Rectified hexacosichoron

The rectified hexacosichoron, or rox, also commonly called the rectified 600-cell, is a convex uniform polychoron that consists of 600 regular octahedra and 120 regular icosahedra. Two icosahedra and 5 octahedra join at each pentagonal prismatic vertex. As the name suggests, it can be obtained by rectifying the hexacosichoron.

Blending 10 rectified hexacosichora results in the small disnub dishexacosichoron, which is uniform.

It is also isogonal under H4/5 symmetry, where it can be called the snub tetrahedral hecatonicosachoron. In this symmetry the icosahedra have the symmetry of snub tetrahedra and 480 of the octahedra have triangular antiprismatic symmetry only. In fact each individual component in the blend of 10 rectified hexacosichora has this symmetry only.

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

along with even permutations of:
 * $$\left(0,\,0,\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$$
 * $$\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,±\frac{5+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right).$$

Representations
A rectified hexacosichoron has the following Coxeter diagrams:


 * o5o3x3o (full symmetry)
 * DCBAVFfxoo ooxooxxoof5oxoofoxxoo3xoxFofofVx5&#zx (H3×A1 symmetry)
 * AoooFxoxVofoFofxxf5oAooxFxooVofoFxfxf ooAoxoFxofVoxfFofx5oooAoxxFfooVfxoFfx&#zx (H2×H2 symmetry)
 * ooxooxxo(of)oxxooxoo5oxoofoxx(oo)xxofooxo3xoxFofof(Vx)fofoFxox&#xt (H3 axial, icosahedron-first)

Related polychora
The rectified hexacosichoron is the colonel of a regiment with 15 members. Of these, one other besideds the colonel itself is Wythoffian (the rectified faceted hexacosichoron), two are hemi-Wythoffian (the small prismatohecatonicosachoron and small pentagonal retroantiprismatoverted dishecatonicosachoron), and one is noble (the small retropental hecatonicosachoron).

The segmentochoron icosahedron atop icosidodecahedron can be obtained as a cap of the rectified hexacosichoron in icosahedron-first orientation; the second segment in this orientation is icosidodecahedron atop small rhombicosidodecahedron.

It is also possible to diminish the recitfied hexacosichoron by removing pentagonal prismatic pyramids from the vertices. If 120 vertices forming an inscribed hexacosichoron are removed, the result is the scaliform swirlprismatodiminished rectified hexacosichoron.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Icosahedron (120): Hexacosichoron
 * Octahedron (600): Hecatonicosachoron
 * Triangle (1200): Rectified hecatonicosachoron
 * Triangle (2400): Semi-uniform small disprismatohexacosihecatonicosachoron
 * Edge (3600): Uniform small rhombated hexacosichoron