Packings with horo- and hyperballs generated by simple frustum orthoschemes

Abstract

In this paper we deal with the packings derived by horo-and hyperballs (briefly hyp-hor packings) in the n-dimensional hyperbolic spaces H n (n = 2, 3) which form a new class of the classical packing problems.We construct in the 2− and 3−dimensional hyperbolic spaces hyphor 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 3 π and in H 3 the optimal configuration belongs to the [7,3,6] Coxeter tiling with density ≈ 0.83267.Moreover, we study the hyp-hor packings in truncated orthoschemes [p, 3, 6] (6 < p < 7, p ∈ R) 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 ≈ 0.85397. That means that these locally optimal hyp-hor configurations provide larger densities that the Böröczky-Florian density upper bound (≈

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

“…Furthermore, in [19,20] we have found that, by allowing horoballs of different types at each vertex of a totally asymptotic simplex and generalizing the simplicial density function to H n for (n ≥ 2), the Böröczky-type density upper bound is not valid for the fully asymptotic simplices for n ≥ 4. For example, in H 4 the locally optimal simplicial packing density is 0.77038 .…”

“…In periodic ball or horoball packings, the local density described below can be extended to the entire hyperbolic space and it is related to the simplicial density function that we generalized in [19] and [20]. In this paper, we shall use such definition of packing density by [24].…”

Decomposition method related to saturated hyperball packings

“…What are the optimal packing and covering arrangements using noncompact balls (horoballs and hyperballs) and what are their densities? These are the so-called hyp-hor packings and coverings (see [17]).…”

“…In [17] we deal with the packings derived by horo-and hyperballs (briefly hyp-hor packings) in n-dimensional hyperbolic spaces H n (n = 2, 3) which form a new class of the classical packing problems. We constructed in the 2− and 3−dimensional hyperbolic spaces hyp-hor packings that are generated by complete Coxeter tilings of degree 1 i.e.…”

“…1), which are denoted by T a . The fundamental domain of T a is a Lambert quadrilateral A 0 A 1 P 0 P 1 (see [17]) that is denoted by F a . It is derived by the truncation of the orthoscheme A 0 A 1 A 2 by the polar line π of the outer vertex A 2 .…”