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

In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular tetrahedron tilings. These are derived from the Coxeter simplex tilings $\{p,3,3\}$ $(7\le p \in \mathbb{N})$ and $\{5,3,3,3,3\}$ in $3$ and $5$-dimensional hyperbolic space. We determine the densest hyperball packing arrangements related to the above tilings. We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings.Comment: 24 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:1505.03338, arXiv:1312.2328, arXiv:1405.024

Triangle angle sums related to translation curves in \SOL geometry

After having investigated the geodesic and translation triangles and their angle sums in \NIL and \SLR geometries we consider the analogous problem in \SOL space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of translation triangles in \SOL geometry and prove that it can be larger or equal than $\pi$. In our work we will use the projective model of \SOL described by E. Moln\'ar in \cite{M97},Comment: 13 pages, 4 figure

Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 33-space

In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of \HYP 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 $\approx 0.81335$ that is -- by our conjecture -- the upper bound density of the relating non-congruent hyperball packings too.Comment: 24 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1803.0494

Packings with horo- and hyperballs generated by simple frustum orthoschemes

In this paper we deal with the packings derived by horo- and hyperballs (briefly hyp-hor packings) in the $n$-dimensional hyperbolic spaces \HYN ($n=2,3$) which form a new class of the classical packing problems. We construct in the $2-$ and $3-$dimensional hyperbolic spaces hyp-hor packings that are generated by complete Coxeter tilings of degree $1$ i.e. the fundamental domains of these tilings are simple frustum orthoschemes and we determine their densest packing configurations and their densities. We prove that in the hyperbolic plane ($n=2$) the density of the above hyp-hor packings arbitrarily approximate the universal upper bound of the hypercycle or horocycle packing density $\frac{3}{\pi}$ and in \HYP the optimal configuration belongs to the $[7,3,6]$ Coxeter tiling with density $\approx 0.83267$. Moreover, we study the hyp-hor packings in truncated orthosche\-mes $[p,3,6]$ (6< p < 7, ~ p\in \bR) whose density function is attained its maximum for a parameter which lies in the interval $[6.05,6.06]$ and the densities for parameters lying in this interval are larger that $\approx 0.85397$. That means that these locally optimal hyp-hor configurations provide larger densities that the B\"or\"oczky-Florian density upper bound $(\approx 0.85328)$ for ball and horoball packings but these hyp-hor packing configurations can not be extended to the entirety of hyperbolic space $\mathbb{H}^3$.Comment: 27 pages, 9 figures. arXiv admin note: text overlap with arXiv:1312.2328, arXiv:1405.024

The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds

The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived from two compact Coxeter polytops which are combinatorially simplicial prisms (or complete orthoschemes of degree $d=1$) in the five dimensional hyperbolic space $\mathbf{H}^5$ (see \cite{BE} and \cite{EK}). The corresponding hyperbolic tilings are generated by reflections through their delimiting hyperplanes those involve the study of the relating densest hyperball (hypersphere) packings with congruent hyperballs. The analogous problem was discussed in \cite{Sz06-1} and \cite{Sz06-2} in the hyperbolic spaces $\mathbf{H}^n$ $(n=3,4)$. In this paper we extend this procedure to determine the optimal hyperball packings to the above 5-dimensional prism tilings. We compute their metric data and the densities of their optimal hyperball packings, moreover, we formulate a conjecture for the candidate of the densest hyperball packings in the 5-dimensional hyperbolic space $\mathbf{H}^5$.Comment: 15 pages, 4 figure

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

In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ 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 \cite{M94} and \cite{Sz14}, 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 $\approx 0.86338$. 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.Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1709.04369, arXiv:1811.03462, arXiv:1803.04948, arXiv:1405.024

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

The aim of this paper 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 $\bar{\mathbf{H}}^3$ 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 $\bar{\mathbf{H}}^n, ~(n \ge 2)$, and prove that, in this sense, {\it the well known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\bar{\mathbf{H}}^3$ does not remain valid to the fully asymptotic tetrahedra.} The density of this locally densest packing is $\approx 0.874994$, may be surprisingly larger than the B\"or\"oczky--Florian density upper bound $\approx 0.853276$ but our local ball arrangement seems not to have extension to the whole hyperbolic space.Comment: 20 pages, 8 figure

On lattice coverings of Nil space by congruent geodesic balls

The Nil geometry, which is one of the eight 3-dimensional Thurston geometries, can be derived from {W. Heisenberg}'s famous real matrix group. The aim of this paper to study {\it lattice coverings} in Nil space. We introduce the notion of the density of considered coverings and give upper and lower estimations to it, moreover we formulate a conjecture for the ball arrangement of the least dense lattice-like geodesic ball covering and give its covering density $\Delta\approx 1.42900615$. The homogeneous 3-spaces have a unified interpretation in the projective 3-sphere and in our work we will use this projective model of the Nil geometry.Comment: 23 pages, 7 figure

Regular prism tilings in \SLR space

\SLR geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. Our aim is to describe and visualize the {\it regular infinite (torus-like) or bounded} $p$-gonal prism tilings in \SLR space. For this purpose we introduce the notion of the infinite and bounded prisms, prove that there exist infinite many regular infinite $p$-gonal face-to-face prism tilings \cT^i_p(q) and infinitely many regular (bounded) $p$-gonal non-face-to-face \SLR prism tilings \cT_p(q) for parameters $p \ge 3$ where $\frac{2p}{p-2} < q \in \mathbb{N}$. Moreover, we develope a method to determine the data of the space filling regular infinite and bounded prism tilings. We apply the above procedure to \cT^i_3(q) and \cT_3(q) where $6< q \in \mathbb{N}$ and visualize them and the corresponding tilings. E. Moln\'ar showed, that the homogeneous 3-spaces have a unified interpretation in the projective 3-space \mathcal{P}^3(\bV^4,\BV_4, \mathbf{R}). In our work we will use this projective model of \SLR geometry and in this manner the prisms and prism tilings can be visualized on the Euclidean screen of computer.Comment: 15 pages, 7 figure

Bisector surfaces and circumscribed spheres of tetrahedra derived by translation curves in \SOL geometry

In the present paper we study the \SOL geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. Moreover, we develop a method to determine the centre and the radius of the circumscribed translation sphere of a given {\it translation tetrahedron}. In our work we will use for computations and visualizations the projective model of \SOL described by E. Moln\'ar in \cite{M97}.Comment: 17 pages, 8 figures. arXiv admin note: text overlap with arXiv:1703.0664