The densest geodesic ball packing by a type of Nil lattices

Jen\H o Szirmai

Abstract: W. Heisenberg's famous real matrix group provides a non-commutative translation group of an affine 3-space. The $\NIL$ geometry which is one of the eight homogeneous Thurston 3-geometries, can be derived from this matrix group. E. Molnár proved in \cite{M97}, that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere $\mathcal{PS}^3(\bV^4,\BV_4, \mathbb{R})$. 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 lattice-like ball packing. Moreover, we determine the densest lattice-like 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.

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