Sets of tiles with a prescribed number of tilings
A tile is a closed subset of \(E^ 2\) homeomorphic to the unit disk. A tiling T is a family of tiles with pairwise disjoint interiors which cover \(E^ 2\). A set S of prototiles is a finite set of incongruent tiles. S admits a tiling if there is a tiling T for which every element of T is congruent to some element of S. If S contains k elements the tiling is k-hedral. If S admits r incongruent tilings then it is r- morphic. The author shows that there are k-hedral r-morphic tilings for every \(k\geq 2\) and \(r\geq 1\). The construction of the author's examples derives from tilings by copies of a large rectangle and copies of a small rectangle having a common dimension and arrayed in periodic strips, each shifted from the next by the same length. These rectangles are furnished with small keys, keyholes and lids as required.
sets of tiles with a prescribed number of tilings
scientific article; zbMATH DE number 4021160