The enumeration of normal 2-homeohedral tilings (Q1062287)

A tiling of the Euclidean plane is called homeohedral if all the tiles form one transitivity class under topological self-transformations of the plane that map the tiling onto itself. A tiling is called 2-homeohedral if the tiles form two transitivity classes under this same group of transformations. In Section 1 we find basic results on tilings, and particularly on homeohedral tilings. In Section 2 the authors present the fundamental diophantine relation between the valences of vertices and the numbers of sides of the two types of tiles in a 2-homeohedral tiling. Section 3 is concerned with methods of determining the 2-homeohedral tilings. The authors enumerate the 508 types of 2-homeohedral tilings known at the present time and illustrate them by isohedral representatives.