Interesting locally optimal ball and horoball arrangements in n-dimensional hyperbolic spaces

2022. 04. 26. 10:30
Arnasli Yahya

We consider the ball and horoball packings belonging to n-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. We determine the optimal ball and horoball packing arrangements and their densities. The centers of horoballs are required to lie at ideal vertices of polyhedral cells constituting the tiling.

In 3-dimensional hyperbolic space, we found the densest packing is realized by the horoballs related to (infinity, 3, 6, infinity) and (infinity, 6, 3, infinity) tilings with density ~0.8413392. It is relatively near the upper bound, 0.85327...

We are interested to investigate a similar structure in higher dimensions. We are curious that the density would be the second-largest (or even be the largest).