{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T14:12:01Z","timestamp":1780582321557,"version":"3.54.1"},"reference-count":6,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2011,12,12]],"date-time":"2011-12-12T00:00:00Z","timestamp":1323648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with p3m1, p4m, or p6m symmetry groups that have polyominoes or polyiamonds as fundamental domains. We display the algorithms\u2019 output and give enumeration tables for small values of n. This expands earlier works [1,2] and is a companion to [3].<\/jats:p>","DOI":"10.3390\/sym3040828","type":"journal-article","created":{"date-parts":[[2011,12,12]],"date-time":"2011-12-12T11:07:38Z","timestamp":1323688058000},"page":"828-851","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry"],"prefix":"10.3390","volume":"3","author":[{"given":"Hiroshi","family":"Fukuda","sequence":"first","affiliation":[{"name":"College of Liberal Arts and Sciences, Kitasato University, 1-15-1 Kitasato, Sagamihara, Kanagawa 252-0373, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chiaki","family":"Kanomata","sequence":"additional","affiliation":[{"name":"School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nobuaki","family":"Mutoh","sequence":"additional","affiliation":[{"name":"School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gisaku","family":"Nakamura","sequence":"additional","affiliation":[{"name":"School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Doris","family":"Schattschneider","sequence":"additional","affiliation":[{"name":"Mathematics Department PPHAC Moravian College, 1200 Main Street, Bethlehem, 18018-6650 PA, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2011,12,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s00373-007-0719-y","article-title":"A method to generate polyominoes and polyiamonds for tilings with rotational symmetry","volume":"23","author":"Fukuda","year":"2007","journal-title":"Graphs Comb."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/978-3-540-89550-3_7","article-title":"Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry","volume":"Volume 4535","author":"Fukuda","year":"2008","journal-title":"Computational Geometry and Graph Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"325","DOI":"10.3390\/sym3020325","article-title":"Polyominoes and polyiamonds as fundamental domains for isohedral tilings of crystal class D2","volume":"3","author":"Fukuda","year":"2011","journal-title":"Symmetry"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.cam.2004.05.002","article-title":"Tilings by polyomoinoes, polyhexes, and polyiamonds","volume":"174","author":"Rhoads","year":"2005","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1007\/PL00009442","article-title":"Isohedral polyomino tiling of the plane","volume":"21","author":"Keating","year":"1999","journal-title":"Discret. Comput. Geom."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1080\/00029890.1978.11994612","article-title":"The plane symmetry groups: Their recognition and notation","volume":"85","author":"Schattschneider","year":"1978","journal-title":"Am. Math. Mon."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/4\/828\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:58:18Z","timestamp":1760219898000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/4\/828"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12,12]]},"references-count":6,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2011,12]]}},"alternative-id":["sym3040828"],"URL":"https:\/\/doi.org\/10.3390\/sym3040828","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,12,12]]}}}