Publications [#248333] of Joshua Socolar
- Socolar, JES; Taylor, JM, An aperiodic hexagonal tile,
Journal of Combinatorial Theory, Series A, vol. 118 no. 8
pp. 2207-2231, Elsevier BV, Orlando, FL, USA, ISSN 0097-3165 [doi]
(last updated on 2019/06/16)
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 2na, 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. © 2011 Elsevier Inc.
Aperiodic, Matching rules, Substitution, Tiling