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- W2102712619 abstract "After having investigated the regular prisms and prism tilings in the $SLR$ space in the previous work cite{Sz13-1} of the second author, we consider the problem of geodesic ball packings related to those tilings and their symmetry groups $mathbf{pq2_1}$. $SLR$ is one of the eight Thurston geometries that can be derived from the 3-dimensional Lie group of all $2times 2$ real matrices with determinant one. In this paper we consider geodesic spheres and balls in $SLR$ (even in $mathbf{SL_{mathrm{2}}R})$, if their radii $rhoin [0, frac{pi}{2})$, and determine their volumes. Moreover, we consider the prisms of the above space and compute their volumes, define the notion of the geodesic ball packing and its density. We develop a procedure to determine the densities of the densest geodesic ball packings for the tilings, or in this paper more precisely, for their generating groups $mathbf{pq2_1}$ (for integer rotational parameters $p,q$; $3le p, frac{2p}{p-2} <q$). We look for those parameters $p$ and $q$ above, where the packing density large enough as possible. Now our record is $0.567362$ for $(p, q) = (8, 10)$. These computations seem to be important, since we do not know optimal ball packing, namely in the hyperbolic space $HYP$. We know only the density upper bound 0.85326, realized by horoball packing of $HYP$ to its ideal regular simplex tiling. Surprisingly, for the so-called translation ball packings under the same groups $mathbf{pq2_1}$ in cite{MSzV13} we have got larger density $0.841700$ for $(p, q) = (5, 10000 rightarrow infty)$ close to the above upper bound. We use for the computation and visualization of the $SLR$ space its projective model introduced by the first author in cite{M97}." @default.
- W2102712619 created "2016-06-24" @default.
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- W2102712619 date "2013-04-02" @default.
- W2102712619 modified "2023-10-18" @default.
- W2102712619 title "Volumes and geodesic ball packings to the regular prism tilings in $widetilde{mathbf{S}mathbf{L}_2mathbf{R}}$ space" @default.
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