The topology of the minimal regular covers of the Archimedean tessellations
type: Item
instance of: scholarly article
title: The topology of the minimal regular covers of the Archimedean tessellations
zbMATH Open document ID: 1309.52007
author: Thierry Coulbois
author: Daniel Pellicer
author: M. Raggi
author: Camilo Ramírez
author: Ferrán Valdez
publication date: 2015-01-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.1518
review text: A map \(M=(\Gamma,i,S)\) is a 2-cell embedding \(i\) of a locally finite graph \(\Gamma\) on a surface \(S\). \textit{M. I. Hartley} [Discrete Comput. Geom. 21, No. 2, 289--298 (1999; Zbl 0927.52017)] provided a procedure to describe an arbitrary map \(M\) algebraically in terms of a regular cover \(\widetilde{M}=(\widetilde{\Gamma}, \widetilde{i},\widetilde{S})\) and subgroup of its authomorphisms. In the reviewed paper it is proved that if \(M=(\Gamma,i,\mathbb{E}^2)\) is a periodic map in Euclidian space with convex faces such that it contains at least two faces with different number of edges, and \(\widetilde{M}=(\widetilde{\Gamma}, \widetilde{i},\widetilde{S})\) is its minimal regular cover, then \(\widetilde{S}\) is homeomorphic to the topological surface obtained by gluing infnitely many torii along a ray, known as Loch Ness Monster.
Mathematics Subject Classification ID: 52B15
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 52C20
zbMATH DE Number: 6387844
zbMATH Keywords: Archimedean tessellation
zbMATH Keywords: Loch Ness monster
zbMATH Keywords: minimal regular cover
reviewed by: Anton Shutov
MaRDI profile type: Publication
OpenAlex ID: W2963340788
arXiv ID: 1210.1518
DOI: 10.1515/ADVGEOM-2014-0023
published in: advg
description: scientific article