Hecatonicosachoron

The hecatonicosachoron, or hi, also commonly called the 120-cell, is one of the 6 convex regular polychora. It has 120 dodecahedra as cells, joining 3 to an edge and 4 to a vertex. It is the first in an infinite family of isochoric dodecahedral swirlchora.

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


 * $$\left(±\frac{5+3\sqrt{5}}{4},\,±\frac{3+\sqrt{5}}{4},\,±\frac{3+\sqrt{5}}{4},\,±\frac{3+\sqrt{5}}{4}\right),$$


 * $$\left(±\frac{2+\sqrt{5}}{2},\,±\frac{2+\sqrt{5}}{2},\,±\frac{2+\sqrt{5}}{2},\,±\frac{1}{2}\right),$$


 * $$\left(±\frac{7+3\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4}\right),$$

together with all the even permutations of:
 * $$\left(±\frac{7+3\sqrt{5}}{4},\,±\frac{3+\sqrt{5}}{4},\,±\frac{1}{2},\,0\right),$$


 * $$\left(±\frac{2+\sqrt{5}}{2},\,±\frac{5+3\sqrt{5}}{4},\,0,\,±\frac{1+\sqrt{5}}{4}\right),$$


 * $$\left(±\frac{2+\sqrt{5}}{2},\,±\frac{3+\sqrt{5}}{4},\,±\frac{3+\sqrt{5}}{2},\,±\frac{1+\sqrt{5}}{4}\right),$$

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Dodecahedron (120): Hexacosichoron
 * Pentagon (720): Rectified hexacosichoron
 * Edge (1200): Rectified hecatonicosachoron

Representations
A hecatonicosachoron has the following Coxeter diagrams:


 * x5o3o3o (full symmetry)
 * xofoFofFxFfBo5oxofoFfxFfFoB BoFfFxfoFofox5oBfFxFfFofoxo&#zx (H2×H2 symmetry)
 * ooCfoBxoFf3oooooofffx3CooBfoFxof *b3oCooBfoFxf&#zx (D4 sykmmetry, C=2F)
 * xfooofFxFfooofx5oofxfooooofxfoo3ooofxfoFofxfooo&#xt (H3 axial, cell-first)