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

Szirmai, Jenő

doi:10.1007/s12215-017-0316-8

Cited by 14 publications

(14 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, we showed that the densest known hyperball packing, dually related to the regular prism tilings, introduced in [12], can be realized by a regular truncated tetrahedron tiling with density ≈ 0.82251. [18] we discussed the problem of congruent and non-congruent hyperball (hypersphere) packings to each truncated regular tetrahedron tiling. These are derived from the Coxeter simplex tilings {p, 3, 3} and {5, 3, 3, 3, 3} in the 3 and 5-dimensional hyperbolic space.…”

Section: Remark 33mentioning

confidence: 99%

See 1 more Smart Citation

Decomposition method related to saturated hyperball packings

Szirmai

2019

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

In this paper we study the problem of hyperball (hypersphere) packings in 3-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new procedure to get a decomposition of 3-dimensional hyperbolic space H 3 into truncated tetrahedra. Therefore, 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.

show abstract

Section: Remark 33mentioning

confidence: 99%

“…In [18] we discussed 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 3and 5-dimensional hyperbolic space.…”

mentioning

confidence: 99%

Decomposition method related to saturated hyperball packings

Szirmai

2019

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…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%

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

Szirmai

2019

Acta Universitatis Sapientiae, Mathematica

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In [25] we discussed congruent and non-congruent hyperball 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.…”

Section: Hyperball (Hypersphere) Packingsmentioning

confidence: 99%

Upper bound of density for packing of congruent hyperballs in hyperbolic $3-$space

Szirmai¹

2018

Preprint

Self Cite

View full text Add to dashboard Cite

In [26] we proved that to each saturated congruent hyperball packing exists a decomposition of 3-dimensional hyperbolic space H 3 into truncated tetrahedra. Therefore, 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. In this paper we prove, using the above results and the results of papers [16] and [23], that the density upper bound of the saturated congruent hyperball (hypersphere) packings related to the corresponding truncated tetrahedron cells is realized in a regular truncated tetrahedra with density ≈ 0.86338. Furthermore, we prove that the density of locally optimal congruent hyperball arrangement in regular truncated tetrahedron is not a monotonically increasing function of the height (radius) of corresponding optimal hyperball, contrary to the ball (sphere) and horoball (horosphere) packings.

show abstract

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

Cited by 14 publications

References 21 publications

Decomposition method related to saturated hyperball packings

Decomposition method related to saturated hyperball packings

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

Upper bound of density for packing of congruent hyperballs in hyperbolic $3-$space

Contact Info

Product

Resources

About