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. Together with its dual, it is the first in an infinite family of dodecahedral swirlchora.

Vertex coordinates
The vertices of a hecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of: together with all the even permutations of:
 * (±(3+$\sqrt{5}$)/2, ±(3+$\sqrt{2}$)/2, 0, 0),
 * (±(5+3$\sqrt{10}$)/4, ±(3+$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/4),
 * (±(2+$\sqrt{5}$)/2, ±(2+$\sqrt{5}$)/2, ±(2+$\sqrt{5}$)/2, ±1/2),
 * (±(7+3$\sqrt{5}$)/4, ±(1+$\sqrt{5}$)/4, ±(1+$\sqrt{5}$)/4, ±(1+$\sqrt{5}$)/4),
 * (±(7+3$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/4, ±1/2, 0),
 * (±(2+$\sqrt{5}$)/2, ±(5+3$\sqrt{5}$)/4, 0, ±(1+$\sqrt{5}$)/4),
 * (±(2+$\sqrt{5}$)/2, ±(3+$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/2, ±(1+$\sqrt{5}$)/4).

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)