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 (the dodecaswirlic hecatonicosachoron), as its cells form 12 rings of 10 cells. It is also the first in a series of isochoric rhombic triacontahedral swirlchora (the rhombitriacontaswirlic hecatonicosachoron).

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

Surtope angles
The surtope angle represents the fraction of solid space occupied by the angle.


 * A2: 0:48.00.00 = 144°   =2/5  Dichoral or Margin angle.  There is a decagon of dodecahedra girthing the figure.
 * A3: 0:42.00.00 =  252° E   =7/20
 * A4  0:38.24.00  = 191/600

The higher order angles might be derived from the tiling x5o3o3o5/2o, which is piecewise-finite (ie any surtope can be 'completed')

Representations
A hecatonicosachoron has the following Coxeter diagrams:


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

Related polychora
Uniform polychoronc ompounds composed of hecatonicosachora include:


 * Chirododecahedral hyperprismatochoron (6)
 * Disdodecahedral hyperprismatochoron (12)

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