
Journal for Geometry and Graphics 26 (2022), No. 1, 051064 Copyright Heldermann Verlag 2022 Packings with Geodesic and Translation Balls and Their Visualizations in SL2R Space Emil Molnár Department of Geometry, Institute of Mathematics, University of Technology and Economics, Budapest, Hungary emolnar@math.bme.hu Jenö Szirmai Department of Geometry, Institute of Mathematics, University of Technology and Economics, Budapest, Hungary szirmai@math.bme.hu [Abstractpdf] Remembering our friendly cooperation between the Geometry Departments of the Technical Universities of Budapest and Vienna (also under different names) a nice topic comes into my memory: the “Gum fibre model”, a model made of fibres and two disks of the hyperbolic base plane as it is wellknown as the surface of a cooling tower of a power plant. \par One point of view is the socalled kinematic geometry by the Vienna colleagues, e.g., as in a paper by H. Stachel [{\it Flexible octahedra in hyperbolic space}, in: {\it NonEuclidean Geometries}, A. Prekopa and E. Molnar (eds.), Janos Bolyai Memorial Volume 581, Springer, Boston (2006) 209225], but also in a very general context. The other point is the socalled $\mathbf{H}^2\times\mathbf{R}$ geometry and $\widetilde{\mathbf{SL}_2\mathbf{R}}$ geometry where  roughly  two hyperbolic planes as circle discs are connected with gum fibres, first: in a simple way, second: in a twisted way. \par This second homogeneous (Thurston) geometry will be our topic (initiated by some Budapest colleagues, and discussed also in international cooperations). We use for the computation and visualization of $\widetilde{\mathbf{SL}_2\mathbf{R}}$ its projective model, as in some previous papers. We found a seemingly extremal geodesic ball packing for the $\widetilde{\mathbf{SL}_2\mathbf{R}}$ group $\mathbf{pq}_k\mathbf{o}_\ell$ ($p = 9$, $q = 3$, $k = 1$, $o = 2$, $\ell = 1$) with density $\approx 0.787758$. A much better translation ball packing was found for the group $\mathbf{pq}_k\mathbf{o}_\ell$ ($p = 11$, $q = 3$, $k = 1$, $o = 2$, $\ell = 1$) with density $\approx 0.845306$. Keywords: Thurston geometries, SL2R geometry, density of ball packing under space group, regular prism tiling, volume in SL2R. MSC: 51C17; 52C22, 52B15, 53A35, 51M20. [ Fulltextpdf (488 KB)] 