Edge-to-Edge Tilings with Vertex, Edge and Tile Transivity Properties

Edge-to-edge tilings with vertex, edge and tile transitivity properties An (a, b, c) tiling T is an a-isogonal, b-isotoxal and c-isohedral tiling, that is, T forms, respectively, a orbits of vertices, b orbits of edges and c orbits of tiles under its symmetry group. This study gives a method of constructing an (a, b, c) tiling of the Euclidean (E 2 ), hyperbolic (H2 ) or spherical (S 2 ) plane. The method allows for a complete enumeration of the (a, 2, c) tilings of E 2 and S 2 as well as a listing of (a, 3, c) tilings of E 2 derived from tilings by regular polygons, which will be presented. Moreover, classes of (1 , b, c) tilings of E 2 , H2 and S 2 will be shown, particularly those derived from [p n ] tilings