Hecatonicosachoron

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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
Edge radius
Face radius
Inradius
Hypervolume
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).

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

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

  • ,
  • ,
  • ,
  • ,

together with all the even permutations of:

  • ,
  • ,
  • .

Surtope angles[edit | edit source]

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[edit | edit source]

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[edit | edit source]

Uniform polychoron compounds composed of hecatonicosachora include:

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]