Truncated hexacosichoron

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Truncated hexacosichoron
Rank4
TypeUniform
Notation
Bowers style acronymTex
Coxeter diagramo5o3x3x ()
Elements
Cells600 truncated tetrahedra, 120 icosahedra
Faces2400 triangles, 1200 hexagons
Edges720+3600
Vertices1440
Vertex figurePentagonal pyramid, edge lengths 1 (base) and 3 (sides)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral anglesTut–6–tut:
 Ike–3–tut:
Central density1
Number of external pieces729
Level of complexity4
Related polytopes
ArmyTex
RegimentTex
DualDodecakis hecatonicosachoron
ConjugateTruncated grand hexacosichoron
Abstract & topological properties
Flag count57600
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

The truncated hexacosichoron, or tex, also commonly called the truncated 600-cell, is a convex uniform polychoron that consists of 120 regular icosahedra and 600 truncated tetrahedra. 1 icosahedron and five truncated tetrahedra join at each vertex. As the name suggests, it can be obtained as the truncation of a hexacosichoron.

It is also isogonal under H4/5 symmetry, with the icosahedra having the symmetry of snub tetrahedra, and 480 of the truncated tetrahedra having trigonal symmetry only.

Cross-sections[edit | edit source]

Vertex coordinates[edit | edit source]

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

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

Semi-uniform variant[edit | edit source]

The truncated hexacosichoron has a semi-uniform variant of the form o5o3y3x that maintains its full symmetry. This variant uses 120 icosahedra of size y and 600 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of icosahedra), its circumradius is given by .

Related polychora[edit | edit source]

The truncated hexacosichoron is the colonel of a two-member regiment that also includes the truncated faceted hexacosichoron.

Uniform polychoron compounds composed of truncated hexacosichora include:

External links[edit | edit source]