The 14 infinite families of isotoxal tilings in the planes of constant curvature
type: Item
instance of: scholarly article
title: The 14 infinite families of isotoxal tilings in the planes of constant curvature
zbMATH Open document ID: 0924.52014
DOI: 10.1007/BF01876875
author: V. Pereyra
published in: Periodica Mathematica Hungarica
publication date: 1995-04-11T00:00:00Z
review text: By a plane \(\Pi\) we mean any complete, simply connected 2-dimensional Riemannian manifold of constant curvature. A tiling in the plane \(\Pi\) is a collection \({\mathcal T}=\{T_i:i\in I=\{1,2,\dots\}\}\) of closed topological disks which covers \(\Pi\) and of which the individual tiles \(T_i\) have disjoint interiors. We assume that intersection of any set of tiles of \({\mathcal T}\) is a connected (possibly empty) set. The vertices of \({\mathcal T}\) are the points which belong to three or more tiles, and the edges of \({\mathcal T}\) are the arcs into which the vertices partition the boundaries of the tiles. A tile is called an \(n\)-gon (although it is not necessarily a polygon whose edges are straight line segments) if it has \(n\) vertices and \(n\) edges on its boundary. We assume the absence of 2-gons in a tiling \({\mathcal T}\). If there is a motif or marking in each tile of \({\mathcal T}\) that tiling is called marked tiling. Associated with every (marked) tiling \({\mathcal T}\) is the group of all isometrics of \(\Pi\) which map \({\mathcal T}\) onto itself. This is known as the symmetry group of \({\mathcal T}\) and is denoted by Sym \({\mathcal T}\). A plane tiling \({\mathcal T}\) is called isotoxal if its symmetry group is transitive on the set of edges. If Sym \({\mathcal T}\) is transitive on the set of tiles (resp. vertices) a tiling \({\mathcal T}\) is called isohedral (resp. isogonal). A complete enumeration of all ``types'' of euclidean and spherical (marked) tilings whose symmetry groups are transitive on the edges is given by Grünbaum and Shepard. Using the method of combinatorial classification of fundamental domains for planar discontinuous groups it is possible to classify all such tilings not only in the euclidean plane and on the sphere but also in the hyperbolic plane. In this article we present the complete list of 14 infinite families of isotoxal marked tilings in the plane of constant curvature. Well known 30 euclidean and 26 s…
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 736865
zbMATH Keywords: 2-dimensional Riemannian manifold of constant curvature
zbMATH Keywords: isotoxal marked tilings
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 736865