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 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’ 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 [3–5].
Highlights
Polyominoes and polyiamonds and their tiling properties have been the subject of research and recreation for more than 50 years [6,7,8,9,10]
We continue our investigations of a different question: how many different n-ominoes and n-iamonds can serve as fundamental domains for isohedral tilings of the plane with respect to a given plane symmetry group? In our earlier work [3,4,5] we addressed this question for symmetry groups with 3, 4, or 6-fold rotational symmetries
We have described computer algorithms that can enumerate and display isohedral tilings by n-omino or n-iamond tiles for given n in which the tiles are fundamental domains and the tilings have pmm, pmg, pgg and cmm symmetry
Summary
Polyominoes and polyiamonds and their tiling properties have been the subject of research and recreation for more than 50 years [6,7,8,9,10]. Rotations of 180◦ about the black and white 2-fold centers on the edges of a polyomino tile T in Tn fills out a strip tiled by copies of T and bounded by the two reflection axes placed at the outset Repeated reflections of this strip in the reflection axes on its boundary produces an isohedral tiling T having T as a fundamental domain for the pmg group G.
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