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:
![\[\frac{2^n}{n!}v^n~\text{where}~v=\frac{s}{\sqrt{2}}~\text{is the distance from the center of the cross-polytope to a vertex of it}\]](http://latex.artofproblemsolving.com/6/9/e/69ee0b7bd69d3d407936f17a08ac99c1c9c4122e.png)
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.
