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)

Related polychora
The rectified pentachoron can be diminished by cutting off triangular prismatic pyramids, with each removing 2 tetrahedra and diminishing the octahedra down to square pyramids. If one pyramid is removed, the result is the triangular antifastegium. If two non-adjacent pyramids are removed, such that one of the octahedra gets reduced down to an equatorial square only, the result is the bidiminished rectified pentachoron.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Octahedron (5): Pentachoron
 * Tetrahedron (5): Pentachoron
 * Triangle (10): Rectified pentachoron
 * Triangle (20): Semi-uniform small prismatodecachoron
 * Edge (30): Semi-uniform small rhombated pentachoron