Journal for Geometry and Graphics 26 (2022), No. 1, 051--064
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
[Abstract-pdf]
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 well-known as the surface of a cooling tower of a power plant. \par One point of view is the so-called kinematic geometry by the Vienna colleagues, e.g., as in a paper by H. Stachel [{\it Flexible octahedra in hyperbolic space}, in: {\it Non-Euclidean Geometries}, A. Prekopa and E. Molnar (eds.), Janos Bolyai Memorial Volume 581, Springer, Boston (2006) 209--225], but also in a very general context. The other point is the so-called $\mathbf{H}^2\times\mathbf{R}$ 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.
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