# Hecatonicosachoron

(Redirected from Hi)
Hecatonicosachoron
Rank4
TypeRegular
Notation
Bowers style acronymHi
Coxeter diagramx5o3o3o ()
Schläfli symbol{5,3,3}
Elements
Cells120 dodecahedra
Faces720 pentagons
Edges1200
Vertices600
Vertex figureTetrahedron, edge length (1+5)/2
Edge figuredoe 5 doe 5 doe 5
Measures (edge length 1)
Circumradius${\displaystyle {\frac {3{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 3.70246}$
Edge radius${\displaystyle {\frac {2{\sqrt {3}}+{\sqrt {15}}}{2}}\approx 3.66854}$
Face radius${\displaystyle {\sqrt {\frac {65+29{\sqrt {5}}}{10}}}\approx 3.60341}$
Inradius${\displaystyle {\frac {7+3{\sqrt {5}}}{4}}\approx 3.42705}$
Hypervolume${\displaystyle 15{\frac {105+47{\sqrt {5}}}{4}}\approx 787.85698}$
Dichoral angle144°
Central density1
Number of external pieces120
Level of complexity1
Related polytopes
ArmyHi
RegimentHi
DualHexacosichoron
κ ?Kappa 120-cell
ConjugateGreat grand stellated hecatonicosachoron
Abstract & topological properties
Flag count14400
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
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:

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

together with all the even permutations of:

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

## 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:

## Isogonal derivatives

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