Packings with horo- and hyperballs generated by simple frustum orthoschemes. (English) Zbl 1399.52039
Summary: We deal with the packings derived by horo- and hyperballs (briefly hyp-hor packings) in \(n\)-dimensional hyperbolic spaces \(\mathbb H^n\) \((n = 2, 3)\) which form a new class of the classical packing problems. We construct in the 2- and 3-dimensional hyperbolic spaces hyp-hor packings that are generated by complete Coxeter tilings of degree 1 i.e. the fundamental domains of these tilings are simple frustum orthoschemes and we determine their densest packing configurations and their densities.
We prove using also numerical approximation methods that in the hyperbolic plane (\(n = 2\)) the density of the above hyp-hor packings arbitrarily approximate the universal upper bound of the hypercycle or horocycle packing density \(3/\pi\) and in \(\mathbb H^3\) the optimal configuration belongs to the [7,3,6] Coxeter tiling with density \(\approx 0.83267\).
Moreover, we study the hyp-hor packings in truncated orthoschemes [\(p\), 3, 6] \((6 < p < 7, p \in\mathbb R)\) whose density function attains its maximum for a parameter which lies in the interval [6.05, 6.06] and the densities for parameters lying in this interval are larger that \(\approx 0.85397\). That means that these locally optimal hyp-hor configurations provide larger densities that the Böröczky-Florian density upper bound (\(\approx 0.85328\)) for ball and horoball packings [K. Böröczky and A. Florian, Acta Math. Acad. Sci. Hung. 15, 237–245 (1964; Zbl 0125.39803)] but these hyp-hor packing configurations can not be extended to the entirety of hyperbolic space \(\mathbb H^3\).
We prove using also numerical approximation methods that in the hyperbolic plane (\(n = 2\)) the density of the above hyp-hor packings arbitrarily approximate the universal upper bound of the hypercycle or horocycle packing density \(3/\pi\) and in \(\mathbb H^3\) the optimal configuration belongs to the [7,3,6] Coxeter tiling with density \(\approx 0.83267\).
Moreover, we study the hyp-hor packings in truncated orthoschemes [\(p\), 3, 6] \((6 < p < 7, p \in\mathbb R)\) whose density function attains its maximum for a parameter which lies in the interval [6.05, 6.06] and the densities for parameters lying in this interval are larger that \(\approx 0.85397\). That means that these locally optimal hyp-hor configurations provide larger densities that the Böröczky-Florian density upper bound (\(\approx 0.85328\)) for ball and horoball packings [K. Böröczky and A. Florian, Acta Math. Acad. Sci. Hung. 15, 237–245 (1964; Zbl 0125.39803)] but these hyp-hor packing configurations can not be extended to the entirety of hyperbolic space \(\mathbb H^3\).
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 |
Citations:
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