# Truncated hecatonicosachoron

Truncated hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymThi
Coxeter diagramx5x3o3o ()
Elements
Cells600 tetrahedra, 120 truncated dodecahedra
Faces2400 triangles, 720 decagons
Edges1200+3600
Vertices2400
Vertex figureTriangular pyramid, edge lengths 1 (base) and (5+5)/2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {34+15{\sqrt {5}}}}\approx 8.21833}$
Hypervolume${\displaystyle 25{\frac {1577+704{\sqrt {5}}}{4}}\approx 19694.9491}$
Dichoral anglesTid–3–tet: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Tid–10–tid: 144°
Central density1
Number of external pieces720
Level of complexity4
Related polytopes
ArmyThi
RegimentThi
DualTetrakis hexacosichoron
ConjugateQuasitruncated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count57600
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

The truncated hecatonicosachoron, or thi, also commonly called the truncated 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra and 120 truncated dodecahedra. 1 tetrahedron and three truncated dodecahedra join at each vertex. As the name suggests, it can be obtained by truncating the hecatonicosachoron.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}$,

along with all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$.

## Semi-uniform variant

The truncated hecatonicosachoron has a semi-uniform variant of the form x5y3o3o that maintains its full symmetry. This variant uses 600 tetrahedra of size y and 120 semi-uniform truncated dodecahedra of form x5y3o as cells, with 2 edge lengths. With edges of length a (surrounded by truncated dodecahedra only) and b (of tetrahedra), its circumradius is given by ${\displaystyle {\sqrt {\frac {14a^{2}+21b^{2}+33ab+(6a^{2}+9b^{2}+15ab){\sqrt {5}}}{2}}}}$.