Penrose tiling (PT) is well known for its non-periodic pattern. But I am just wondering if it is possible at all to create a Moiré pattern that possesses a periodic pattern by combing two or more than two Penrose tiling.
There is a youtube video shows Roger Penrose using two paper sheets with PT on a projector and he got a nice crossing lines. https://www.youtube.com/watch?v=th3YMEamzmw&t=2120s The limited size of the paper sheet he used does not allow one to know whather those patterns are repeated periodically or not.
Penrose tilings are covered in a recent Veritasium episode. Check this part: https://youtu.be/48sCx-wBs34?t=834.
The lines mentioned in the video are very similar to the lines that appear in the Moiré pattern. Both of them have sets of parallel lines with short and long gaps.
It is not a mathematical proof for your question, but the similarity is quite interesting.
I created an online version of the Penrose tiling Moiré pattern demonstration, you can check and play with it here: https://csfulop.github.io/PenroseMoire/?x=212?y=0?r=0