Enumerating isotopy classes of tilings guided by the symmetry of triply
- We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems.
Author details: | Benedikt Maximilian KolbeGND, Myfanwy E. EvansORCiD |
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DOI: | https://doi.org/10.1137/20M1358943 |
ISSN: | 2470-6566 |
Title of parent work (English): | Siam journal on applied algebra and geometry |
Publisher: | Society for Industrial and Applied Mathematics |
Place of publishing: | Philadelphia |
Publication type: | Article |
Language: | English |
Date of first publication: | 2022/02/28 |
Publication year: | 2022 |
Release date: | 2023/04/19 |
Tag: | Delaney--Dress; algebraic systems; group; hyperbolic tilings; isotopic tiling theory; mapping class group; orbifolds; presentations; representations of groups as automorphism groups of; tiling theory; triply periodic minimal surface; two-dimensional topology |
Volume: | 6 |
Issue: | 1 |
Article number: | 1358943 |
Number of pages: | 40 |
First page: | 1 |
Last Page: | 40 |
Funding institution: | Emmy Noether Programme of the Deutsche Forschungsgemeinschaft; Deutscher; Akademischer Austauschdienst |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access / Hybrid Open-Access |
License (German): | CC-BY - Namensnennung 4.0 International |