The least dense hyperball covering of regular prism tilings in hyperbolic $$n$$ n -space

Rend. Circ. Mat. Palermo, II. Ser

2017

Self Cite

In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular tetrahedron tilings. These are derived from the Coxeter simplex tilings {p, 3, 3} (7 ≤ p ∈ N) and {5, 3, 3, 3, 3} in 3 and 5-dimensional hyperbolic space. We determine the densest hyperball packing arrangements related to the above tilings. We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings.

Section: Remark 11mentioning

confidence: 99%

Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings

Rend. Circ. Mat. Palermo, II. Ser

2017

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“…Similarly to the former cases (see [21], [22], [24], [14], [16], [17]) it is interesting to study and to construct locally optimal congruent and non-congruent hyperball packings relating to suitable truncated polyhedron tilings in 3-and higher dimensions as well. This study fits into our program to look for the upper bound density of the congruent and non-congruent hyperball packings in H n .…”

Section: On Hyperball Packings In a Doubly Truncated Orthoschemementioning

confidence: 99%

“…In [21] and [22] we analysed the regular prism tilings (simply truncated Coxeter orthoscheme tilings) and the corresponding optimal hyperball packings in H n (n = 3, 4) and we extended the method -developed in the former paper [22] -to 5-dimensional hyperbolic space (see [23]). In paper [24] we studied the n-dimensional hyperbolic regular prism honeycombs and the corresponding coverings by congruent hyperballs and we determined their least dense covering densities. Furthermore, we formulated conjectures for candidates of the least dense hyperball covering by congruent hyperballs in 3-and 5-dimensional hyperbolic spaces.…”

Section: Introductionmentioning

confidence: 99%

Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 3-space

Acta Universitatis Sapientiae, Mathematica

2019

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In [17] we considered hyperball packings in 3-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of H 3 into truncated tetrahedra. In order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. Therefore, in this paper we examine the doubly truncated Coxeter orthoscheme tilings and the corresponding congruent and non-congruent hyperball packings. We proved that related to the mentioned Coxeter tilings the density of the densest congruent hyperball packing is ≈ 0.81335 that is -by our conjecture -the upper bound density of the relating non-congruent hyperball packings, too. * Mathematics Subject Classification 2010: 52C17, 52C22, 52B15.

“…In H 2 the universal lower bound of the hypercycle covering density is π determined by I. Vermes in [34]. In the paper [28] we have studied the n-dimensional (n ≥ 3) hyperbolic regular prism honeycombs and the corresponding coverings by congruent hyperballs and we have determined their least dense covering densities. Moreover, we have formulated a conjecture for the candidate of the least dense hyperball covering by congruent hyperballs in the 3and 5-dimensional hyperbolic space.…”

Section: On Hyperball Packingsmentioning

confidence: 99%

Packings with horo- and hyperballs generated by simple frustum orthoschemes

2017

Acta Math. Hungar.

In this paper we deal with the packings derived by horo-and hyperballs (briefly hyp-hor packings) in the n-dimensional hyperbolic spaces 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 hyphor 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 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 π and in H 3 the optimal configuration belongs to the [7,3,6] Coxeter tiling with density ≈ 0.83267.Moreover, we study the hyp-hor packings in truncated orthoschemes [p, 3, 6] (6 < p < 7, p ∈ R) whose density function is attained 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 ≈ 0.85397. That means that these locally optimal hyp-hor configurations provide larger densities that the Böröczky-Florian density upper bound (≈