Geodesic ball packings in S² × R space for generalized Coxeter space groups

Abstract

W. Thurston classified the eight simply connected three-dimensional maximal homogeneous Riemannian geometries (see Thurston and Levy 1997, Scott 1983). One of these is the S ² × R geometry which is the direct product of the spherical plane S ² and the real line R. The complete list of the space groups of S ² × R is given by Farkas (Beitr Algebra Geom 42:235–250, 2001). Farkas and Molnár (Proceedings of the Colloquium on Differential Geometry, Debrecen, Hungary, pp 105–118, 2001) have classified the S ² × R manifolds by similarity and diffeomorphism. In this paper we investigate the geodesic balls of S ² × R and compute their volume, define in this space the notion of geodesic ball packing and its density. Moreover, we determine the densest geodesic ball packing for generalized Coxeter space groups of S ² × R. The density of the densest packing for these space groups is ≈ 0.82445423. Surprisingly, the kissing number of the balls in this packing is only 2 (!!). Molnár (Beitr Algebra Geom 38(2):261–288, 1997) has shown that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere ${\mathcal{PS}^3({\bf V}^4,{\varvec V}_4, \mathbb{R})}$. In our work we shall use this projective model of S ² × R geometry and in this manner the geodesic lines, geodesic spheres can be visualized on the Euclidean screen of the computer. This visualization will show also the arrangement of some geodesic ball packings for the above space groups.