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

Jenő Szirmai

doi:10.2478/ausm-2019-0032

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.

References

[1] K. Bezdek, Sphere Packings Revisited, Eur. J. Combin., 27/6 (2006), 864–883.10.1016/j.ejc.2005.05.001
Search in Google Scholar Back to article
[2] J. Bolyai, Appendix, Scientiam spatii absolute veram exhibens, Marosvásárhely, (1831).
Search in Google Scholar Back to article
[3] K. Böröczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar., 32 (1978), 243–261.10.1007/BF01902361
Search in Google Scholar Back to article
[4] K. Böröczky, A. Florian,Über die dichteste Kugelpackung im hyperbolis-chen Raum, Acta Math. Acad. Sci. Hungar., 15 (1964), 237–245.10.1007/BF01897041
Search in Google Scholar Back to article
[5] G. Fejes Tóth, G. Kuperberg, W. Kuperberg, Highly Saturated Packings and Reduced Coverings, Monatsh. Math., 125/2 (1998), 127–145.10.1007/BF01332823
Search in Google Scholar Back to article
[6] L. Fejes Tóth, Regular Figures, Macmillan (New York), 1964.
Search in Google Scholar Back to article
[7] H. -C. Im Hof, Napier cycles and hyperbolic Coxeter groups, Bull. Soc. Math. Belgique, 42 (1990), 523–545.
Search in Google Scholar Back to article
[8] R. Kellerhals, On the volume of hyperbolic polyhedra, Math. Ann., 285 (1989), 541–569.10.1007/BF01452047
Search in Google Scholar Back to article
[9] R. T. Kozma, J. Szirmai, New Lower Bound for the Optimal Ball Packing Density of Hyperbolic 4-space, Discrete Comput. Geom., 53/1 (2015), 182-198, DOI: 10.1007/s00454-014-9634-1.10.1007/s00454-014-9634-1
Search in Google Scholar Back to article
[10] E. Molnár, The Projective Interpretation of the eight 3-dimensional homogeneous geometries, Beitr. Algebra Geom.,38/2 (1997), 261–288.
Search in Google Scholar Back to article
[11] E. Molnár, J. Szirmai, Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups, Publications de l’Institut Mathématique, 103 (117) (2018), 129–146, DOI: 10.2298/PIM1817129M, arXiv: 1612.04541.10.2298/PIM1817129M
Search in Google Scholar Back to article
[12] M. Stojanović, Coxeter Groups as Automorphism Groups of Solid Transitive 3-simplex Tilings, Filomat,28/3 (2014), 557–577, DOI 10.2298/FIL1403557S.10.2298/FIL1403557S
Search in Google Scholar Back to article
[13] M. Stojanović, Hyperbolic space groups and their supergroups for fundamental simplex tilings, Acta Math. Hungar.,153/2 (2017), 276–288, DOI: 10.1007/s10474-017-0761-z.10.1007/s10474-017-0761-z
Search in Google Scholar Back to article
[14] J. Szirmai, Hyperball packings in hyperbolic 3-space, Mat. Vesn., 70/3 (2018), 211–221.
Search in Google Scholar Back to article
[15] J. Szirmai, Packings with horo- and hyperballs generated by simple frustum orthoschemes, Acta Math. Hungar., 152/2 (2017), 365–382, DOI:10.1007/s10474-017-0728-0.10.1007/s10474-017-0728-0
Search in Google Scholar Back to article
[16] J. Szirmai, Density upper bound of congruent and non-congruent hyper-ball packings generated by truncated regular simplex tilings, Rendiconti del Circolo Matematico di Palermo Series 2, 67 (2018), 307–322, DOI: 10.1007/s12215-017-0316-8, arXiv:1510.03208.10.1007/s12215-017-0316-8
Search in Google Scholar Back to article
[17] J. Szirmai, Decomposition method related to saturated hyperball packings, Ars Math. Contemp., 16 (2019), 349–358.10.26493/1855-3974.1485.0b1
Search in Google Scholar Back to article
[18] J. Szirmai, The optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic d-space, Beitr. Algebra Geom.,48/1 (2007), 35–47.
Search in Google Scholar Back to article
[19] J. Szirmai, Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space, Aequat. Math., 85 (2013), 471-482, DOI: 10.1007/s00010-012-0158-6.10.1007/s00010-012-0158-6
Search in Google Scholar Back to article
[20] J. Szirmai, Horoball packings and their densities by generalized simplicial density function in the hyperbolic space, Acta Math. Hungar.,136/1-2 (2012), 39–55, DOI: 10.1007/s10474-012-0205-8.10.1007/s10474-012-0205-8
Search in Google Scholar Back to article
[21] J. Szirmai, The p-gonal prism tilings and their optimal hypersphere packings in the hyperbolic 3-space, Acta Math. Hungar., 111 (1-2) (2006), 65–76.10.1007/s10474-006-0034-8
Search in Google Scholar Back to article
[22] J. Szirmai, The regular prism tilings and their optimal hyperball packings in the hyperbolic n-space, Publ. Math. Debrecen, 69 (1-2) (2006), 195–207.10.1007/s10474-006-0034-8
Search in Google Scholar Back to article
[23] J. Szirmai, The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds, Kragujevac J. Math.40 (2) (2016), 260-270, DOI:10.5937/KgJMath1602260S.10.5937/KgJMath1602260S
Search in Google Scholar Back to article
[24] J. Szirmai, The least dense hyperball covering to the regular prism tilings in the hyperbolic n-space, Ann. Mat. Pur. Appl.195/1 (2016), 235–248, DOI: 10.1007/s10231-014-0460-0.10.1007/s10231-014-0460-0
Search in Google Scholar Back to article
[25] J. Szirmai, A candidate for the densest packing with equal balls in Thurston geometries, Beitr. Algebra Geom., 55/2 (2014), 441-452, DOI: 10.1007/s13366-013-0158-2.10.1007/s13366-013-0158-2
Search in Google Scholar Back to article
[26] J. Szirmai, Hyperball packings related to cube and octahedron tilings in hyperbolic space, Contributions to Discrete Mathematics, (to appear), (2020).
Search in Google Scholar Back to article
[27] J. Szirmai, Upper bound of density for packing of congruent hyperballs in hyperbolic 3−space, Submitted manuscript, (2019).
Search in Google Scholar Back to article
[28] I. Vermes,Über die Parkettierungsmöglichkeit des dreidimensionalen hyperbolischen Raumes durch kongruente Polyeder, Studia Sci. Math. Hungar., 7 (1972), 267–278.
Search in Google Scholar Back to article
[29] I. Vermes, Ausfüllungen der hyperbolischen Ebene durch kongruente Hyperzykelbereiche, Period. Math. Hungar., 10/4 (1979), 217–229.10.1007/BF02020020
Search in Google Scholar Back to article
[30] I. Vermes,Über reguläre Überdeckungen der Bolyai-Lobatschewskischen Ebene durch kongruente Hyperzykelbereiche, Period. Math. Hungar., 25/3 (1981), 249–261.
Search in Google Scholar Back to article

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

Abstract

Paradigm

My account