Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

doi:10.3390/sym3040828

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Symmetry 2011, 3(4), 828-851; doi:10.3390/sym3040828

Article

Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

Hiroshi Fukuda^1,* , Chiaki Kanomata², Nobuaki Mutoh², Gisaku Nakamura² and Doris Schattschneider³

¹ College of Liberal Arts and Sciences, Kitasato University, 1-15-1 Kitasato, Sagamihara, Kanagawa 252-0373, Japan ² School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan ³ Mathematics Department PPHAC Moravian College, 1200 Main Street, Bethlehem, 18018-6650 PA, USA

* Author to whom correspondence should be addressed.

Received: 4 August 2011; in revised form: 29 November 2011 / Accepted: 2 December 2011 / Published: 12 December 2011

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Abstract: 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’ output and give enumeration tables for small values of n. This expands earlier works [1,2] and is a companion to [3].

Keywords: polyominoes; polyiamonds; isohedral tilings; two-dimensional symmetry groups; fundamental domains

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MDPI and ACS Style

Fukuda, H.; Kanomata, C.; Mutoh, N.; Nakamura, G.; Schattschneider, D. Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry. Symmetry 2011, 3, 828-851.

AMA Style

Fukuda H., Kanomata C., Mutoh N., Nakamura G., Schattschneider D. Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry. Symmetry. 2011; 3(4):828-851.

Chicago/Turabian Style

Fukuda, Hiroshi; Kanomata, Chiaki; Mutoh, Nobuaki; Nakamura, Gisaku; Schattschneider, Doris. 2011. "Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry." Symmetry 3, no. 4: 828-851.

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