Coincidences related to a dissection of the regular octagon

Behind this dissection of the regular octagon lies the combinatorial structure of the $2$-dimensional faces of the $4$-dimensional hypercube.

More generally, consider a $d$-dimensional hypercube. The vertices of this shape are given by sequences of length $d$ consisting of $0$s and $1$s. By projecting the point $e_k$, whose only nonzero coordinate is the $k$-th one, to $\left(\cos \frac{\pi (k-1)}{d}, \sin \frac{\pi (k-1)}{d}\right)$, the hypercube is projected onto a regular $2d$-gon. By appropriately choosing the 2-dimensional faces of this hypercube, we obtain a tiling of the regular $2d$-gon by rhombi. Below are illustrations for the cases $d=4$ and $d=5$.

There are multiple possible ways to choose the appropriate faces, but a standard choice is as follows: take all $2$-dimensional faces of the form

$$ \{(0,\dots,0,x,1,\dots,1,y,0,\dots,0)\mid 0\leq x,y\leq 1\}. $$

The figures above correspond to this particular choice. It is a fun puzzle to verify that the projections of these faces do not overlap and together form a tiling of the $2d$-gon.