Penrose tilings as a cross section of a $5$-dimensional regular tiling Could somebody explain to me how a penrose tiling , which is not periodic, can be a cross section of a regular tiling in $5$ dimensions, which is periodic? It does not make sense to me how a periodic tiling can produce an aperiodic cross-section.
Also, are there any examples of periodic $3$-dimensional tilings that can produce an aperiodic $2$-dimensional cross section? That would greatly help to visualize the previous question. Thanks!

To give a 3-dimensional example: Let $v$ be the vector $\left(1,\frac{\sqrt{2}}2,\frac{\pi}4\right)$ and $P$ be the plane through the origin orthogonal to $v$. Take the points in $\mathbb{Z}^3$ within distance $0.5$ of $P$, and project them down onto $P$. Then we get the following planar arrangement of points, where the edges are those of $\mathbb{Z}^3$ colored according to their direction:
     
Here is a GIF of this projection as the thickness of the "slab" of points we project grows from $0$ to $2.5$. (The file was too large to embed within this post, so I have linked to Imgur directly.)
