A plane tiling by the copies of a polyomino is called isohedral if every pair of copies in the tiling has a symmetry of the tiling that maps one copy to the other. We show that, for every $n$-omino (i.e., polyomino consisting of n cells), the number of non-equivalent isohedral tilings generated by 90 degree rotations, so called p4-tilings or quarter-turn tilings, is bounded by a constant (independent of n). The proof relies on the analysis of the factorization of the boundary word of a polyomino.
@InProceedings{amano_et_al:LIPIcs.ISAAC.2017.5, author = {Amano, Kazuyuki and Haruyama, Yoshinobu}, title = {{On the Number of p4-Tilings by an n-Omino}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {5:1--5:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.5}, URN = {urn:nbn:de:0030-drops-82498}, doi = {10.4230/LIPIcs.ISAAC.2017.5}, annote = {Keywords: polyomino, plane tiling, isohedral tiling, word factorization} }
Feedback for Dagstuhl Publishing