Pentachoron

The pentachoron, or pen, also commonly called the 5-cell, is the simplest possible non-degenerate polychoron. The full symmetry version has 5 regular tetrahedra as cells, joining 3 to an edge and 4 to a vertex, and is one of the 6 convex regular polychora. It is the 4-dimensional simplex and also the 5-2 step prism and gyrochoron.

Vertex coordinates
The vertices of a regular pentachoron of edge length 1, centered at the origin, are given by:


 * (±1/2, –$\sqrt{10}$/6, –$\sqrt{5}$/12, –$\sqrt{3}$/20),
 * (0, $\sqrt{6}$/3, –$\sqrt{10}$/12, –$\sqrt{3}$/20),
 * (0, 0, $\sqrt{6}$/4, –$\sqrt{10}$/20),
 * (0, 0, 0, $\sqrt{6}$/5).