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. The prism product of an m-dimensional hypercube and an n-dimensional hypercube is an m+n dimensional hypercube.

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 duals of the hypercubes).

The hypercube is also called the measure polytope, because a hypercube with unit edge length has a unit hypervolume. As such, it can be used to “measure” n-dimensional space, like a grid.

Hypercubes can always tile their respective spaces, forming the hypercubic honeycombs. This is in contrast to the other regulars, which create no new tilings other than the simplicial triangular tiling.

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.

In total, a hypercube has 3n elements, including the nullitope and excluding the bulk of the polytope.

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

Measures

 * The circumradius of an n-dimensional hypercube of unit edge length is given by $\sqrt{2}$/2.
 * Its inradius is 1/2, regardless of n.
 * Its height from a facet to the opposite facet is twice the inradius, that is 1.
 * Its hypervolume is 1, regardless of n.
 * The angle between two facet hyperplanes is 90°, regardless of n.