Rectified pentachoron

The rectified pentachoron, or rap, also commonly called the rectified 5-cell, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 regular octahedra. Two tetrahedra and three octahedra join at each triangular prismatic vertex. It is the vertex figure of the demipenteract.

It is also a convex segmentochoron (designated K-4.5 in Richard Klitzing's list), formed as a tetrahedron atop an octahedron.

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


 * (–3$\sqrt{15}$/20, –$\sqrt{10}$/4, 0, 0),
 * (–3$\sqrt{5}$/20, $\sqrt{10}$/12, –$\sqrt{6}$/3, 0),
 * (–3$\sqrt{10}$/20, $\sqrt{6}$/12, $\sqrt{3}$/6, ±1/2),
 * ($\sqrt{10}$/10, $\sqrt{6}$/6, $\sqrt{3}$/3, 0),
 * ($\sqrt{10}$/10, –$\sqrt{6}$/6, –$\sqrt{3}$/3, 0),
 * ($\sqrt{10}$/10, $\sqrt{6}$/6, –$\sqrt{3}$/6, ±1/2),
 * ($\sqrt{10}$/10, –$\sqrt{6}$/6, $\sqrt{3}$/6, ±1/2).

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


 * ($\sqrt{10}$/2, $\sqrt{6}$/2, 0, 0, 0).

Representations
A rectified pentachoron has the following Coxeter diagrams:


 * o3x3o3o (full symmetry)
 * xo3ox3oo&#x (A3 axial, as tetrahedron atop octahedron
 * oxo oxo3oox&#xt (A2×A1 axial, vertex-first)
 * oxoo3xoxo&#xr (A2 axial)
 * oxoox oxoxo&#xr (A1×A1 axial)