Penrose tilings as a cross section of a $5$-dimensional regular tiling

The key to this is that the projection is onto a 2 dimensional space which sits at an angle of "irrational slope" with respect to the period lattice of the tiling

The following picture from the Tiling Encyclopedia should emphasize well what happens, it is a 1-dimensional projection of a lattice in 2 dimensions.

Note that the key is the fact that since the line on which you project has an irrational slope with respect to $\mathbb Z^2$, it is impossible for the orange strip to have any period. Any period of the lattice will take some points in the orange strip, outside of the strip. And this leads to the aperiodicity of the Fibonacci tiling. Exactly the same thing happens with the Penrose.

P.S. The picture shows for me, if it doesn't load, here is the link to the Tiling Encyclopedia:

Fibonacci Pic

More About Fibonacci