{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:10:38Z","timestamp":1760220638301,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2011,6,9]],"date-time":"2011-06-09T00:00:00Z","timestamp":1307577600000},"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 pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are members of the crystal class D2 among the 17 two-dimensional symmetry groups [2]. We display the algorithms\u2019 output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups p3, p31m, p3m1, p4, p4g, p4m, p6, and p6m [3\u20135].<\/jats:p>","DOI":"10.3390\/sym3020325","type":"journal-article","created":{"date-parts":[[2011,6,9]],"date-time":"2011-06-09T15:42:46Z","timestamp":1307634166000},"page":"325-364","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D2"],"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":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chiaki","family":"Kanomata","sequence":"additional","affiliation":[{"name":"School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nobuaki","family":"Mutoh","sequence":"additional","affiliation":[{"name":"School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gisaku","family":"Nakamura","sequence":"additional","affiliation":[{"name":"School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Doris","family":"Schattschneider","sequence":"additional","affiliation":[{"name":"Mathematics Department PPHAC Moravian College, 1200 Main Street Bethlehem, 18018-6650 PA, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2011,6,9]]},"reference":[{"key":"ref_1","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."},{"key":"ref_2","unstructured":"Coxeter, H.S.M., and Moser, W.O.J. (1965). Generators and Relations for Discrete Groups, Springer-Verlag."},{"key":"ref_3","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_4","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":"4535","author":"Fukuda","year":"2008","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Fukuda, H., Mutoh, N., Nakamura, G., and Schattschneider, D. (2011, June 02). Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry. Available online: http:\/\/arnetminer.org\/viewpub.do?pid=2855920.","DOI":"10.3390\/sym3040828"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Golomb, S.W. (1994). Polyominoes: Puzzles, Patterns, Problems, and Packings, Princeton University Press. [2nd ed.].","DOI":"10.1515\/9780691215051"},{"key":"ref_7","unstructured":"Gr\u00fcnbaum, B., and Shephard, G. (1987). Tilings and Patterns, W.H. Freeman."},{"key":"ref_8","unstructured":"Martin, G.E. (1991). Polyominoes: A guide to Puzzles and Problems in Tiling, Mathematical Association of America."},{"key":"ref_9","unstructured":"Myers, J. (2011, June 02). Polyomino, polyhex and polyiamond tiling. Available online: www.srcf.ucam.org\/jsm28\/tiling\/."},{"key":"ref_10","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_11","unstructured":"In [10], G. Rhoads has used the term \u201cfundamental domain\u201d in a non-traditional way; what he calls a fundamental domain is more commonly called a unit cell, or translation unit, which is a smallest patch that can tile the plane using only translations. He has investigated the question of which polyominoes, polyhexes, and polyiamonds can serve as translation units for isohedral tilings. In many cases, these isohedral tilings will not have a polyomino or polyiamond fundamental domain in the traditional sense."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/2\/325\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:56:21Z","timestamp":1760219781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/2\/325"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,6,9]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2011,6]]}},"alternative-id":["sym3020325"],"URL":"https:\/\/doi.org\/10.3390\/sym3020325","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2011,6,9]]}}}