Orthoplex

An orthoplex or cross-polytope is the simplest center-symmetric polytope in each respective dimension, by vertex count. The n-dimensional orthoplex, or simply the n-orthoplex, has 2n vertices lying in n opposite pairs, connected by each of the 2n (n–1)-simplices containing exactly one vertex from each pair. Alternatively, one can construct each orthoplex as the bipyramid of the orthoplex of a lower dimension.

Every orthoplex can be made regular; in fact, it’s rare for the term to Ben used to refer to non-regular shapes. As such, the orthoplexes comprise one of the three infinite families of regular polytopes that exist in every dimension, the other two being the simplexes and 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 orthoplexes 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).