# Movies of some neat cubical complexes.

##### 2008-02-25 05:27:32 +0000personalmathematics

I made some movies of some of my favorite complexes: let $I^n$ be the $n$-dimensional cube, and let $e_1, \ldots, e_n$ be the $n$ edges around the origin, and let $e_i e_j$ be the square face containing the edges $e_i$ and $e_j$. Define a subcomplex $\Sigma^2_n \subset I^n$ consisting of the squares $e_1 e_2, e_2 e_3, \ldots, e_{n-1} e_n, e_n e_1$ and all the squares in $I^n$ parallel to these. It turns out that $\Sigma^2_n$ is a surface with a lot of symmetries.

In particular $\Sigma^2_4$ is a torus in $\R^4$, 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 $\Sigma^2_5$ is a genus five surface in $\R^5$, and here is a movie of it spinning:

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