Some tilings of the plane whose singular points form a perfect set

Abstract:

Let $\mathcal {J}$ be a tiling of the plane such that for every tile $T$ of $\mathcal {J}$ there correspond a tile $T’$ of $\mathcal {J}$ (not necessarily unique) and an integer $k(T,T’)$ (depending on $T$ and $T’$), $2 < k$, such that $T$ meets $T’$ in $k(T,T’)$ connected components. Then the set of singular points of $\mathcal {J}$ is a nowhere dense, perfect set.