Lattice-like Packings and Coverings with Congruent Translation Balls and Cylinders in Sol geometry

TL;DR

This paper investigates optimal lattice-like packings and coverings with congruent translation balls and cylinders in Sol geometry, providing density estimates and exact solutions for specific configurations.

Contribution

It introduces density notions for these packings and coverings in Sol space and determines exact optimal densities for certain cylinder packings.

Findings

Upper bounds for ball coverings based on radii and volumes.

Exact optimal packing densities for specific cylinder packings.

Density estimates for coverings with translation balls and cylinders.

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 {\it translation tetrahedra}. Moreover we determine the exact optimal packing and covering densities of a type of cylinder packings belonging to the fundamental lattices.