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 ℓ

_{1}-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*(2

*n*,

*n*).

**Higher dimensions**

List of regular polytopes

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