Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 51, No. 2, pp. 353371 (2010) 

The densest translation ball packing by fundamental lattices in $\SOL$ spaceJen\H o SzirmaiBudapest University of Technology and Economics Institute of Mathematics, Department of Geometry, H1521 Budapest, Hungary, email: szirmai@@math.bme.huAbstract: In the eight homogeneous Thurston 3geometries  $\mathbf E^3$, $\mathbf S^3$, $\mathbf H^3$, $\mathbf S^2\!\times\!\mathbf R}$, $\mathb H^2\!\times\!\mathbf R$, $\widetilde{\bmathbf S\mathbf L_2\mathbf R}$, $\mathbf{Nil}$, $\mathbf{Sol}$  the notions of translation curves and translation balls can be introduced in a unified way by initiative of E. Molnár (see [1], [2]). P. Scott in [3] defined $\SOL$ lattices to which latticelike translation ball packings can be defined. In our joint work [4] with E. Molnár we have studied the relation between $\SOL$ lattices and lattices of the pseudoeuclidean (or Minkowskian) plane (see [5], [6]). In the present paper the translation balls of $\SOL$ geometry are investigated, their volume is computed, and the notions of $\SOL$ parallelepiped and density of the latticelike ball packing are defined. Moreover, the densest translation ball packing by socalled fundamental lattices, which is one (Type {\bf I/1}) of the 17 Bravaistype of $\SOL$lattices described in [4] is determined. It turns out that the optimal arrangement has a richer symmetry group (in Type {\bf I/2}) for $N=4$. This density is $\delta \approx 0.56405083$ and the kissing number of the balls to this packing is 6. In our work we shall use the affine model of the $\SOL$ space through affineprojective homogeneous coordinates introduced by E. Molnár in [7]. \smallskip [1] Bölcskei, A.; Szil'agyi, B.: Visualization of curves and spheres in Sol geometry. KoG {\bf 10} (2006), 2732. [2] Moln{á}r, E.; Szilágyi, B.: Translation curves and their spheres in homogeneous geometries. Manuscript to Publicationes Math. Debrecen, 2009. [3] Scott, P.: The geometries of 3manifolds. Bull. Lond. Math. Soc. {\bf 15} (1983), 401487. [4] Moln{á}r, E.; Szirmai, J.: Classification of $\SOL$ lattices. Manuscript to Geom. Dedicata, 2009. [5] Alpers, K.; Quaisser, E.: Lattices in the pseudoeuclidean plane. Geom. Dedicata, {\bf 72} (1998), 129141. [6] Baltag, I. A.; Garit, V. I.: Dvumernye diskretnye affinnye gruppy. Izdat. \v Stiinca, Ki\v sinev, 1981. [7] Moln{á}r, E.: The projective interpretation of the eight $3$dimensional homogeneous geometries. Beitr. Algebra Geom. {\bf 38}(2) (1997), 261288. Full text of the article (for subscribers):
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