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)

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.

• Item Type: Articles
• Additional Information: This research was partially supported by EPSRC grant EP/R013691/1.
• Status: Early Online Publication
• Refereed: Yes
• 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: 1474-7480
• 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

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
300655		Applications of space filling curves to substitution tilings	Michael Whittaker	Engineering and Physical Sciences Research Council (EPSRC)	EP/R013691/1	M&S - Mathematics