# Small rhombated great hecatonicosachoron

Small rhombated great hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymSirghi
Coxeter diagramx5o5/2x5o ()
Elements
Cells
Faces
Edges3600+7200
Vertices3600
Vertex figureWedge, edge lengths (1+5)/2 (two edges of base rectangle and top edge), (5–1)/2 (other 2 base edges), and 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {7+2{\sqrt {5}}}}\approx 3.38705}$
Hypervolume${\displaystyle 75{\frac {54+41{\sqrt {5}}}{2}}\approx 5462.95452}$
Dichoral angleDid–5–pip: 162°
Raded–4–pip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Related polytopes
ArmySemi-uniform Srix, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (icosidodecahedra), ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (main triangles)
RegimentSirghi
ConjugateSmall rhombated grand stellated hecatonicosachoron
Abstract & topological properties
Flag count129600
Euler characteristic–960
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits9
ConvexNo
NatureTame

The small rhombated great hecatonicosachoron, or sirghi, is a nonconvex uniform polychoron that consists of 720 pentagonal prisms, 120 dodecadodecahedra, and 120 rhombidodecadodecahedra. 1 dodecadodecahedron, 2 pentagonal prisms, and 2 rhombidodecadodecahedra join at each vertex. it can be obtained by cantellating the great hecatonicosachoron.

## Vertex coordinates

Coordinates for the vertices of a small rhombated great hecatonicosachoron of edge length 1 are given by all permutations of:

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

together with all even permutations of:

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

## Related polychora

The small rhombated great hecatonicosachoron is the colonel of a regiment with 7 members. Its other members include the small retrosphenoverted hecatonicosidishecatonicosachoron, rhombic small dishecatonicosachoron, pseudorhombic small hecatonicosihexacosichoron, grand rhombic small hecatonicosihexacosichoron, small hecatonicosihexacosintercepted dishecatonicosachoron, and small hexacosintercepted prismatodishecatonicosachoron.