Recently (despite the one year of active participation) I noticed that our nice Penrose background consists of identical blocks with visible joints between them:
Of course, it is not a critical problem. But sometimes I become a perfectionist so I ask the question: is it possible to correct this? Are we interested to correct this?
This Penrose tiling was generated by Mathematica. See the Verbeia's post for the details. Unfortunately this nice tiling was cropped roughly with visible imperfections.
Szabolcs had already shown that we can obtain the periodicity in one direction with several deformed tiles. In addition, I noticed that one can find a two dimensional periodic approximant in the center of Verbeia's generator:
The horizontal periodicity is perfect which is enough. It has also the vertical periodicity with several mismatches of directions of tiles (near the bottom). It is not possible to do the periodic approximant without mismatches at all. If you are interesting in the theory of the periodic approximants you can read O. Entin-Wohlman, M. Kléman, and A. Pavlovitch, Penrose tiling approximants, J. Phys. France 49, 587-598 (1988) (link) or similar articles.
Clickable svg files for the periodic approximants of different sizes: