<s>[INST] 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?
 [/INST] 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.
</s>