Nonexpansive directions in the Jeandel-Rao Wang shift

TL;DR

This paper identifies the set of slopes of nonexpansive directions in the Jeandel-Rao Wang shift, showing it as a topological invariant and analyzing the combinatorial structure of related tilings using irrational rotations.

Contribution

It determines the exact set of nonexpansive directions for the Jeandel-Rao Wang shift and describes their combinatorial structure in terms of irrational rotations.

Findings

Set of nonexpansive directions includes four specific slopes involving the golden mean.

The set serves as a topological invariant distinguishing the Jeandel-Rao shift.

Descriptions of tiling structures along nonexpansive directions using irrational rotations.

Abstract

We show that $\{0,\varphi+3,-3\varphi+2,-\varphi+\frac{5}{2}\}$ is the set of slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao Wang shift, where $\varphi=(1+\sqrt{5})/2$ is the golden mean. This set is a topological invariant allowing to distinguish the Jeandel-Rao Wang shift from other subshifts. Moreover, we describe the combinatorial structure of the two resolutions of the Conway worms along the nonexpansive directions in terms of irrational rotations of the unit interval. The introduction finishes with pictures of nonperiodic Wang tilings corresponding to what Conway called the cartwheel tiling in the context of Penrose tilings. The article concludes with open questions regarding the description of octopods and essential holes in the Jeandel-Rao Wang shift.