# Hecatonicosachoron

(Redirected from 120 cell)
Hecatonicosachoron
Rank4
TypeRegular
SpaceSpherical
Bowers style acronymHi
Info
Coxeter diagramx5o3o3o
Schläfli symbol{5,3,3}
SymmetryH4, order 14400
ArmyHi
RegimentHi
Elements
Vertex figureTetrahedron, edge length (1+5)/2
Cells120 dodecahedra
Faces720 pentagons
Edges1200
Vertices600
Measures (edge length 1)
Circumradius${\displaystyle \frac{3\sqrt2+\sqrt{10}}{2} ≈ 3.70246}$
Edge radius${\displaystyle \frac{2\sqrt3+\sqrt{15}}{2} ≈ 3.66854}$
Face radius${\displaystyle \sqrt{\frac{65+29\sqrt5}{10}} ≈ 3.60341}$
Inradius${\displaystyle \frac{7+3\sqrt5}{4} ≈ 3.42705}$
Hypervolume${\displaystyle 15\frac{105+47\sqrt5}{4} ≈ 787.85698}$
Dichoral angle144°
Central density1
Euler characteristic0
Number of pieces120
Level of complexity1
Related polytopes
DualHexacosichoron
ConjugateGreat grand stellated hecatonicosachoron
Properties
ConvexYes
OrientableYes
NatureTame

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:

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

together with all the even permutations of:

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

## 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)

## Related polychora

o5o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Hecatonicosachoron hi {5,3,3}
Truncated hecatonicosachoron thi t{5,3,3}
Rectified hecatonicosachoron rahi r{5,3,3}
Hexacosihecatonicosachoron xhi 2t{5,3,3}
Rectified hexacosichoron rox r{3,3,5}
Truncated hexacosichoron tex t{3,3,5}
Hexacosichoron ex {3,3,5}
Small rhombated hecatonicosachoron srahi rr{5,3,3}
Great rhombated hecatonicosachoron grahi tr{5,3,3}
Small rhombated hexacosichoron srix rr{3,3,5}
Great rhombated hexacosichoron grix tr{3,3,5}
Small disprismatohexacosihecatonicosachoron sidpixhi t0,3{5,3,3}
Prismatorhombated hexacosichoron prix t0,1,3{5,3,3}
Prismatorhombated hecatonicosachoron prahi t0,1,3{3,3,5}
Great disprismatohexacosihecatonicosachoron gidpixhi t0,1,2,3{5,3,3}

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: