Coverings with congruent and non-congruent hyperballs generated by doubly truncated Coxeter orthoschemes

Abstract

After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, we consider the corresponding covering problems. In \cite{MSSz} the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of truncated tetrahedra, and determined the optimal congruent hyperball packing and covering configurations belonging to some of these classes. In this paper we compliment these results with the investigation of the non-congruent covering cases, and the remainig congruent cases. We prove, that between congruent and non-congruent hyperball coverings the thinnest belongs to the $\{7,3,7\}$ Coxeter tiling with density $\approx 1.26829$. This covering density is smaller than the conjectured lower bound density of L.~Fejes~T\'oth for coverings with balls and horoballs. We also study the local packing arrangements related to $\{u,3,7\}$ $(6< u < 7, ~ u\in \mathbb{R})$ doubly truncated orthoschemes and the corresponding hyperball coverings. We prove, that these coverings are achieved their minimum density at parameter $u\approx 6.45953$ with covering density $\approx 1.26454$ which is smaller then the above record-small density, but this hyperball arrangement related to this locally optimal covering can not be extended to the entire $\mathbb{H}^3$. Moreover we see, that in the hyperbolic plane $\mathbb{H}^2$ the universal lower bound of the congruent circle, horocycle, hypercycle covering density $\frac{\sqrt{12}}{\pi}$ can be approximated arbitrarily well also with non-congruent hypercycle coverings generated by doubly truncated Coxeter orthoschemes.