$\SLR$空间中无限正棱柱镶嵌的非周期测地球填充
Non-periodic geodesic ball packings to infinite regular prism tilings in $\SLR$ space
摘要：
在\cite{Sz13-1}中，我们定义并描述了$\SLR$空间中的{ \it正则无限或有界} $p$-棱柱镶嵌.证明了对于整数参数$p,q;~3 \le p$，$ \frac{2p}{p-2} < q$，存在无穷多个正则无限$p$ -π面-面棱柱镶嵌$\cT^i_p(q)$和无穷多个正则有界$p$-π非面-面棱柱镶嵌$\cT_p(q)$.此外，在\cite{MSz14}和\cite{MSzV13}中，我们通过$\cT_p(q)$的指数为2的旋转子群确定了它的对称群，记为$\mathbf{pq2_1}$，并研究了相应的测地线和平移球填充.本文研究了正则无穷或有界$p$ -棱柱镶嵌的结构，证明了对于每个参数，其基图的侧曲线都是欧氏圆的弧。此外，我们还研究了由$\SLR$几何中的正无穷$p$ -正无穷面-面棱柱镶嵌$\cT^i_p(q)$导出的全等正非周期棱柱镶嵌的非周期测地球填充。我们发展了一个确定上述非周期最优测地球填充的密度的方法，并将此算法应用于它们。我们寻找上面的参数$p$和$q$，其中堆积密度尽可能大。现在，我们得到了$(p, q) = (29,3)$的最大密度$\approx 0.626606$，它大于相应的周期测地球填充在群$\mathbf{pq2_1}$下的最大密度.在我们的工作中，我们将使用由E. {Moln在\cite{M97}。
英文摘要：
In \cite{Sz13-1} we defined and described the {\it regular infinite or bounded} $p$-gonal prism tilings in $\SLR$ space. We proved that there exist infinitely many regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ and infinitely many regular bounded $p$-gonal non-face-to-face prism tilings $\cT_p(q)$ for integer parameters $p,q;~3 \le p$, $ \frac{2p}{p-2} < q$. Moreover, in \cite{MSz14} and \cite{MSzV13} we have determined the symmetry group of $\cT_p(q)$ via its index 2 rotational subgroup, denoted by $\mathbf{pq2_1}$ and investigated the corresponding geodesic and translation ball packings. In this paper we study the structure of the regular infinite or bounded $p$-gonal prism tilings, prove that the side curves of their base figurs are arcs of Euclidean circles for each parameter. Moreover, we examine the non-periodic geodesic ball packings of congruent regular non-periodic prism tilings derived from the regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ in $\SLR$ geometry. We develop a procedure to determine the densities of the above non-periodic optimal geodesic ball packings and apply this algorithm to them. We look for those parameters $p$ and $q$ above, where the packing density large enough as possible. Now, we obtain larger density $\approx 0.626606$ for $(p, q) = (29,3)$ then the maximal density of the corresponding periodical geodesic ball packings under the groups $\mathbf{pq2_1}$. In our work we will use the projective model of $\SLR$ introduced by E. {Molnár} in \cite{M97}.
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