In geometry, a cross-polytope, hyperoctahedron, orthoplex, 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 . 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: 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 convex regular polyhedra known as the Platonic solids. This can be generalised to higher dimensions with an n-orthoplex being constructed as a bipyramid with an (n−1)-orthoplex base. 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). 4 dimensions The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of the six convex regular 4-polytopes. These 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Higher dimensions The cross polytope family is one of three regular polytope families, labeled by Coxeter as βn, the other two being the hypercube family, labeled as γn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn. The n-dimensional cross-polytope has 2n vertices, and 2n facets ((n − 1)-dimensional components) all of which are (n − 1)-simplices. The vertex figures are all (n − 1)-cross-polytopes. The Schläfli symbol of the cross-polytope is {3,3,...,3,4}. The dihedral angle of the n-dimensional cross-polytope is . This gives: δ2 = arccos(0/2) = 90°, δ3 = arccos(−1/3) = 109.47°, δ4 = arccos(−2/4) = 120°, δ5 = arccos(−3/5) = 126.87°, ... δ∞ = arccos(−1) = 180°. The hypervolume of the n-dimensional cross-polytope is For each pair of non-opposite vertices, there is an edge joining them. More generally, each set of k + 1 orthogonal vertices corresponds to a distinct k-dimensional component which contains them. The number of k-dimensional components