Packings with Geodesic and Translation Balls and Their Visualizations in SL2R Space

E. Molnár ; J. Szirmai

Journal for geometry and graphics, Tome 26 (2022) no. 1, pp. 51-64

Résumé

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 $\approx 0.845306$.

Détail

Classification : 51C17, 52C22, 52B15, 53A35, 51M20
Mots-clés : Thurston geometries, SL2R geometry, density of ball packing under space group, regular prism tiling, volume in SL2R