Stripes on rectangular tilings
We consider a class of cut-and-project sets $Λ= Λ_F \times \zahl$ in the plane. Let $L=Λ+w\real$, $w\in\real^2$, 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 $[0,1] \times \real$.
math.MG↗