Structure and Visualization of Optimal Horoball Packings in 3 -dimensional Hyperbolic Space
Abstract
Four packings of hyperbolic 3-space are known to yield the optimal packing density of
0.85328…
. They are realized in the regular tetrahedral and cubic Coxeter honeycombs with Schläfli symbols
{3,3,6}
and
{4,3,6}
. These honeycombs are totally asymptotic, and the packings consist of horoballs (of different types) centered at the ideal vertices. We describe a method to visualize regular horoball packings of extended hyperbolic 3-space
H
¯
3
using the Beltrami-Klein model and the Coxeter group of the packing. We produce the first known images of these four optimal horoball packings.