In geometry, a cross-polytope,[1] hyperoctahedron, orthoplex,[2] staurotope,[3] or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.
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 n-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ1-norm on Rn: In 1 dimension the cross-polytope is simply the line segment [−1, +1], in 2 dimensions it is a square (or diamond) with vertices {(±1, 0), (0, ±1)}.
This can be generalised to higher dimensions with an n-orthoplex being constructed as a bipyramid with an (n−1)-orthoplex base.
The hypervolume of the n-dimensional cross-polytope is 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): The extended f-vector for an n-orthoplex can be computed by (1,2)n, like the coefficients of polynomial products.
[7] Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), βpn = 2{3}2{3}...2{4}p, or ... Real solutions exist with p = 2, i.e. β2n = βn = 2{3}2{3}...2{4}2 = {3,3,..,4}.
An orthogonal projection can be defined that maps all the vertices equally-spaced on a circle, with all pairs of vertices connected, except multiples of n. The regular polygon perimeter in these orthogonal projections is called a petrie polygon.