Abstract
AbstractWe 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.
Highlights
The fact that periodically arranged structures can be enforced by local rules is familiar to everyone
It was a great surprise to crystallographers in the 1980s when Dan Shechtman discovered a metal alloy whose diffraction pattern implied a great deal of structural order but had rotational symmetry precluding periodicity [20]
(1) We denote the collection of standard R1-tilings by Ωa because these tilings are seen to be mutually local derivable (MLD; see [2]) to the arrowed hex tilings [1]
Summary
The fact that periodically arranged structures can be enforced by local rules is familiar to everyone. The patterns of tile parities (given by labeling hexagons only with the information of whether the tile of Figure 1 or its mirror image is used) are very different, and closely follow the structure of the R1-edges, which are forced to carry the same parities across them. This observation leads to a remarkably simple proof of aperiodicity, presented in full detail in §2 and which we briefly outline now. The substitutional hull (modulo charge-flip) factors almost everywhere 1-to-1 to its maximal equicontinuous factor and so has pure point dynamical spectrum and the structure of a regular model set
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Journal of the Institute of Mathematics of Jussieu
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
