= Face of a cube
{title2=$2^{n-k}\binom nk$}
= Faces of a cube
{synonym}
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 are
$$
2^{n-k}\binom nk
$$
faces 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.
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