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

Walton, James J.; Whittaker, Michael F.

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


Authors

JAMES WALTON JAMES.WALTON@NOTTINGHAM.AC.UK
Assisstant Professor

Michael F. Whittaker



Abstract

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.

Citation

Walton, J. J., & Whittaker, M. F. (2021). An aperiodic tile with edge-to-edge orientational matching rules. Journal of the Institute of Mathematics of Jussieu, 22(4), 1727-1755. https://doi.org/10.1017/S1474748021000517

Journal Article Type Article
Acceptance Date Sep 22, 2021
Online Publication Date Oct 18, 2021
Publication Date Oct 18, 2021
Deposit Date Oct 20, 2021
Publicly Available Date Oct 22, 2021
Journal Journal of the Institute of Mathematics of Jussieu
Print ISSN 1474-7480
Electronic ISSN 1475-3030
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 22
Issue 4
Pages 1727-1755
DOI https://doi.org/10.1017/S1474748021000517
Keywords General Mathematics
Public URL https://nottingham-repository.worktribe.com/output/6504589
Publisher URL https://www.cambridge.org/core/journals/journal-of-the-institute-of-mathematics-of-jussieu/article/an-aperiodic-tile-with-edgetoedge-orientational-matching-rules/235CF72DFC014CCE924F8965B72FE433
Additional Information Copyright: © The Author(s), 2021. Published by Cambridge University Press; License: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.; Free to read: This content has been made available to all.

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