Jenő Szirmai

Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types

The goal of this paper is to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space

Isoptic surfaces of polyhedra

{\mathbb{H}^3}

Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings

. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known Böröczky–Florian density upper bound for “congruent horoball” packings of

Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space

{\mathbb{H}^3}

Projective metric realizations of cone-manifolds with singularities along 2-bridge knots and links

remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.

The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space

Nil geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from W. Heisenberg’s famous real matrix group. The aim of this paper is to study lattice-like ball coverings in Nil space. We introduce the notion of the density of the considered coverings and give upper and lower estimates to it, moreover in Section 3, we formulate a conjecture for the ball arrangement of the least dense latticelike geodesic ball covering and give its covering density