).
For \(n=5\) there is a tiling Schläfli sympol \(\{5,3,3,3,3\}\). Corresponding upper bound is \(\approx 0.50514\).
There are only two types of truncated regular tetrahedron tilings of \(\mathbb{H}^n\).
For \(n=3\) there is a family of tilings with Schläfli sympols \(\{p,3,3\}\). In this case the density function is maximal for \(p=7\) and corresponding upper bound is \(\approx 0.82251\).
For \(n=5\) there is a tiling Schläfli sympol \(\{5,3,3,3,3\}\). Corresponding upper bound is \(\approx 0.50514\).
Reviewer: Anton Shutov (Vladimir)
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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