Cross-polytope
A cross-polytope, also known as a orthoplex or hyper-octahedron is a higher-dimension analog for the octahedron. When no dimension is specified, it is assumed to be 4. The graph of the nD cross-polytope can be formed by the following method:
1. Take the K 2n complete graph.
2. Find n pairs of vertices so no two pairs share a common vertex.
3. Remove all edges between two vertices in the same pair
The area of an nD cross-polytope with side length s is given by: This formula can be derived by splitting the cross-polytope into many hyper-tetrahedra.
16-cell
A 16-cell is the 4th dimensional cross-polytope. It is made by combining two cubes. The net of a tesseract is composed of 8 cubes. It has the Schlaefli symbol . One simple coordinate system for its vertices are . The dual of the 16-cell is the tesseract.