# Truncated great hecatonicosachoron

Truncated great hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymTighi
Coxeter diagramx5x5/2o5o ()
Elements
Cells
Faces
Edges720+3600
Vertices1440
Vertex figurePentagonal pyramid, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (base) and ${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{2}}}}$ (side)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {13+5{\sqrt {5}}}{2}}}\approx 3.47709}$
Hypervolume${\displaystyle {\frac {2925+1260{\sqrt {5}}}{2}}\approx 2871.22283}$
Dichoral angleSissid–5/2–tigid: 144°
Tigid–10–tigid: 144°
Central density6
Number of external pieces3720
Level of complexity15
Related polytopes
ArmySemi-uniform Tex, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (icosahedra), 1 (surrounded by truncated tetrahedra)
RegimentTighi
ConjugateQuasitruncated grand stellated hecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count57600
Euler characteristic–960
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits4
ConvexNo
NatureTame

The truncated great hecatonicosachoron, or tighi, is a nonconvex uniform polychoron that consists of 120 small stellated dodecahedra and 120 truncated great dodecahedra. One small stellated dodecahedron and five truncated great dodecahedra join at each vertex. As the name suggests, it can be obtained by truncating the great hecatonicosachoron.

## Vertex coordinates

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

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

plus all even permutations of:

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

## Related polychora

The truncated great hecatonicosachoron is the colonel of a two-member regiment that also includes the truncated grand hecatonicosachoron.