Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 48, No. 2, pp. 383397 (2007) 

The densest geodesic ball packing by a type of Nil latticesJen\H o SzirmaiBudapest University of Technology and Economics, Institute of Mathematics, Department of GeometryAbstract: W. Heisenberg's famous real matrix group provides a noncommutative translation group of an affine 3space. The $\NIL$ geometry which is one of the eight homogeneous Thurston 3geometries, can be derived from this matrix group. E. Molnár proved in \cite{M97}, that the homogeneous 3spaces have a unified interpretation in the projective 3sphere $\mathcal{PS}^3(\bV^4,\BV_4, \mathbb{R})$. In our work we will use this projective model of the $\NIL$ geometry. In this paper we investigate the geodesic balls of the $\NIL$ space and compute their volume (see (2.4)), introduce the notion of the $\NIL$ lattice, $\NIL$ parallelepiped (see Section 3) and the density of the latticelike ball packing. Moreover, we determine the densest latticelike geodesic ball packing (Theorem 4.2). The density of this densest packing is $\approx 0.78085$, may be surprising enough in comparison with the Euclidean result $\frac{\pi}{\sqrt{18}}\approx 0.74048$. The kissing number of the balls in this packing is 14. Full text of the article:
Electronic version published on: 7 Sep 2007. This page was last modified: 28 Jun 2010.
© 2007 Heldermann Verlag
