# Truncated hexacosichoron

Truncated hexacosichoron
Rank4
TypeUniform
SpaceSpherical
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${\displaystyle \sqrt{\frac{23+9\sqrt5}{2}} ≈ 4.64352}$
Hypervolume${\displaystyle 25\frac{161+80\sqrt5}{4} ≈ 2124.28399}$
Dichoral anglesTut–6–tut: ${\displaystyle \arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751°}$
Ike–3–tut: ${\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124°}$
Central density1
Number of external pieces729
Level of complexity4
Related polytopes
ArmyTex
RegimentTex
DualDodecakis hecatonicosachoron
ConjugateTruncated grand hexacosichoron
Abstract & topological properties
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.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±1,\,±\frac{1+\sqrt5}{2},\,±(2+\sqrt5)\right),}$
• ${\displaystyle \left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{4+\sqrt5}{2}\right),}$
• ${\displaystyle \left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{7+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±3\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right).}$

## Semi-uniform variant

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 ${\displaystyle \sqrt{\frac{3a^2+10b^2+10ab+(a^2+4b^2+4ab)\sqrt5}{2}}}$.

## Related polychora

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

o5o3o3o truncations
Name OBSA CD diagram Picture
Hecatonicosachoron hi x5o3o3o
Truncated hecatonicosachoron thi x5x3o3o
Rectified hecatonicosachoron rahi o5x3o3o
Hexacosihecatonicosachoron xhi o5x3x3o
Rectified hexacosichoron rox o5o3x3o
Truncated hexacosichoron tex o5o3x3x
Hexacosichoron ex o5o3o3x
Small rhombated hecatonicosachoron srahi x5o3x3o
Great rhombated hecatonicosachoron grahi x5x3x3o
Small rhombated hexacosichoron srix o5x3o3x
Great rhombated hexacosichoron grix o5x3x3x
Small disprismatohexacosihecatonicosachoron sidpixhi x5o3o3x
Prismatorhombated hexacosichoron prix x5x3o3x
Prismatorhombated hecatonicosachoron prahi x5o3x3x
Great disprismatohexacosihecatonicosachoron gidpixhi x5x3x3x