Stripes on rectangular tilings
Akio Hizume, Yoshikazu Yamagishi

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 in the plane. Let , , be a countable union of parallel lines. Then either (1) is a discrete family of lines, (2) is a dense subset of , or (3) each connected component of the closure of is homeomorphic to .
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