The densest translation ball packing by fundamental lattices in $\SOL$ space

Jen\H o Szirmai

Abstract: In the eight homogeneous Thurston 3-geometries -- $\mathbf E^3$, $\mathbf S^3$, $\mathbf H^3$, $\mathbf S^2\!\times\!\mathbf R}$, $\mathb H^2\!\times\!\mathbf R$, $\widetilde{\bmathbf S\mathbf L_2\mathbf R}$, $\mathbf{Nil}$, $\mathbf{Sol}$ -- 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 lattice-like 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 lattice-like ball packing are defined. Moreover, the densest translation ball packing by so-called fundamental lattices, which is one (Type {\bf I/1}) of the 17 Bravais-type 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 affine-projective 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), 27--32. [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 3-manifolds. Bull. Lond. Math. Soc. {\bf 15} (1983), 401--487. [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), 129--141. [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), 261--288.

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