Regular tessellation of the Lobachevskii plane

P.I. Troshin

Kazan Federal University, Kazan, 420008 Russia

E-mail: paul.troshin@gmail.com

Received March 11, 2019

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DOI: 10.26907/2541-7746.2019.4.591-605

For citation: Troshin P.I. Regular tessellation of the Lobachevskii plane. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, vol. 161, no. 4, pp. 591–605. doi: 10.26907/2541-7746.2019.4.591-605. (In Russian)

Abstract

This paper discusses a new algorithm for construction of regular tessellation of the Lobachevskii plane. The problems of combinatorial and topological arrangement of regular tessellations, finding the number of tiles in each layer of such tessellation, and implementation of the algorithm in the modern computer programming language were studied. The relevance of the study is determined, on the one hand, by the unceasing interest in hyperbolic geometry and, in particular, in tessellations within it. On the other hand, the relevance is due to the insufficient number of published algorithm descriptions and their implementations. The following methods were used:

• implementation of the basic knowledge in the group of motions of the Lobachevskii plane in the Beltrami–Klein model, its trigonometry and isometries to the other known models to construct a prototile and tessellation layers;
• splitting the tessellation by layers and layers by subclasses of tiles, studying the arrangement of each layer with respect to the previous one, finding the number of tiles in the layers with the help of mathematical induction;
• devising an algorithm in the form of a pseudocode and in the programming language of Wolfram Mathematica.

In the course of the study, the following results were obtained:

• an algorithm for regular tessellation of the Lobachevskii plane, which produces the tessellation layer by layer, without repetition of the tiles, by means of proper rigid motions applied to the initial prototile;
• the algorithm implemented in the programming language of Wolfram Mathematica;
• formulas for estimation of the number of tiles in layers for the suggested algorithm.

The obtained results and observations made in this paper are important for construction of tessellations in hyperbolic geometry.

Keywords: regular tessellation, tiling, Lobachevskii plane, hyperbolic geometry, Schla¨fli symbol, group of motions, Beltrami–Klein model, tile, prototile

Acknowledgments. We thank the reviewers for their constructive and helpful feedback on the style of the manuscript. The study was supported by the Russian Foundation for Basic Research (project no. 1831-00295).

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