hyperoctahedron

A hyperoctahedron, also known as a cross-polytope or n-orthoplex, is a type of regular polytope in n-dimensional Euclidean space. It is the dual of the hypercube. A hyperoctahedron is defined by the absolute values of its coordinates summing to a constant. For example, in 2D, the hyperoctahedron is a square, and in 3D, it is an octahedron.

The general form of a hyperoctahedron in n dimensions is defined by the equation |x1| + |x2| + ... + |xn| <= 1. The vertices of a hyperoctahedron are located at positions (±1, 0, ..., 0), (0, ±1, ..., 0), ..., (0, 0, ..., ±1). This means there are 2n vertices. The number of facets (faces of dimension n-1) of an n-dimensional hyperoctahedron is 2n. Each facet is an (n-1)-dimensional simplex.

The hyperoctahedron is one of the five regular cross-polytopes, which are the n-dimensional analogues of the triangle, square, tetrahedron, cube, and octahedron. The name "hyperoctahedron" is derived from its 3D analogue, the octahedron. The properties of hyperoctahedra are studied in the field of geometry, particularly in higher-dimensional spaces.