scholarly journals Top dense hyperbolic ball packings and coverings for complete coxeter orthoscheme groups

2018 ◽  
Vol 103 (117) ◽  
pp. 129-146
Author(s):  
Emil Molnár ◽  
Jenő Szirmai
Keyword(s):  
Hyperbolic Space ◽  
Coxeter Group ◽  
Upper Bound ◽  
Dihedral Angles ◽  
Ball Packings ◽  
Ball Packing ◽  
Packing Densities

In n-dimensional hyperbolic space Hn (n > 2), there are three types of spheres (balls): the sphere, horosphere and hypersphere. If n = 2, 3 we know a universal upper bound of the ball packing densities, where each ball?s volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g., in H3 a densest (not unique) horoball packing is derived from the {3,3,6} Coxeter tiling consisting of ideal regular simplices T? reg with dihedral angles ?/3. The density of this packing is ??3 ? 0.85328 and this provides a very rough upper bound for the ball packing densities as well. However, there are no ?essential" results regarding the ?classical" ball packings with congruent balls, and for ball coverings either. The goal of this paper is to find the extremal ball arrangements in H3 with ?classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called complete Coxeter orthoschemes and their extended groups. In Theorems 1.1 and 1.2 we formulate also conjectures for the densest ball packing with density 0.77147... and the loosest ball covering with density 1.36893..., respectively. Both are related with the extended Coxeter group (5,3,5) and the so-called hyperbolic football manifold. These facts can have important relations with fullerenes in crystallography.

scholarly journals The volume of a compact hyperbolic antiprism

2018 ◽  
Vol 27 (13) ◽  
pp. 1842010
Author(s):  
Nikolay Abrosimov ◽  
Bao Vuong
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Space ◽  
Convex Polyhedron ◽  
Rotational Symmetry ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dihedral Angles ◽  
Integral Formulas ◽  
Necessary And Sufficient

We consider a compact hyperbolic antiprism. It is a convex polyhedron with [Formula: see text] vertices in the hyperbolic space [Formula: see text]. This polyhedron has a symmetry group [Formula: see text] generated by a mirror-rotational symmetry of order [Formula: see text], i.e. rotation to the angle [Formula: see text] followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.

scholarly journals Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

2021 ◽  
Author(s):  
Nikolay Abrosimov ◽  
Bao Vuong
Keyword(s):  
Convex Hull ◽  
Hyperbolic Space ◽  
General Type ◽  
Integral Formula ◽  
Sufficient Conditions ◽  
Gram Matrix ◽  
Dihedral Angles ◽  
Hyperbolic Tetrahedron ◽  
Volume Formula ◽  
Necessary And Sufficient

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

The boundedness of a weighted Coxeter group with non-3-edge-labeling graph

2019 ◽  
Vol 18 (05) ◽  
pp. 1950085
Author(s):  
Yan Li ◽  
Jian-Yi Shi
Keyword(s):  
Parabolic Subgroup ◽  
Coxeter Group ◽  
Upper Bound ◽  
Hecke Algebras ◽  
Product Formula ◽  
Edge Labeling

Let [Formula: see text] be a weighted Coxeter group such that the order [Formula: see text] of the product [Formula: see text] is not 3 for any [Formula: see text] and that [Formula: see text], where [Formula: see text] is the longest element in the parabolic subgroup [Formula: see text] of [Formula: see text] generated by [Formula: see text]. We prove that [Formula: see text] is bounded with [Formula: see text] an upper bound in the sense of Lusztig in Sec. 13.2 of [Hecke Algebras with Unequal Parameters, arXiv:math/0208154 v2 [math.RT] 10 Jun 2014], verifying a conjecture of Lusztig in our case (see Conjecture 13.4 in loc. cite).

On Ford isometric spheres in complex hyperbolic space

1994 ◽  
Vol 115 (3) ◽  
pp. 501-512 ◽  
Author(s):  
John R. Parker
Keyword(s):  
Hyperbolic Space ◽  
Upper Bound ◽  
Discrete Subgroup ◽  
Complex Hyperbolic Space

AbstractThe complex hyperbolic version of Shimizu's lemma gives an upper bound on the radii of isometric spheres of maps in a discrete subgroup of PU(n, 1) containing a vertical Heisenberg translation. The purpose of this paper is to show that in a neighbourhood of this bound radii of isometric spheres only take values in a particular discrete set. When the group contains certain ellipto-parabolic maps this upper bound can be improved and the set of values of the radii is more restricted. Examples are given that show that these results cannot be improved.

scholarly journals On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Representation Theory ◽  
Generating Function ◽  
Coxeter Group ◽  
Upper Bound ◽  
Open Problem ◽  
Special Class ◽  
Exact Enumeration ◽  
Restricted Permutations ◽  
International Audience ◽  
Freely Braided

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

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

2019 ◽  
Vol 11 (2) ◽  
pp. 437-459
Author(s):  
Jenő Szirmai
Keyword(s):  
Hyperbolic Space ◽  
Upper Bound ◽  
Decomposition Algorithm ◽  
3 Dimensional

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.

scholarly journals Boyd-Maxwell ball packings

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Hao Chen ◽  
Jean-Philippe Labbé
Keyword(s):  
Coxeter Group ◽  
Lorentz Space ◽  
Affine Space ◽  
Root Systems ◽  
Fundamental Result ◽  
Geometric Representation ◽  
Ball Packings ◽  
Fractal Patterns ◽  
International Audience

International audience In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In fact, the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: given a geometric representation of a Coxeter group in Lorentz space, the set of limit directions of weights equals the set of limit roots.

ON FREE PLANES IN LATTICE BALL PACKINGS

2002 ◽  
Vol 34 (3) ◽  
pp. 284-290
Author(s):  
MARTIN HENK ◽  
GÜNTER M. ZIEGLER ◽  
CHUANMING ZONG
Keyword(s):  
Lattice Plane ◽  
Plane Parallel ◽  
Dimensional Plane ◽  
Ball Packings ◽  
Ball Packing

This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d-dimensional packing lattice of balls one can find a four- dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d.

scholarly journals Even More Infinite Ball Packings from Lorentzian Root Systems

10.37236/4989 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Hao Chen
Keyword(s):  
Root System ◽  
Root Systems ◽  
Theoretical Level ◽  
Coxeter Graph ◽  
Partial Classification ◽  
Ball Packings ◽  
Ball Packing ◽  
Level 2 ◽  
Extreme Rays

Boyd (1974) proposed a class of infinite ball packings that are generated by inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms of root systems in Lorentz spaces. In particular, he showed that the space-like weight vectors correspond to a ball packing if and only if the associated Coxeter graph is of "level 2"'. In Maxwell's work, the simple roots form a basis of the representations space of the Coxeter group. In several recent studies, the more general based root system is considered, where the simple roots are only required to be positively independent. In this paper, we propose a geometric version of "level'' for root systems to replace Maxwell's graph theoretical "level''. Then we show that Maxwell's results naturally extend to the more general root systems with positively independent simple roots. In particular, the space-like extreme rays of the Tits cone correspond to a ball packing if and only if the root system is of level $2$. We also present a partial classification of level-$2$ root systems, namely the Coxeter $d$-polytopes of level-$2$ with $d+2$ facets.

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