Hecatonicosachoron

Hecatonicosachoron
(OFF file)
Rank	4
Type	Regular
Notation
Bowers style acronym	Hi
Coxeter diagram	x5o3o3o ()
Schläfli symbol	{5,3,3}
Elements
Cells	120 dodecahedra
Faces	720 pentagons
Edges	1200
Vertices	600
Vertex figure	Tetrahedron, edge length (1+√5)/2
Edge figure	doe 5 doe 5 doe 5
Measures (edge length 1)
Circumradius	${\frac {3{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 3.70246$
Edge radius	${\frac {2{\sqrt {3}}+{\sqrt {15}}}{2}}\approx 3.66854$
Face radius	${\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\approx 3.60341$
Inradius	${\frac {7+3{\sqrt {5}}}{4}}\approx 3.42705$
Hypervolume	$15{\frac {105+47{\sqrt {5}}}{4}}\approx 787.85698$
Dichoral angle	144°
Central density	1
Number of external pieces	120
Level of complexity	1
Related polytopes
Army	Hi
Regiment	Hi
Dual	Hexacosichoron
κ ?	Kappa 120-cell
Conjugate	Great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count	14400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

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).

Gallery

• 

  Rotating hecatonicosachoron

• 

  Net

• 

  Cross-sections

• 

  Cross-section animation

Vertex coordinates

The vertices of a hecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of:

• $\left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,0,\,0\right)$,
• $\left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)$,

together with all the even permutations of:

• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)$.

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 (i.e. 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 polychoron compounds 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

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#5).

• Klitzing, Richard. "hi".
• Quickfur. "The 120-Cell".

• Nan Ma. "120-cell {5, 3, 3}".

• Wikipedia contributors. "120-cell".
• Hi.gher.Space Wiki Contributors. "Cosmochoron".