Face of a cube ($2^{n-k}\left(\genfrac{}{}{0ex}{}{n}{k}\right)$)

A $k$-dimensional face of the standard $n$-cube $[0,1{]}^n$ is obtained by choosing $k$ coordinates to vary freely and fixing each of the other $n-k$ coordinates to either zero or one. It is therefore isometric to $[0,1{]}^k$.

There arefaces of dimension $k$: choose the free coordinates and then their complementary endpoint values. For the three-dimensional cube, this gives eight vertices, twelve edges and six square faces. The full cube is the unique $n$-dimensional face. Some conventions also adjoin an empty face of dimension $-1$.

In a cubical complex, characteristic maps attach these faces to lower-dimensional cubes by isometries. Distinct abstract faces can be identified in the quotient complex, so a cell's face count in its model cube need not equal the number of distinct lower-dimensional cells in the whole complex.