An aperiodic hexagonal tile

Details An aperiodic hexagonal tile An aperiodic hexagonal tile

2010-03-22

arxiv.org/abs/1003.4279v2

Journal of Combinatorial Theory, Series A 118 (2011) pp. 2207-2231

Mathematics

Combinatorics

32 pages, 24 figures; submitted to Journal of Combinatorial Theory Series A. Version 2 is a major revision. Parts of Version 1

Scientific paper

10.1016/j.jcta.2011.05.001

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 $2^n a$, 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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