<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In [17] we considered hyperball packings in 3-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing has provided a decomposition of ℍ<sup>3</sup> into truncated tetrahedra. Thus, 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 prove 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.</p></abstract>

In [17] we considered hyperball packings in 3-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing has provided a decomposition of ℍ3 into truncated tetrahedra. Thus, 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 prove 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.

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

Department of Geometry, Institute of Mathematics

Budapest University of Technology and Economics

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In [17] we considered hyperball packings in 3-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing...

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

Published Online: Feb 27, 2020

Page range: 437 - 459

Received: Jan 24, 2019

DOI: https://doi.org/10.2478/ausm-2019-0032

Keywords
Hyperbolic geometry, hyperball packings, packing density, Coxeter tilings

© 2019 Jenő Szirmai, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: Feb 27, 2020

Page range: 437 - 459

Received: Jan 24, 2019

DOI: https://doi.org/10.2478/ausm-2019-0032

KeywordsHyperbolic geometry, hyperball packings, packing density, Coxeter tilings

© 2019 Jenő Szirmai, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Hyperbolic geometry, hyperball packings, packing density, Coxeter tilings