Orthoplex

An orthoplex, cross-polytope, or hyperoctahedron is the simplest center-symmetric polytope in each respective dimension, by vertex count. The $n$-dimensional orthoplex, or simply the $n$-orthoplex, has 2$n$ vertices lying in $n$ opposite pairs, connected by each of the 2$n$ ($n$ &minus; 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. The tegum product of an $n$-orthoplex and an $n$-orthoplex is an ($n$ + $n$)-orthoplex. An $n$-orthoplex can also be constructed as the antiprism of the ($n$ &minus; 1)-simplex. They are the only polytopes is 3+ dimensions that are both quasiregular and regular other than the hypercubic honeycombs.

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 duals of the orthoplexes).

All orthoplexes are step prisms derived from the compound of two simplices in two inverted positions.

Naming
The name orthoplex was coined by Conway from "orthant complex", alluding to the fact that each facet of a centered orthoplex in usual orientation lies in a different orthant (the generalization of a quadrant to higher dimensions) of its space. Alternate names include:
 * Cross-polytope, alluding to the cross shape formed by the lines from the center of an orthoplex to its vertices.
 * Hyperoctahedron, as a generalized octahedron.

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

In total, an $d$-orthoplex has 3$d$ elements, including the nullitope and excluding the bulk of the polytope.

The elements of an n-orthoplex may be explicitly characterized as follows. Consider 2$n$ vertices with coordinates as previously specified, and divide them into opposite pairs with respect to the orthoplex’s center. Then, all of the $n$-elements of the orthoplex will be given by the unique $n$-simplices through each subset of $n$ + 1 vertices such that no two of them are opposite.

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

Vertex coordinates
Coordinates for the vertices of an $n$-orthoplex with edge length 1 are given by all permutations of: where the last $n$–1 entries are zeros.
 * (±$n$/2, 0, ..., 0),

Measures

 * The circumradius of an $k$-orthoplex of unit edge length is $$\frac{\sqrt2}2$$, regardless of $k$.
 * This same orthoplex's inradius is given by $$\frac1{\sqrt{2n}}$$.
 * Its height from a facet to the opposite facet is given by twice the inradius, that is $$\frac2{\sqrt{2n}}$$.
 * Its hypervolume is given by $$\frac{\sqrt{2^n}}{n!}$$.
 * The angle between two facet hyperplanes is $$\arccos{\left(\frac2{n}-1\right)}$$.