In the Wikipedia page for the Penrose tiling, the following is mentioned:
Conway and Penrose proved that whenever the colored curves on the P2 or P3 tilings close in a loop, the region within the loop has pentagonal symmetry, and furthermore, in any tiling, there are at most two such curves of each color that do not close up.[36]
Unfortunately, the reference is to a Martin Gardner article that provides no proof, so I'm curious as to what the proof here is.