# 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$\sqrt{\frac{23+9\sqrt5}{2}} ≈ 4.64352$ Hypervolume$25\frac{161+80\sqrt5}{4} ≈ 2124.28399$ Dichoral anglesTut–6–tut: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751^\circ$ Ike–3–tut: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$ 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.

## Vertex coordinates

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

• $\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),$ • $\left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$ • $\left(0,\,±1,\,±\frac{1+\sqrt5}{2},\,±(2+\sqrt5)\right),$ • $\left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{4+\sqrt5}{2}\right),$ • $\left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{7+\sqrt5}{4}\right),$ • $\left(±\frac12,\,±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$ • $\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$ • $\left(±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±3\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right).$ ## 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 $\sqrt{\frac{3a^2+10b^2+10ab+(a^2+4b^2+4ab)\sqrt5}{2}}$ .

## Related polychora

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

Uniform polychoron compounds composed of truncated hexacosichora include: