An aperiodic tile with edge-to-edge orientational matching rules

Walton, J. J. and Whittaker, M. F. (2021) An aperiodic tile with edge-to-edge orientational matching rules. Journal of the Institute of Mathematics of Jussieu, (doi: 10.1017/S1474748021000517) (Early Online Publication)

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We present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar–Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.

Item Type:Articles
Additional Information:This research was partially supported by EPSRC grant EP/R013691/1.
Status:Early Online Publication
Glasgow Author(s) Enlighten ID:Whittaker, Professor Mike
Authors: Walton, J. J., and Whittaker, M. F.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of the Institute of Mathematics of Jussieu
Publisher:Cambridge University Press
ISSN (Online):1475-3030
Published Online:18 October 2021
Copyright Holders:Copyright © 2021 The Authors
First Published:First published in Journal of the Institute of Mathematics of Jussieu 2021
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
300655Applications of space filling curves to substitution tilingsMichael WhittakerEngineering and Physical Sciences Research Council (EPSRC)EP/R013691/1M&S - Mathematics