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The set of possible kit-and-dart Penrose tilings is uncountably infinite. It would be very helpful to have some natural probability distribution $\mu$ over this set; such a distribution would allow ...

Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...