Stripes on rectangular tilings

TL;DR

This paper investigates the geometric structure of certain cut-and-project sets in the plane, classifying the possible configurations of lines formed by their unions and their topological properties.

Contribution

It provides a classification of the geometric and topological nature of unions of lines derived from cut-and-project sets in the plane.

Findings

L can be a discrete family of lines

L can be dense in the plane

Connected components of the closure of L are homeomorphic to [0,1]×R

Abstract

We consider a class of cut-and-project sets $\Lambda = \Lambda_F \times \zahl$ in the plane. Let $L=\Lambda+w\real$, $w\in\real^2$, be a countable union of parallel lines. Then either (1) $L$ is a discrete family of lines, (2) $L$ is a dense subset of $\real^2$, or (3) each connected component of the closure of $L$ is homeomorphic to $[0,1] \times \real$.