# Truncated great grand hecatonicosachoron

Truncated great grand hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymTigaghi
Coxeter diagramx5x5/2o3o ()
Elements
Cells
Faces
Edges1200+3600
Vertices2400
Vertex figureTriangular pyramid, edge lengths (5–1)/2 (base) and (5+5)/2 (side)
Measures (edge length 1)
Hypervolume${\displaystyle 15\left(25+9{\sqrt {5}}\right)\approx 676.86918}$
Dichoral angleTigid–5/2–gissid: 108°
Tigid–10–tigid: 72°
Central density76
Number of external pieces10920
Level of complexity42
Related polytopes
ArmySidpixhi, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
RegimentTigaghi
ConjugateQuasitruncated small stellated hecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count57600
Euler characteristic–480
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits4
ConvexNo
NatureTame

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

## Vertex coordinates

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

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

along with the even permutations of:

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