After going through the following question on Penrose Tiling and reading de Bruijn's papers on the subject, I came accross Grünbaum and Shephardbook "Tilings and Patterns", p. 543, where they say that there are only two Penrose tilings with global 5-rotational symmetry. Where can one find a proof for that (that these are the only ones)?
Moreover, I then saw in Wikipedia the following picture, which somehow implies that there are more than two tilings with a global 5-rotational symmetry... so what is in fact wrong here? can it be that most of the tilings in this picture are do not have a global 5-rotational symmetry and it is only an illusion? Thank you, Thomas.
The list of patches in the wikipedia link are not Penrose tilings as considered by Grumbaum & Shephard, just one of them is. The former are obtained via the "cut-and-project" method, while the latter are restricted to tilings that can be obtained via inflation/deflation procedure described in the book.
In this context, the proof of the claim you ask for relies on the process of deflation: by deflating repeatedly in a 5-fold tiling you will eventually reach one of the two vertices with 5-fold symmetry (in the case of Kite and Dart tilings, either a sun or a star).