SYMMETRIES OF HOROBALL PACKINGS RELATED TO FAMOUS 3-DIMENSIONAL HYPERBOLIC TILINGS

We discuss the symmetries of optimal packings in 3-dimensional hyperbolic space related to some famous polyhedral tilings, the Coxeter tilings with Schlafli symbols (3, 3, 6) and (4, 3, 6), as well as the Lambert-cube tilings with (p, q) = (3, 6) and (4, 4). We introduce the simplicial density function to reveal that the optimal ball packings of hyperbolic 3-space feature limiting objects called horoballs. We show that for both types of tilings there are multiple optimal packing arrangements with distinct symmetries and various horoball types. Finally, we mention implications to and recent results in higher dimensional hyperbolic spaces.