In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope consist of all permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. (Note: some authors define a cross-polytope only as the boundary of this region.)
The n-dimensional cross-polytope can also be defined as the closed unit ball in the ℓ₁-norm on R:

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)}. In 3 dimensions it is an octahedron—one of the five regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these.
The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T(2n,n).

Higher dimensions

List of regular polytopes