A cube's vertices can be trivially divided into two sets, each forming the vertices of a regular tetrahedron. I was wondering if such a construction could be generalized to higher dimensions.
I've found here a set of coordinates for the $7$-simplex, which imply that it can be inscribed in a $7$-hypercube. (There’s a pretty cool connection to projective geometry there). So, I'm only aware of the cases $n=3,7$ at the moment.
For a particular dimension $n$, we could check the cases for all possible edge lengths of the $n$-simplex: if we choose a vertex and an edge length, we can check whether our remaining points allow us to build the $(n-1)$-simplex of correct edge length we need to build our $n$-simplex. I've checked $n=4$, doesn't seem to work. But this approach doesn't seem generalizable.
We could also try to construct examples by playing around with the coordinates of the hypercube $(\pm1,\pm1,\ldots,\pm1)$, but so far I haven't found anything.
So, for which dimensions can a regular $n$-simplex be built from the vertices of a regular $n$-hypercube?