# Cross-polytope

In geometry, a **cross-polytope**,^{[1]} **hyperoctahedron**, **orthoplex**,^{[2]} or **cocube** is a regular, convex polytope that exists in *n*-dimensions. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. The *n*-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ_{1}-norm on **R**^{n}:

The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a *n*-dimensional cross-polytope is a Turán graph *T*(2*n*, *n*).

The 4-dimensional cross-polytope also goes by the name **hexadecachoron** or **16-cell**. It is one of the six convex regular 4-polytopes. These 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.

The **cross polytope** family is one of three regular polytope families, labeled by Coxeter as *β _{n}*, the other two being the hypercube family, labeled as

*γ*, and the simplices, labeled as

_{n}*α*. A fourth family, the infinite tessellations of hypercubes, he labeled as

_{n}*δ*.

_{n}^{[3]}

For each pair of non-opposite vertices, there is an edge joining them. More generally, each set of *k*+1 orthogonal vertices corresponds to a distinct *k*-dimensional component which contains them. The number of *k*-dimensional components (vertices, edges, faces, ..., facets) in an *n*-dimensional cross-polytope is thus given by (see binomial coefficient):

There are many possible orthographic projections that can show the cross-polytopes as 2-dimensional graphs. Petrie polygon projections map the points into a regular 2*n*-gon or lower order regular polygons. A second projection takes the 2(*n*−1)-gon petrie polygon of the lower dimension, seen as a bipyramid, projected down the axis, with 2 vertices mapped into the center.

The vertices of an axis-aligned cross polytope are all at equal distance from each other in the Manhattan distance (L^{1} norm). Kusner's conjecture states that this set of 2*d* points is the largest possible equidistant set for this distance.^{[5]}

Cross-polytopes can be combined with their dual cubes to form compound polytopes: