Truncated hexacosichoron

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Truncated hexacosichoron
Schlegel half-solid truncated 600-cell.png
Bowers style acronymTex
Coxeter diagramo5o3x3x (CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
Cells600 truncated tetrahedra, 120 icosahedra
Faces2400 triangles, 1200 hexagons
Vertex figurePentagonal pyramid, edge lengths 1 (base) and 3 (sides)
Measures (edge length 1)
Dichoral anglesTut–6–tut:
Central density1
Number of external pieces729
Level of complexity4
Related polytopes
DualDodecakis hecatonicosachoron
ConjugateTruncated grand hexacosichoron
Abstract & topological properties
Flag count57600
Euler characteristic0
SymmetryH4, order 14400

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]

Tex sections Bowers.png Tex-slices.gif

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:

o5o3o3o truncations
Name OBSA CD diagram Picture
Hecatonicosachoron hi x5o3o3o
Schlegel wireframe 120-cell.png
Truncated hecatonicosachoron thi x5x3o3o
Schlegel half-solid truncated 120-cell.png
Rectified hecatonicosachoron rahi o5x3o3o
Hexacosihecatonicosachoron xhi o5x3x3o
Rectified hexacosichoron rox o5o3x3o
Rectified 600-cell schlegel halfsolid.png
Truncated hexacosichoron tex o5o3x3x
Schlegel half-solid truncated 600-cell.png
Hexacosichoron ex o5o3o3x
Schlegel wireframe 600-cell.png
Small rhombated hecatonicosachoron srahi x5o3x3o
Great rhombated hecatonicosachoron grahi x5x3x3o
Cantitruncated 120-cell.png
Small rhombated hexacosichoron srix o5x3o3x
Great rhombated hexacosichoron grix o5x3x3x
Cantitruncated 600-cell.png
Small disprismatohexacosihecatonicosachoron sidpixhi x5o3o3x
Runcinated 120-cell.png
Prismatorhombated hexacosichoron prix x5x3o3x
Runcitruncated 120-cell.png
Prismatorhombated hecatonicosachoron prahi x5o3x3x
Runcitruncated 600-cell.png
Great disprismatohexacosihecatonicosachoron gidpixhi x5x3x3x
Omnitruncated 120-cell wireframe.png

External links[edit | edit source]

  • Klitzing, Richard. "tex".