Hecatonicosachoron

{{Infobox polytope 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.
 * type=Regular
 * dim = 4
 * obsa = Hi
 * off=Hecatonicosachoron.off
 * img=Schlegel_wireframe_120-cell.png
 * cells = 120 dodecahedra
 * faces = 720 pentagons
 * edges = 1200
 * vertices = 600
 * verf = Tetrahedron, edge length (1+$\sqrt{5}$)/2
 * schlafli = {5,3,3}
 * coxeter = x5o3o3o
 * army=Hi
 * reg=Hi
 * symmetry = H4, order 14400
 * circum = $$\frac{3\sqrt2+\sqrt{10}}{2} ≈ 3.70246$$
 * rad1 = $$\frac{2\sqrt3+\sqrt{15}}{2] ≈ 3.66854$$
 * rad2 = $$\sqrt{\frac{65+29\sqrt5}{10}} ≈ 3.60341$$
 * inrad = $$\frac{7+3\sqrt5}{4} ≈ 3.42705$$
 * hypervolume = $$15\frac{105+47\sqrt5}{4} ≈ 787.85698$$
 * dich = 144°
 * pieces = 120
 * loc = 1
 * dual=Hexacosichoron
 * conjugate=Great grand stellated hecatonicosachoron
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

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

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 sykmmetry, C=2F)
 * xfooofFxFfooofx5oofxfooooofxfoo3ooofxfoFofxfooo&#xt (H3 axial, cell-first)

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