An aperiodic hexagonal tile


Journal Article

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.

Full Text

Duke Authors

Cited Authors

  • Socolar, JES; Taylor, JM

Published Date

  • November 1, 2011

Published In

Volume / Issue

  • 118 / 8

Start / End Page

  • 2207 - 2231

Electronic International Standard Serial Number (EISSN)

  • 1096-0899

International Standard Serial Number (ISSN)

  • 0097-3165

Digital Object Identifier (DOI)

  • 10.1016/j.jcta.2011.05.001

Citation Source

  • Scopus