An aperiodic hexagonal tile |
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Authors: | Joshua ES Socolar |
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Institution: | a Physics Department, Duke University, Durham, NC 27514, United States b P.O. Box U91, Burnie, Tas. 7320, Australia |
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Abstract: | We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space-filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of n2a, where a sets the scale of the most dense lattice and n takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these tilings and a previous construction by Penrose. Alternative forms of the prototile enforce the local matching rules by shape alone, one using a prototile that is not a connected region and the other using a three-dimensional prototile. |
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Keywords: | Tiling Aperiodic Substitution Matching rules |
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