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 the lower dimension.

Every orthoplex can be made regular; in fact, it’s rare for the term to be 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 dual of an orthoplex is a hypercube.

If a line is drawn between the vertices in each pair of opposite vertices, all the lines are perpendicular to one another. This is the origin of the name "cross-polytope."

Elements
All of the elements of an orthoplex are simplexes. The number of d-dimensional elements in an n-orthoplex is given by the the binomial coefficient 2d+1C(n, d+1). This is because any choice of d+1 vertices, no two opposite, define a unique d-dimensional simplex. In particular, an n-dimensional orthoplex has 2n vertices and 2n facets, each shaped like an (n–1)-dimensional simplex, with the vertex figure being the orthoplex of the previous dimension.

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

Measures

 * The circumradius of an n-orthoplex of unit edge length is $\sqrt{2}$/2, regardless of n.
 * This same orthoplex's inradius is given by 1/$\sqrt{2n}$.
 * Its height from a facet to the opposite facet is given by twice the inradius, that is 2/$\sqrt{2n}$.
 * Its hypervolume is given by $\sqrt{2^{n}}$/n!.
 * The angle between two facet hyperplanes is acos(2/n–1).