New horoball packing density lower bound in hyperbolic 5-space

Abstract

Koszul type Coxeter Simplex tilings exist in hyperbolic space H n for 2 ≤ n ≤ 9, and their horoball packings have the highest known regular ball packing densities for 3 ≤ n ≤ 5. In this paper we determine the optimal horoball packings of Koszul type Coxeter simplex tilings of n-dimensional hyperbolic space for 6 ≤ n ≤ 9, which give new lower bounds for packing density in each dimension. The symmetries of the packings are given by Coxeter simplex groups.

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

“…Theorem 2.1 The volume of a three-dimensional hyperbolic complete orthoscheme (except Lambert cube cases) S is expressed with the essential angles α 01 , α 12 , α 23 ,…”

“…The set of all horoball types (they are congruent) at a vertex is a one-parameter family. In our investigations we allow horoballs in different types (see [20], [21], [22]).…”

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

“…In [24], we reported [13] and [14] and considered the Coxeter tilings in H 3 where the generating orthoscheme was a simple truncated one with some parallel faces i.e. their dihedral angle is zero (symbol ∞).…”

“…We determined their optimal ball and horoball packings, proved that the densest packing was realized at tilings (∞, 3, 6, ∞), and (∞; 6; 3; ∞) with density ≈ 0.8413392, see Fig. 1, 12, 19 and [20, 21, 22] and [14,15,16] for further connections.…”

Visualization of sphere and horosphere packings related to Coxeter tilings generated by simply truncated orthoschemes with parallel faces

“…However these ball packing configurations are only locally optimal and cannot be extended to the entirety of the ambient space H n . In [21] In [22], [23] we extend our study of horoball packings to H n (5 ≤ n ≤ 9)…”