Hecatonicosachoron
Hecatonicosachoron  

Rank  4 
Type  Regular 
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  
Edge radius  
Face radius  
Inradius  
Hypervolume  
Dichoral angle  144° 
Central density  1 
Number of external pieces  120 
Level of complexity  1 
Related polytopes  
Army  Hi 
Regiment  Hi 
Dual  Hexacosichoron 
κ ^{?}  Kappa 120cell 
Conjugate  Great grand stellated hecatonicosachoron 
Abstract & topological properties  
Flag count  14400 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  H_{4}, order 14400 
Convex  Yes 
Nature  Tame 
The hecatonicosachoron, or hi, also commonly called the 120cell, 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]

Rotating hecatonicosachoron

Net

Crosssection animation
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]
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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 piecewisefinite (i.e. any surtope can be 'completed').
Representations[edit  edit source]
A hecatonicosachoron has the following Coxeter diagrams:
 x5o3o3o () (full symmetry)
 xofoFofFxFfBo5oxofoFfxFfFoB BoFfFxfoFofox5oBfFxFfFofoxo&#zx (H_{2}×H_{2} symmetry)
 ooCfoBxoFf3oooooofffx3CooBfoFxof *b3oCooBfoFxf&#zx (D_{4} symmetry, C=2F)
 xfooofFxFfooofx5oofxfooooofxfoo3ooofxfoFofxfooo&#xt (H_{3} axial, cellfirst)
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:
 Dodecahedron (120): Hexacosichoron
 Pentagon (720): Rectified hexacosichoron
 Edge (1200): Rectified hecatonicosachoron
External links[edit  edit source]
 Bowers, Jonathan. "Category 1: Regular Polychora" (#5).
 Klitzing, Richard. "hi".
 Quickfur. "The 120Cell".
 Nan Ma. "120cell {5, 3, 3}".
 Wikipedia contributors. "120cell".
 Hi.gher.Space Wiki Contributors. "Cosmochoron".
 Schläfli type 5,3,3
 Articles needing cleanup
 All pages needing cleanup
 Cleanup tagged articles with a reason field
 Polytope Wiki pages needing cleanup
 H4 symmetry
 Hi army
 Rhombic triacontahedral swirlchora
 Dodecahedral swirlchora
 Isogonal swirlchora
 Isochoric swirlchora
 Noble swirlchora
 Convex regular polychora