Some 2-isohedral tilings of the plane (Q807981)

A 2-isohedral tiling is one in which the tiles form two transitivity classes under the symmetry group. The author considers all the 2- isohedral tilings of the plane when (i) all tiles have four edges and (ii) four tiles meet at each vertex (tilings of topological type \([4^ 4])\). Isohedral tilings are described by \textit{B. Grünbaum} and \textit{G. C. Shephard} [Tilings and Patterns (1987; Zbl 0601.05001)]. Furthermore A. W. M. Dress and D. Huson [\textit{A. W. M. Dress} and \textit{D. Huson}, Geom. Dedicata 24, 295-310 (1987; Zbl 0634.52011); \textit{D. Huson}, Diplomarbeit Bielefeld (1986)] have classified all the 2-isohedral tilings of the plane using Delaney symbols. Using adjacency symbols like Grünbaum and Shephard the author describes and classifies the 2-isohedral tilings of the plane of type \([4^ 4]\). He can show that there are exactly 69 such tilings. The shape of all these possible tiles and special properties of them are illustrated by figures too. In Huson's list of all the 2-isohedral tilings of the plane there are precisely the same 69 tilings of the considered type. The author shows connections between his listed tilings. At the end he describes which of his tilings can be constructed by dissecting an isohedral tiling.