i-Block Transitive Tilings by Convex Pentagons

The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. There are 14 known distinct kinds of pentagons that admit tilings of the plane. Five of these known types admit tile-transitive tilings. The remaining 9 known types admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form a tile-transitive tiling. In this talk, we present some combinatorial results concerning pentagons that admit i-block transitive tilings. These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings. We will present the methods of this algorithm and the results of the computer searches so far.