Hypercube

A hypercube is the simplest center-symmetric polytope in each respective dimension, by facet count. Hypercubes are a direct generalization of squares and cubes to higher dimensions. The n-dimensional hypercube, or simply the n-hypercube, has 2n vertices, such that for every of n directions, half the vertices lie on one side, and half lie on the other. Its facets are the 2n hypercubes defined by the vertices on each side in each direction. Alternatively, one can construct each hypercube as the prism of the hypercube of the lower dimension.

Every hypercube can be made regular. As such, the hypercubes comprise one of the three infinite families of polytopes that exist in every dimension, the other two being the simplexes and the orthoplexes.

The dual of a hypercube is an orthoplex.

Elements
All of the elements of a hypercube are hypercubes themselves. The number of d-dimensional elements of an n-hypercube is given by the binomial coefficient 2n–dC(n, d). This is because for each choice of n–d of the hypercube’s n directions, and for each of the subsequent 2n–d choices of sides, the vertices on these sides define a unique d-dimensional simplex. In particular, an n-dimensional hypercube has 2n vertices and 2n facets, and its vertex figure is the simplex of the previous dimension.

Examples
Excluding the degenerate point, the hypercubes up to 10D are the following:

Measures

 * The circumradius of a D-dimensional orthoplex of unit edge length is given by $\sqrt{D}$/2.
 * The same hypercube's inradius is 1/2, regardless of dimension.
 * Its height from a facet to the opposite facet is twice the inradius, that is 1.
 * Its hypervolume is 1, regardless of dimension.
 * The angle between two facet hyperplanes is 90º, once again in all dimensions.