The 14 infinite families of isotoxal tilings in the planes of constant curvature
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 spherical isotoxal marked tilings appear as starting examples in these 14 families.
2-dimensional Riemannian manifold of constant curvature
scientific article; zbMATH DE number 736865