Hexacosihecatonicosachoron

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Hexacosihecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymXhi
Coxeter diagramo5x3x3o ()
Elements
Cells600 Truncated tetrahedra, 120 truncated icosahedra
Faces1200 triangles, 720 pentagons, 2400 hexagons
Edges3600+3600
Vertices3600
Vertex figureDigonal disphenoid, edge lengths 1 (base 1), (1+5)/2 (base 2) and 3 (sides)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral anglesTut–3–tut:
 Ti–6–tut:
 Ti–5–ti: 144°
Central density1
Number of external pieces720
Level of complexity6
Related polytopes
ArmyXhi
RegimentXhi
DualDisphenoidal trischiliahexacosichoron
ConjugateGreat hexacosihecatonicosachoron
Abstract & topological properties
Flag count86400
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits6
ConvexYes
NatureTame

The hexacosihecatonicosachoron, or xhi, also commonly called the bitruncated 120-cell, is a convex uniform polychoron that consists of 600 truncated tetrahedra and 120 truncated icosahedra. 2 truncated tetrahedra and 2 truncated icosahedra join at each vertex. It is the medial stage of the truncation series between a hecatonicosachoron and its dual hexacosichoron. As such, it could also be called a bitruncated 600-cell.

Cross-sections[edit | edit source]

Card with cell counts, vertex figure, and cross-sections.

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a hexacosihecatonicosachoron of edge length 1 are given by all permutations of:

  • ,
  • ,

together with all even permutations of:

  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • .

Semi-uniform variant[edit | edit source]

The hexacosihecatonicosachoron has a semi-uniform variant of the form o5x3y3o that maintains its full symmetry. This variant uses 600 semi-uniform truncated tetrahedra of form x3y3o and 120 semi-uniform truncated icosahedra of form o5x3y as cells, with 2 edge lengths.

With edges of length a (of pentagonal faces) and b (of triangular faces), its circumradius is given by .

External links[edit | edit source]