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

الأصل · EN

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 H³ using the Beltrami-Klein model and the Coxeter group of the packing. We produce the first known images of these four optimal horoball packings.