Lattice coverings by congruent translation balls using translation-like bisector surfaces in Nil geometry

TL;DR

This paper explores lattice coverings in Nil geometry using translation-like bisector surfaces, determining key geometric properties, and estimating the minimal density of such coverings with congruent translation balls.

Contribution

It introduces a method to analyze translation bisectors, centers, and radii of translation spheres in Nil space, and provides the first estimate of minimal covering density.

Findings

Derived equations for translation-like bisectors in Nil geometry.

Showed that isosceles property does not imply equal angles in translation triangles.

Estimated minimal covering density as approximately 1.42783.

Abstract

In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. We develop a method to determine the centre and the radius of the circumscribed translation sphere of a given {\it translation tetrahedron}. A further aim of this paper is to study lattice-like coverings with congruent translation balls in Nil space. We introduce the notion of the density of the considered coverings and give upper estimate to it using the radius amd the volume of the circumscribed translation sphere of a given {\it translation tetrahedron}. The found minimal upper bound density of the translation ball coverings $\Delta \approx 1.42783$. In our work we will use for computations and visualizations the projective model of Nil described by E. Moln\'ar in \cite{M97}.