Cross-polytope

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In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. 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 ℓ1-norm on Rn, those points x = (x1, x2..., xn) satisfying | x 1 | + | x 2 | + ⋯ + | x n | ≤ 1. {\displaystyle |x_{1}|+|x_{2}|+\cdots +|x_{n}|\leq 1.} An n-orthoplex can be constructed as a bipyramid with an (n−1)-orthoplex base. The cross-polytope is the dual polytope of the hypercube. The vertex-edge graph of an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph ).