Medial hexacosichoron

The medial hexacosichoron, or mix, is a regular compound polychoron. It is a compound of 120 pentachora. It has 600 tetrahedra as cells, with 4 cels joining at each vertex. It can also be seen as a compound f 60 stellated decachora.

Vertex coordinates
The vertices of a medial hexacosichoron of edge length 1, centered at the origin, are given by all permutations of:
 * $$\left(±\frac{\sqrt{5}}{5},\,±\frac{\sqrt{5}}{5},\,0,\,0\right),$$
 * $$\left(±\frac{1}{2},\,±\frac{\sqrt{5}}{10},\,±\frac{\sqrt{5}}{10},\,±\frac{\sqrt{5}}{10}\right),$$
 * $$\left(±\frac{5+\sqrt{5}}{20},\,±\frac{5+\sqrt{5}}{20},\,±\frac{5+\sqrt{5}}{20},\,±\frac{3\sqrt{5}-5}{20}\right),$$
 * $$\left(±\frac{5+3\sqrt{5}}{20},\,±\frac{5-\sqrt{5}}{20},\,±\frac{5-\sqrt{5}}{20},\,±\frac{5-\sqrt{5}}{20}\right),$$

together with all the even permutations of:
 * $$\left(±\frac{\sqrt{5}}{10},\,±\frac{5+3\sqrt{5}}{20},\,±\frac{3\sqrt{5}-5}{20},\,0\right),$$
 * $$\left(±\frac{1}{2},\,±\frac{5+\sqrt{5}}{20},\,0,\,±\frac{5-\sqrt{5}}{20}\right),$$
 * $$\left(±\frac{\sqrt{5}}{10},\,±\frac{5+\sqrt{5}}{20},\,±\frac{\sqrt{5}}{5},\,±\frac{5-\sqrt{5}}{20}\right),$$