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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.

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  • $\begingroup$ Just download the book Penrose Tiles to Trapdoor Ciphers, Cambridge University Press $\endgroup$
    – BooleanCoder
    Apr 19, 2021 at 4:18
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    $\begingroup$ That is the Martin Gardner article that provides no proof. $\endgroup$
    – W. Cadegan-Schlieper
    Apr 19, 2021 at 14:06
  • $\begingroup$ Branko Grünbaum and Geoffrey Shephard's "Tilings and Patterns" may be a helpful source. Especially, around page $543$. If you open a free account you can borrow the book at archive.org/details/isbn_0716711931/page/542/mode/2up $\endgroup$
    – krazy-8
    Jun 3, 2021 at 14:12
  • $\begingroup$ Okay that looks promising. $\endgroup$
    – W. Cadegan-Schlieper
    Jun 4, 2021 at 18:39
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    $\begingroup$ ...it leaves a proof as an exercise. Does say that it can be based on making the closed curves smaller and smaller through composition though. $\endgroup$
    – W. Cadegan-Schlieper
    Jun 5, 2021 at 20:24

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