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

Walton, James J.; Whittaker, Michael F.

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.

• 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.