Hypercube

Example

For $d \ge 1$, the $d$-dimensional hypercube is $$C_d = [0,1]^d = \set{x \in \R^d : 0 \le x_i \le 1 \text{ for } i = 1, \dots, d}.$$ It is a convex polytope: it is the convex hull of the $2^d$ points with all coordinates in $\set{0, 1}$, and these are its vertices. Its facets are the $2d$ sets where one coordinate is fixed to $0$ or to $1$.

A face of $C_d$ is obtained by fixing some coordinates to $0$ or $1$ and leaving the others free. A $k$-dimensional face leaves $k$ coordinates free, so the number of $k$-faces is $$f_k(C_d) = 2^{d-k} \binom{d}{k}.$$ For $d = 3$ this gives the f-vector $(8, 12, 6)$ of the ordinary cube.

$C_d$ is a simple polytope. At a vertex $v$ the edges change exactly one coordinate of $v$, and there are $d$ coordinates, so every vertex lies in $d$ edges.