Coverings with horo- and hyperballs generated by simply truncated orthoschmes

Szirmai, Jenő; Eper, Miklós

doi:10.30755/nsjom.12680

Cited by 2 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) Notice, that the parameter u, where the minimum is attained ≈ 6.45953, is very close to the corresponding parameter in [8], where it was ≈ 6.45962. (2) The above covering density is smaller than the ≈ 1.280 density belonging to Fejes Tóth László, Böröczky Károly, but this hyperball covering can not be extended to the entire space.…”

Section: Remarkmentioning

confidence: 76%

“…Here we will prove, using the results of our paper [8], that the theoretic lower bound √ 12/π for congruent hypercycle coverings can be arbitrary approximated with non-congruent hypercycle coverings related to doubly truncated orthoschemes. Proof.…”

Section: Hypercycle Covering In Hyperbolic Planementioning

confidence: 81%

Section: Hypercycle Covering In Hyperbolic Planementioning

confidence: 97%

“…So if a → 1, we get a hyp-hor covering, investigated in [8]. We recall Theorem 3.2 of [8] using the denotation of the present paper: that means that density √ 12/π can be arbitrarily approximated with hypercycle coverings related to doubly truncated orthoschemes. □ Remark: The general investigation of planar non-congruent hypersphere packings and coverings related to the doubly truncated orthoschemes is discussed in a forthcoming paper.…”

Section: Hypercycle Covering In Hyperbolic Planementioning

confidence: 97%

“…Other types of ball packings and coverings in H n are the so-called hyphor packings [31] and coverings [8], where in a configuration we use both horoball and hyperball, and the fundamental domain of the tilings are simply truncated simply asymptotic Coxeter orthoschemes. In the hyperbolic plane in both cases (packing and covering) the theoretic bound of the density (as above) can be attained in limit.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Coverings with congruent and non-congruent hyperballs generated by doubly truncated Coxeter orthoschemes

Eper,

Szirmai

2022

CDM

View full text Add to dashboard Cite

After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings [Acta Univ. Sapientiae, Mathematica, 11, 2 (2019), 437–459], we consider the corresponding covering problems. In Non-fundamental trunc-simplex tilings and their optimal hyperball packings and coverings in hyperbolic space the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of truncated tetrahedra, and determined the optimal congruent hyperball packing and covering configurations belonging to some of these classes. In this paper, we complement these results with the investigation of the non-congruent covering cases and the remaining congruent cases. We prove, that between congruent and non-congruent hyperball coverings the thinnest belongs to the $\{7,3,7\}$ Coxeter tiling with density $\approx 1.26829$. This covering density is smaller than the conjectured lower bound density of L. Fejes Tóth for coverings with balls and horoballs. We also study the local packing arrangements related to $\{u,3,7\}$ $(6< u < 7, ~ u\in \mathbb{R})$ doubly truncated orthoschemes and the corresponding hyperball coverings. We prove, that these coverings are achieved their minimum density at parameter $u\approx 6.45953$ with covering density $\approx 1.26454$ which is smaller than the above record-small density, but this hyperball arrangement related to this locally optimal covering can not be extended to the entire $\mathbb{H}^3$. Moreover, we see that in the hyperbolic plane $\mathbb{H}^2$ the universal lower bound of the congruent circle, horocycle, hypercycle covering density $\sqrt{12}/\pi$ can be approximated arbitrarily well also with non-congruent hypercycle coverings generated by doubly truncated Coxeter orthoschemes.

show abstract

Section: Remarkmentioning

confidence: 76%

Section: Hypercycle Covering In Hyperbolic Planementioning

confidence: 81%