The 37 combinatorial types of minimal, non-transitive, equivariant tilings of the Euclidean plane
type: Item
instance of: scholarly article
title: The 37 combinatorial types of minimal, non-transitive, equivariant tilings of the Euclidean plane
zbMATH Open document ID: 0628.05021
DOI: 10.1016/0012-365X(86)90007-5
author: Andreas W. M. Dress
author: Rudolf Scharlau
published in: Discrete Mathematics
publication date: 1986-01-01T00:00:00Z
review text: A tiling of the Euclidean plane is called minimal non-transitive if its symmetry group has exactly two orbits on the vertices, edges, and tiles. A complete enumeration of all equivariant types of minimal non-transitive tilings having identical generalized Schläfli symbol is given. The authors prove the Theorem: There exist precisely 37 equivalence classes of minimal non-transitive equivariant tilings (\({\mathcal T},\Gamma)\) of the Euclidean plane. In all cases, \({\mathcal T}\) can be chosen in such a way that \(\Gamma\) equals the full symmetry group Iso(\({\mathcal T})\). In 13 cases, Iso(\({\mathcal T})\) is the full combinatorial automorphism group Aut(\({\mathcal T})\), so there exist up to isomorphism 13 combinatorial tilings T such that Aut(T) acts minimal non-transitively, and admitting a representation in the Euclidean plane in such a way that Aut(T) acts isometrically.
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 4021161
zbMATH Keywords: non-transitive plane tilings
zbMATH Keywords: enumeration
zbMATH Keywords: minimal non-transitive tilings
zbMATH Keywords: identical generalized Schläfli symbol
MaRDI profile type: Publication
cites work: Q3911038
cites work: Regular polytopes and equivariant tessellations from a combinatorial point of view
cites work: Q3697822
cites work: The eighty-one types of isohedral tilings in the plane
cites work: The Ninety-One Types of Isogonal Tilings in the Plane
cites work: Isotoxal tilings
cites work: A hierarchy of classification methods for patterns*
cites work: The 2-homeotoxal tilings of the plane and the 2-sphere
cites work: The enumeration of normal 2-homeohedral tilings
cites work: 2-isohedral triangulation
full work available at URL: 0012-365x(86)90007-5
OpenAlex ID: W1998421418
description: scientific article; zbMATH DE number 4021161