Movies of some neat cubical complexes.

 February 25, 2008 personal mathematics

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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 \mathbb{R}^4 , and here is a movie of it spinning: