I made some movies of some of my favorite complexes: let be the -dimensional cube, and let be the edges around the origin, and let be the square face containing the edges and . Define a subcomplex consisting of the squares and all the squares in parallel to these. It turns out that is a surface with a lot of symmetries.

In particular is a torus in , and here is a quicktime movie of it spinning:

I’m particularly fond of this, as you can really see that four squares are coming together at each vertex (hence, it has zero curvature), and you can see the hole in the torus as it spins.

The complex is a genus five surface in , and here is a quicktime movie of it spinning:

I represented the extra dimensions with color—not that it helps much!