Movies of some neat cubical complexes.

 February 25, 2008 5:27 AM personal mathematics

I made some movies of some of my favorite complexes: let In be the n-dimensional cube, and let e1,,en be the n edges around the origin, and let eiej be the square face containing the edges ei and ej. Define a subcomplex Σ2nIn consisting of the squares e1e2,e2e3,,en1en,ene1 and all the squares in In parallel to these. It turns out that Σ2n is a surface with a lot of symmetries.

In particular Σ24 is a torus in R4, and here is a 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 Σ25 is a genus five surface in R5, and here is a movie of it spinning:

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