What's wrong with this Penrose pattern?

Solution 1:

The Wikipedia article states "Therefore, a finite patch cannot differentiate between the uncountably many Penrose tilings, nor even determine which position within the tiling is being shown". Thus they could be the same tiling but differently centered.

In particular, aside from the two dimensional shifting of Penrose tilings, if a higher dimensional lattice is shifted before projecting to two dimensions, then a similar situation holds.

Solution 2:

I believe your constant $\gamma$ to be essentially the same as the constant $y$ in de Bruijn 1981 cited below. I don't use your exact projection matrices though like you I do use matrices involving sines and cosines of multiples of $2\pi/5$. I ran into similar issues, and for me, setting $y_i=\epsilon$ for $\epsilon>0$ as small as I could represent it got rid of the forbidden intersections like the ST intersection. Somewhere, distantly, in my tiling is an ST intersection but I regard that as approximation error and simply work to keep it distant.

So I recommend $\gamma_i=\epsilon$. It may not work because your matrices are in a different basis from mine, but I bet some straightforward combination of $\pm\epsilon$ will do the trick for you.

Additionally if it would be helpful I can dig through my notes to get my derivation.

de Bruijn, N. G. "Algebraic theory of Penrose's non-periodic tilings of the plane. I, II: dedicated to G. Pólya." Indagationes Mathematicae 43.1 (1981): 39-66.