Hecatonicosachoron

Hecatonicosachoron
(OFF file)
Rank	4
Type	Regular
Space	Spherical
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	$\frac{3\sqrt2+\sqrt{10}}{2} ≈ 3.70246$
Edge radius	$\frac{2\sqrt3+\sqrt{15}}{2} ≈ 3.66854$
Face radius	$\sqrt{\frac{65+29\sqrt5}{10}} ≈ 3.60341$
Inradius	$\frac{7+3\sqrt5}{4} ≈ 3.42705$
Hypervolume	$15\frac{105+47\sqrt5}{4} ≈ 787.85698$
Dichoral angle	144°
Central density	1
Number of pieces	120
Level of complexity	1
Related polytopes
Army	Hi
Regiment	Hi
Dual	Hexacosichoron
Conjugate	Great grand stellated hecatonicosachoron
Abstract properties
Flag count	14400
Euler characteristic	0
Topological properties
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:

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

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 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".