Lattice-like packings and coverings with congruent translation balls and cylinders in Sol geometry

Abstract

The aim of this paper is to study lattice-like coverings with congruent translation balls and the packings and coverings with a type of translation cylinders in Sol space related to the fundamental lattices. We introduce the notions of the densities of the considered problems and give upper estimate to ball coverings using the radii and the volumes of the circumscribed translation spheres of given translation tetrahedra. Moreover we determine the exact optimal packing and covering densities of a type of cylinder packings belonging to the fundamental lattices.