Cross Polytope

The cross polytope is the regular polytope in dimensions corresponding to the convex hull of the points formed by permuting the coordinates (, 0, 0, ..., 0). A cross-polytope (also called an orthoplex) is denoted and has vertices and Schläfli symbol . The cross polytope is named because its vertices are located equidistant from the origin along the Cartesian axes in Euclidean space, which each such axis perpendicular to all others. A cross polytope is bounded by -simplexes, and is a dipyramid erected (in both directions) into the th dimension, with an -dimensional cross polytope as its base.

In one dimension, the cross polytope is the line segment . In two dimensions, the cross polytope is the filled square with vertices , , , . In three dimensions, the cross polytope is the convex hull of the octahedron with vertices , , , , , . In four dimensions, the cross polytope is the 16-cell, depicted in the above figure by projecting onto one of the four mutually perpendicular three-spaces within the four-space obtained by dropping one of the four vertex components (R. Towle).

The skeleton of is isomorphic with the circulant graph , also known as the cocktail party graph .

For all dimensions, the dual of the cross polytope is the hypercube (and vice versa). Consequently, the number of -simplices contained in an -cross polytope is .