Hypercube

The hypercube is one of the three infinite families of regular polytopes that exist in every dimension (the other two are the orthoplex and the simplex). The D+1-dimensional hypercube can be constructed as the prism of the D-dimensional hypercube. They are dual to the orthoplices.

The number of N-dimensional elements in a D-dimensional hypercube is given by the coefficient of xn in the full expansion of (x+2)D, all of which are hypercubes of the appropriate dimension. In particular, a D-dimensional hypercube has 2D vertices and 2D facets (each shaped like a (D-1)-dimensional hypercube), with the vertex figure being the simplex of the previous dimension.

The hypercubes up to 10D are:


 * Dyad (1D)
 * Square (2D)
 * Cube (3D)
 * Tesseract (4D)
 * Penteract (5D)
 * Hexeract (6D)
 * Hepteract (7D)
 * Octeract (8D)
 * Enneract (9D)
 * Dekeract (10D)

Measures

 * The circumradius of a D-dimensional orthoplex of unit edge length is given by $\sqrt{D}$/2.
 * The same hypercube'si 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.