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: \[\frac{2^n}{n!}v^n~\text{where}~v=\frac{s}{\sqrt{2}}~\text{is the distance from the center of the cross-polytope to a vertex of it}\] 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 ${3,3,4}$. One simple coordinate system for its vertices are $(\pm1, 0, 0, 0),(0, \pm1, 0, 0),(0, 0, \pm1, 0),(0, 0, 0, \pm1)$. The dual of the 16-cell is the tesseract.

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