Structure and Visualization of Optimal Horoball Packings in $3$-dimensional Hyperbolic Space

Four packings of hyperbolic 3-space are known to yield the optimal packing density of $0.85328\dots$. They are realized in the regular tetrahedral and cubic Coxeter honeycombs with Schl\"afli 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 $\bar{\mathbb{H}}^3$ using the Beltrami-Klein model and the Coxeter group of the packing. We produce the first known images of these four optimal horoball packings.