What's wrong with this Penrose pattern? I programmed the Penrose tiling by projecting a portion of 5D lattice to 2D space, by the "cut and project" method described in

*

*Quasicrystals: projections of 5-D lattice into 2 and 3 dimensions, H. Au-Yang and J. Perk.

*Generalised 2D Penrose tilings, A. Pavlovitch and M. Kléman

The orthonormal basis is chosen as
$$
M=\sqrt{\frac{2}{5}} \begin{bmatrix}
\cos 0 & \cos \frac{2\pi}{5} & \cos \frac{4\pi}{5}& \cos \frac{6\pi}{5}& \cos \frac{8\pi}{5}  \\
\sin 0 & \sin \frac{2\pi}{5} & \sin \frac{4\pi}{5}& \sin \frac{6\pi}{5}& \sin \frac{8\pi}{5}  \\
\cos 0 & \cos \frac{4\pi}{5} & \cos \frac{8\pi}{5}& \cos \frac{12\pi}{5}& \cos \frac{16\pi}{5}  \\
\sin 0 & \sin \frac{4\pi}{5} & \sin \frac{8\pi}{5}& \sin \frac{12\pi}{5}& \sin \frac{16\pi}{5}  \\
\frac{1}{\sqrt {2}} & \frac{1}{\sqrt {2}}  & \frac{1}{\sqrt {2}} & \frac{1}{\sqrt {2}}  & \frac{1}{\sqrt {2}}\\
\end{bmatrix}
$$
Each row presents a basis vector, i.e.
$$
M_i\cdot M_j=0, \;\;\textrm{for } i<j.$$
and
$$||M_i||=1,  \;\;\textrm{for } 1\leq i \leq 5. 
$$
$M$ consists of the parallel operator (representing the physical space)
$$
A=\begin{bmatrix}
M_1\\
M_2 \\
\end{bmatrix}=
\begin{bmatrix}
\cos 0 & \cos \frac{2\pi}{5} & \cos \frac{4\pi}{5}& \cos \frac{6\pi}{5}& \cos \frac{8\pi}{5}  \\
\sin 0 & \sin \frac{2\pi}{5} & \sin \frac{4\pi}{5}& \sin \frac{6\pi}{5}& \sin \frac{8\pi}{5}  \\
\end{bmatrix}
$$
and the perpendicular operator
$$
B=\begin{bmatrix}
M_3\\
M_4 \\
M_5 \\
\end{bmatrix}=\begin{bmatrix}
\cos 0 & \cos \frac{4\pi}{5} & \cos \frac{8\pi}{5}& \cos \frac{12\pi}{5}& \cos \frac{16\pi}{5}  \\
\sin 0 & \sin \frac{4\pi}{5} & \sin \frac{8\pi}{5}& \sin \frac{12\pi}{5}& \sin \frac{16\pi}{5}  \\
\frac{1}{\sqrt {2}} & \frac{1}{\sqrt {2}}  & \frac{1}{\sqrt {2}} & \frac{1}{\sqrt {2}}  & \frac{1}{\sqrt {2}}\\
\end{bmatrix}
$$
The 5D lattice points are integer combinations of basis such as
$$ 
p=i \begin{bmatrix}
1\\
0\\
0\\
0\\
0\\
\end{bmatrix}
+ j\begin{bmatrix}
0\\
1\\
0\\
0\\
0\\
\end{bmatrix}
+\dots, \;\; i,j,\dots \in \mathbb{Z}
$$
A 5D cube (centered at origin) is projected into 3D as polytope
$$
v'= B v, \;\; v\in hypercube
$$
so that I can check whether a $p$ is inside this polytope (20 faces). This is called "cutting" the 5D lattice points.
The resultant 2d projection $Ap$ is

Everything works fine, however, my result differs from the "standard" one (e.g. in wiki page) as follows

Is this a mistake or an alternative view of the same tiling?
Finally, I find this image (from Vertex Frequencies in Generalized Penrose Patterns, by E. Zobetz and A. Preisinger)

where the center of standard tiling exhibits the "S" pattern, while the center of my version has the "ST" pattern. But what does it mean exactly?

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.
