Horoball Packing Research Articles

New lower bounds for optimal horoball packing density in hyperbolic n-space for 6 le n le 9

Koszul type Coxeter simplex tilings exist in hyperbolic n-space mathbb {H}^n up to n = 9, and their horoball packings achieve the highest known regular ball packing densities for n = 3, 4, 5. In this paper we determine the optimal horoball packing densities of Koszul simplex tilings in dimensions 6 le n le 9, which give new lower bounds for optimal packing density in each dimension. The symmetries of the packings are given by Coxeter simplex groups, and a parameter related to the Busemann function gives an isometry invariant description of different optimal horoball packing configurations.

Optimal ball and horoball packings generated by 3-dimensional simply truncated Coxeter orthoschemes with parallel faces

In this paper, we consider the ball and horoball packings belonging to 3-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper is to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space ℍ3. The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices. We prove that the densest packing of the above cases is realized by horoballs related to (∞, 3, 6, ∞) and (∞, 6, 3, ∞) tilings with density ≈ 0.8413392. The determined densest packing configurations give the second-highest density belonging to the Coxeter groups, and second largest locally optimal density are expected in higher dimensions for groups with similar structures.

Visualization of Sphere and Horosphere Packings Related to Coxeter Tilings by Simply Truncated Orthoschemes with Parallel Faces

In this paper we describe and visualize the densest ball and horoball packing configurations to the simply truncated 3-dimensional hyperbolic Coxeter orthoschemes with parallel faces, using the results of [24]. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter reflection group. We use the Beltrami-Cayley-Klein ball model of 3-dimensional hyperbolic space H^3, the images were made by the Python programming language.

New horoball packing density lower bound in hyperbolic 5-space

We describe the optimal horoball packings of asymptotic Koszul type Coxeter simplex tilings of $5$-dimensional hyperbolic space where the symmetries of the packings are generated by Coxeter groups. We find that the optimal horoball packing density of $\delta_{opt}=0.59421\dots$ is realized in an entire commensurability class of arithmetic Coxeter tilings. Eleven optimal arrangements are achieved by placing horoballs at the asymptotic vertices of the corresponding Coxeter simplices that give the fundamental domains. When multiple horoball types are allowed, in the case of the arithmetic Coxeter groups, the relative packing densities of the optimal horoball types are rational submultiples of $\delta_{opt}$, corresponding to the Dirichlet-Voronoi cell densities of the packing. The packings given in this paper are so far the densest known in hyperbolic $5$-space.

Horoball packings related to the 4-dimensional hyperbolic 24 cell honeycomb {3,4,3,4}

In this paper we study the horoball packings related to the hyperbolic 24 cell honeycomb by Coxeter-Schl?fli symbol {3,4,3,4} in the extended hyperbolic 4-spaceH 4 where we allow horoballs in different types centered at the various vertices of the 24 cell. Introducing the notion of the generalized polyhedral density function, we determine the locally densest horoball packing arrangement and its density with respect to the above regular tiling. The maximal density is ? 0:71645 which is equal to the known greatest horoball packing density in hyperbolic 4-space, given in [13].

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

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.

Commensurability of link complements

In 2013, Chesebro and DeBlois constructed a certain family of hyperbolic links whose complements have the same volume, trace field, Bloch invariant, and cusp parameters up to $PGL(2,\mathbb Q)$. In this paper, we show that these link complements are incommensurable to each other. We use horoball packing to prove this.

Geometry of planar surfaces and exceptional fillings

If a hyperbolic 3-manifold admits an exceptional Dehn filling, then the length of the slope of that Dehn filling is known to be at most six. However, the bound of six appears to be sharp only in the toroidal case. In this paper, we investigate slope lengths of other exceptional fillings. We construct hyperbolic 3-manifolds that have the longest known slopes for reducible fillings. As an intermediate step, we show that the problem of finding the longest such slope is equivalent to a problem on the maximal density horoball packings of planar surfaces, which should be of independent interest. We also discuss lengths of slopes of other exceptional Dehn fillings, and prove that six is not realized by a slope corresponding to a small Seifert fibered space filling.

New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-Space

In this paper we consider ball packings in $4$-dimensional hyperbolic space. We show that it is possible to exceed the conjectured $4$-dimensional realizable packing density upper bound due to L. Fejes T\'oth (Regular Figures, 1964). We give seven examples of horoball packing configurations that yield higher densities of $0.71644896\dots$, where horoballs are centered at ideal vertices of certain Coxeter simplices, and are invariant under the actions of their respective Coxeter groups.

A candidate for the densest packing with equal balls in Thurston geometries

The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a \(3\)-dimensional space of constant curvature was settled by Böröczky and Florian for the hyperbolic space \(\mathbf H ^3\), and, with the proof of the famous Kepler conjecture, by Hales for the Euclidean space \(\mathbf E ^3\). The goal of this paper is to extend the problem of finding the densest geodesic ball (or sphere) packing for the other \(3\)-dimensional homogeneous geometries (Thurston geometries) \(\mathbf S ^2\!\times \!\mathbf R , \mathbf H ^2\!\times \!\mathbf R , \) \(\widetilde{\mathbf{S \mathbf L _2\mathbf R }}, \mathbf {Nil} , \mathbf {Sol} . \) In the following a transitive symmetry group of the ball packing is assumed, which is one of the discrete isometry groups of the considered space. Moreover, we describe a candidate of the densest geodesic ball packing. The greatest density until now is \(\approx 0.85327613\) that is not realized by a packing with equal balls of the hyperbolic space \(\mathbf H ^3\). However, that is attained, e.g., by a horoball packing of \(\overline{\mathbf{H }}^3\) where the ideal centres of horoballs lie on the absolute figure of \(\overline{\mathbf{H }}^3\) inducing the regular ideal simplex tiling \((3,3,6)\) by its Coxeter–Schläfli symbol. In this work we present a geodesic ball packing in the \(\mathbf S ^2\times \mathbf R \) geometry whose density is \({\approx }0.87757183\). The extremal configuration is described in Theorem 2.6. A conjecture for the densest ball packing in Thurston geometries and further remarks are summarized in Sect. 1.1, 1.2 and 2.3.

Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space

In Szirmai (Acta Mathematica Hungarica 136/1-2:39–55, 2012) we generalized the notion of simplicial density function for horoballs in the extended hyperbolic space \({\overline{\mathbf{H}}^3}\), where we allowed horoballs in different types centered at various vertices of a totally asymptotic tetrahedron. By this new aspect, in this paper we study the locally densest horoball packing arrangements and their densities with respect to totally asymptotic regular simplices in the hyperbolic n-space \({\overline{\mathbf{H}}^n}\) extended with its absolute figure, where the ideal centers of horoballs lie in the vertices of a totally asymptotic regular tetrahedron. We will prove that, in this sense, the well known Boroczky–Florian density upper bound for “congruent ball and horoball” packings of\({\overline{\mathbf{H}}^3}\)does not remain valid for the analogous packing of\({\overline{\mathbf{H}}^n}\), for n ≥ 4. Although locally optimal ball arrangements do not seem to have extensions to the whole n-dimensional hyperbolic space. Moreover, we determine an explicit formula for the density of the above locally optimal horoball packings, allowing horoballs of different types (i.e. they are differently packed in their ideal simplex).

Horoball packings and their densities by generalized simplicial density function in the hyperbolic space

The aim of this paper is to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space \(\overline{\mathbf{H}}^{3}\) extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymptotic tetrahedron. We allow horoballs of different types at the various vertices. Moreover, we generalize the notion of the simplicial density function in the extended hyperbolic space \(\overline{\mathbf{H}}^{n}\) (n≧2), and prove that, in this sense, the well known Boroczky–Florian density upper bound for “congruent horoball” packings of \(\overline{\mathbf{H}}^{3}\)does not remain valid to the fully asymptotic tetrahedra.

Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types

The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\mathbb{H}^3$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.

The cusped hyperbolic orbifolds of minimal volume in dimensions less than ten

We detect the cusped complete hyperbolic orbifolds of minimal volume in dimensions less than ten. Our method is geometric and uses results from crystallography and the theory of sphere packings. In fact, such an n-orbifold gives rise to a horoball packing whose orthogonal projection to the cusp boundary yields the densest euclidean lattice packing in dimension n − 1 .