$\SLR$空间中的正棱柱镶嵌
Regular prism tilings in $\SLR$ space
摘要：
$\SLR$几何是8个三维Thurston几何之一，它可以从所有行列式为1的$2\times 2$实矩阵的三维李群导出。我们的目标是描述和可视化的{ \it正则无限（环面状）或有界} $p$ -棱镜镶嵌在$\SLR$空间。为此，我们引入了无限有界棱柱的概念，证明了对于参数$p \ge 3$，存在无穷多个正则的无限$p$ -有界的面-面棱柱镶嵌$\cT^i_p(q)$和无穷多个正则的（有界的）$p$ -有界的非面-面$\SLR$棱柱镶嵌$\cT_p(q)$，其中$ \frac{2p}{p-2} < q \in \mathbb{N}$ .此外，我们发展了一种方法来确定空间填充的数据的规则无限和有界棱柱镶嵌。我们将上述过程应用于$\cT^i_3(q)$和$\cT_3(q)$，其中$6< q \in \mathbb{N}$并可视化它们和相应的平铺。E. Molnár表明，齐性3-空间有一个统一的解释，在射影3-空间$\mathcal{P}^3(\bV^4,\BV_4, \mathbf{R})$。在我们的工作中，我们将使用$\SLR$几何的这个投影模型，并且以这种方式，棱镜和棱镜拼接可以在计算机的欧几里德屏幕上可视化。
英文摘要：
$\SLR$ geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. Our aim is to describe and visualize the {\it regular infinite (torus-like) or bounded} $p$-gonal prism tilings in $\SLR$ space. For this purpose we introduce the notion of the infinite and bounded prisms, prove that there exist infinite 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 $\SLR$ prism tilings $\cT_p(q)$ for parameters $p \ge 3$ where $ \frac{2p}{p-2} < q \in \mathbb{N}$. Moreover, we develope a method to determine the data of the space filling regular infinite and bounded prism tilings. We apply the above procedure to $\cT^i_3(q)$ and $\cT_3(q)$ where $6< q \in \mathbb{N}$ and visualize them and the corresponding tilings. E. Molnár showed, that the homogeneous 3-spaces have a unified interpretation in the projective 3-space $\mathcal{P}^3(\bV^4,\BV_4, \mathbf{R})$. In our work we will use this projective model of $\SLR$ geometry and in this manner the prisms and prism tilings can be visualized on the Euclidean screen of computer.
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