Regular prism tilings in $\SLR$ space

TL;DR

This paper explores and visualizes regular prism tilings in the $ ext{SL}_2(R)$ Thurston geometry, establishing their existence, classifying them, and developing methods for their visualization.

Contribution

It introduces the concept of infinite and bounded prisms in $ ext{SL}_2(R)$ space, proves the existence of various regular prism tilings, and provides a visualization method using projective models.

Findings

Existence of infinitely many regular infinite $p$-gonal prism tilings.

Existence of infinitely many regular bounded $p$-gonal prism tilings.

Development of a visualization method for these tilings.

Abstract

$\SLR$ 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. Moln\'ar 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.