More ways to tile with only one shape polygon
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Monotiles with arbitrarily large isohedral number do exist;
The additional topological possibilities afforded in 3D allow construction of a simply connected monotile with a rule enforced by shape only, which is impossible for the hexagonal parquet in 2D;
The precise statement of the tiling problem matters— whether color matching rules are allowed; whether multiply connected shapes are allowed; whether spacefilling is required as opposed to just maximum density. So what about the quest for thek = ∞ monotile? Schmitt.
KeywordsMatching Rule Density Tiling Black Edge Mathematical Intelligencer Figure Tile Edge
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