Orthoplex

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

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

The orthoplices up to 10D are:


 * Dyad (1D)
 * Square (2D)
 * Octahedron (3D)
 * Hexadecachoron (4D)
 * Triacontaditeron (5D)
 * Hexacontatetrapeton (6D)
 * Hecatonicosoctaexon (7D)
 * Diacosipentacontahexazetton (8D)
 * Pentacosidodecayotton (9D)
 * Chilliaicositetraxennon (10D)

Measures

 * The circumradius of a D-dimensional orthoplex of unit edge length is $\sqrt{2}$/2, regardless of dimension.
 * The same orthoplex's inradius can be given by $\sqrt{2D}$/2D.
 * Its height from a facet to the opposite facet is given by twice the inradius, that is $\sqrt{2D}$/D.
 * Its hypervolume is given by $\sqrt{2^{D}}$/D! (where ! means factorial).
 * The angle between two facet hyperplanes is acos(2/D-1).