In a Penrose-like tiling, every patch occurs an infinite number of times, but the larger a patch is, the less frequently it occurs.
I think there are tilings where a distant patch is rotated by an amount that doesn't rationally divide the circle, in which case if you can measure the orientation to arbitrary precision you can find your location within an arbitrarily large area.
I think this is tied to the notion of finite local complexity: https://tilings.math.uni-bielefeld.de/glossary/flc/
