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

Abstract

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 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 H n , (n ≥ 2), and prove that, in this sense, the well known Böröczky-Florian density upper bound for "congruent horoball" packings of H 3 does not remain valid to the fully asymptotic tetrahedra. The density of this locally densest packing is ≈ 0.874994, may be surprisingly larger than the Böröczky-Florian density upper bound ≈ 0.853276 but our local ball arrangement seems not to have extension to the whole hyperbolic space. * Mathematics Subject Classification 2010: 52C17, 52C22, 52B15.

New horoball packing density lower bound in hyperbolic 5-space

“…However, it is often useful to distinguish between certain horoballs of a packing; we shall use the notion of horoball type with respect to the fundamental domain of the given tiling as introduced in [22]. Two horoballs of a horoball packing are said to be of the same type or equipacked if and only if their local packing densities with respect to a particular cell (in our case a Coxeter simplex) are equal, otherwise the two horoballs are of different type.…”

“…This limit is independent of the choice of center C for B n−1 C (r). In the case of periodic ball or horoball packings, this local density defined above can be extended to the entire hyperbolic space, and is related to the simplicial density function (defined below) that we generalized in [21] and [22]. In this paper we shall use such definition of packing density (cf.…”

“…Furthermore, in [21,22] we 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. In H 4 the locally optimal simplicial packing density is 0.77038 .…”

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

“…2. If we allow horoballs of different types at the various vertices of a totally asymptotic simplex and generalize the notion of the simplicial density function in H n , (n ≥ 2) then the Böröczky-Florian type density upper bound does not remain valid for the fully asymptotic simplices [20], [21].…”

“…We note here, that the discussion of the densest horoball packings in the n-dimensional hyperbolic space n ≥ 3 with horoballs of different types has not been settled yet as well (see [13], [14], [20], [21]).…”