# Truncated hexacosichoron

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${\displaystyle {\sqrt {\frac {23+9{\sqrt {5}}}{2}}}\approx 4.64352}$
Hypervolume${\displaystyle 25{\frac {161+80{\sqrt {5}}}{4}}\approx 2124.28399}$
Dichoral anglesTut–6–tut: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Ike–3–tut: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
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
Flag orbits4
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:

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

## 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 ${\displaystyle {\sqrt {\frac {3a^{2}+10b^{2}+10ab+(a^{2}+4b^{2}+4ab){\sqrt {5}}}{2}}}}$.

## 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: