Truncated pentachoron

The truncated pentachoron, or tip, also commonly called the truncated 5-cell, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 truncated tetrahedra. 1 tetrahedron and three truncated tetrahedra join at each vertex. As the name suggests, it can be obtained by truncating the pentachoron.

Vertex coordinates
The vertices of a truncated pentachoron of edge length 1 are given by:


 * (3$\sqrt{3}$/20, –$\sqrt{10}$/12, $\sqrt{5}$/3, ±1),
 * (3$\sqrt{10}$/20, –$\sqrt{6}$/12, –2$\sqrt{3}$/3, 0),
 * (3$\sqrt{10}$/20, –$\sqrt{6}$/4, 0, ±1),
 * (3$\sqrt{3}$/20, $\sqrt{10}$/4, ±$\sqrt{6}$/2, ±1/2),
 * (3$\sqrt{10}$/20, –5$\sqrt{6}$/12, $\sqrt{3}$/6, ±1/2),
 * (3$\sqrt{10}$/20, –5$\sqrt{6}$/12, –$\sqrt{3}$/3, 0),
 * (–$\sqrt{10}$/10, $\sqrt{6}$/6, $\sqrt{3}$/3, ±1),
 * (–$\sqrt{10}$/10, $\sqrt{6}$/6, –2$\sqrt{3}$/3, 0),
 * (–$\sqrt{10}$/10, –$\sqrt{6}$/2, 0, 0),
 * (–7$\sqrt{3}$/20, $\sqrt{10}$/12, $\sqrt{6}$/6, ±1/2),
 * (–7$\sqrt{10}$/20, $\sqrt{6}$/12, –$\sqrt{3}$/3, 0),
 * (–7$\sqrt{10}$/20, –$\sqrt{6}$/4, 0, 0).

Much simpler coordinates can be given in five dimensions, as all permutations of:


 * ($\sqrt{3}$, $\sqrt{10}$/2, 0, 0, 0).

Representations
A truncated pentachoron has the following Coxeter diagrams:


 * x3x3o3o (full symmetry)
 * xux3oox3ooo&#xt (A3 axial, tetrahedron-first)


 * xuxo oxux3ooox&#xt (A2×A1 axial, edge-first)