Hecatonicosachoron

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Hecatonicosachoron
Schlegel wireframe 120-cell.png
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
Edge radius
Face radius
Inradius
Hypervolume
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).

Cross-sections[edit | edit source]

Hi sections Bowers.png

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 (ie 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 sykmmetry, C=2F)
  • xfooofFxFfooofx5oofxfooooofxfoo3ooofxfoFofxfooo&#xt (H3 axial, cell-first)

Related polychora[edit | edit source]

o5o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Hecatonicosachoron hi {5,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 120-cell.png
Truncated hecatonicosachoron thi t{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid truncated 120-cell.png
Rectified hecatonicosachoron rahi r{5,3,3} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Rahi.png
Hexacosihecatonicosachoron xhi 2t{5,3,3} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Xhi.png
Rectified hexacosichoron rox r{3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Rectified 600-cell schlegel halfsolid.png
Truncated hexacosichoron tex t{3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated 600-cell.png
Hexacosichoron ex {3,3,5} CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel wireframe 600-cell.png
Small rhombated hecatonicosachoron srahi rr{5,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Srahi.png
Great rhombated hecatonicosachoron grahi tr{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantitruncated 120-cell.png
Small rhombated hexacosichoron srix rr{3,3,5} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Srix.png
Great rhombated hexacosichoron grix tr{3,3,5} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cantitruncated 600-cell.png
Small disprismatohexacosihecatonicosachoron sidpixhi t0,3{5,3,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcinated 120-cell.png
Prismatorhombated hexacosichoron prix t0,1,3{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated 120-cell.png
Prismatorhombated hecatonicosachoron prahi t0,1,3{3,3,5} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Runcitruncated 600-cell.png
Great disprismatohexacosihecatonicosachoron gidpixhi t0,1,2,3{5,3,3} CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Omnitruncated 120-cell wireframe.png

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]

  • Klitzing, Richard. "Hi".