Rectified hecatonicosachoron

The rectified hecatonicosachoron, or rahi, also commonly called the rectified 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra and 120 icosidodecahedra. Two tetrahedra and three icosidodecahedra join at each triangular prismatic vertex. As the name suggests, it can be obtained by rectifying the hecatonicosachoron.

Vertex coordinates
The vertices of a rectified hecatonicosachoron of edge length 1 are given by all permutations of:

along with all even permutations of:
 * $$\left(0,\,0,\,±\frac{1+\sqrt5}{2},\,±(2+\sqrt5)\right),$$
 * $$\left(0,\,±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),$$


 * $$\left(0,\,±\frac12,\,±\frac{7+3\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{1+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,±(2+\sqrt5),\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right).$$

Representations
A rectified hecatonicosachoron has the following Coxeter diagrams:


 * o5x3o3o (full symmetry)
 * ofxoxooxFf(oV)fFxooxoxfo5xoxfofFxoo(xo)ooxFfofxox3oooxFfofxF(Vo)FxfofFxooo&#xt (H3 axial, icosidodecahedron-first)

Related polychora
The rectified hecatonicosachoron is the colonel of a 3-member regiment that also includes the facetorectified hecatonicosachoron and small hecatonicosintercepted hecatonicosachoron.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Icosidodecahedron (120): Hexacosichoron
 * Tetrahedron (600): Hecatonicosachoron
 * Pentagon (720): Rectified hexacosichoron
 * Triangle (2400): Semi-uniform small disprismatohexacosihecatonicosachoron
 * Edge (3600): Semi-uniform small rhombated hecatonicosachoron