Truncated hecatonicosachoron

From Polytope Wiki
Jump to navigation Jump to search
Truncated hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymThi
Coxeter diagramx5x3o3o ()
Elements
Cells600 tetrahedra, 120 truncated dodecahedra
Faces2400 triangles, 720 decagons
Edges1200+3600
Vertices2400
Vertex figureTriangular pyramid, edge lengths 1 (base) and (5+5)/2 (sides)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral anglesTid–3–tet:
 Tid–10–tid: 144°
Central density1
Number of external pieces720
Level of complexity4
Related polytopes
ArmyThi
RegimentThi
DualTetrakis hexacosichoron
ConjugateQuasitruncated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count57600
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

The truncated hecatonicosachoron, or thi, also commonly called the truncated 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra and 120 truncated dodecahedra. 1 tetrahedron and three truncated dodecahedra join at each vertex. As the name suggests, it can be obtained by truncating the hecatonicosachoron.

Cross-sections[edit | edit source]

Vertex coordinates[edit | edit source]

The vertices of a truncated hecatonicosachoron of edge length 1 are given by all permutations of:

  • ,
  • ,
  • ,

along with all even permutations of:

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

Semi-uniform variant[edit | edit source]

The truncated hecatonicosachoron has a semi-uniform variant of the form x5y3o3o that maintains its full symmetry. This variant uses 600 tetrahedra of size y and 120 semi-uniform truncated dodecahedra of form x5y3o as cells, with 2 edge lengths. With edges of length a (surrounded by truncated dodecahedra only) and b (of tetrahedra), its circumradius is given by .

External links[edit | edit source]