Decomposition method related to saturated hyperball packings

Szirmai, Jenő

doi:10.26493/1855-3974.1485.0b1

Cited by 11 publications

(17 citation statements)

References 15 publications

(29 reference statements)

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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%

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

Szirmai

2019

Acta Universitatis Sapientiae, Mathematica

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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.

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Section: On Hyperball Packings In a Doubly Truncated Orthoschemementioning

confidence: 99%

“…

…”

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confidence: 99%

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

Szirmai

2019

Acta Universitatis Sapientiae, Mathematica

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“…In [26] we considered hyperball packings in 3-dimensional hyperbolic space and developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of 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.…”

Section: Hyperball (Hypersphere) Packingsmentioning

confidence: 99%

“…In [26] we modified the classical definition of saturated packing for noncompact ball packings with generalized balls (horoballs, hyperballs) in ndimensional hyperbolic space H n (n ≥ 2 integer parameter):…”

Section: Hyperball (Hypersphere) Packingsmentioning

confidence: 99%

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

Szirmai¹

2018

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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.

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An upper bound of the density for packing of congruent hyperballs in hyperbolic $$3-$$space

Szirmai

2023

Aequat. Math.

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In Szirmai (Ars Math Contemp 16:349–358, 2019) we proved that to each saturated congruent hyperball packing there exists a decomposition of the 3-dimensional hyperbolic space $$\mathbb {H}^3$$ 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 results of the papers Miyamoto (Topology 33(4): 613–629, 1994) and Szirmai (Mat Vesn 70(3): 211–221, 2018), that the density upper bound of the saturated congruent hyperball (hypersphere) packings related to the corresponding truncated tetrahedron cells is realized in regular truncated tetrahedra with density $$\approx 0.86338$$ ≈ 0.86338 . Furthermore, we prove that the density of locally optimal congruent hyperball arrangement in a regular truncated tetrahedron is not a monotonically increasing function of the height (radius) of the corresponding optimal hyperball, unlike the ball (sphere) and horoball (horosphere) packings.

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Decomposition method related to saturated hyperball packings

Cited by 11 publications

References 15 publications

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

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

An upper bound of the density for packing of congruent hyperballs in hyperbolic $$3-$$space

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