Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube 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 1-norm on Rn.