# Small rhombated grand hexacosichoron

Small rhombated grand hexacosichoron
Rank4
TypeUniform
Notation
Bowers style acronymSirgax
Coxeter diagramo5/2x3o3x ()
Elements
Cells600 cuboctahedra, 720 pentagrammic prisms, 120 great icosidodecahedra
Faces1200+2400 triangles, 3600 squares, 1440 pentagrams
Edges3600+7200
Vertices3600
Vertex figureWedge, edge lengths 1 (two edges of base rectangle and top edge), 2 (side edges), and (5–1)/2 (remaining base edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {19-8{\sqrt {5}}}}\approx 1.05425}$
Hypervolume${\displaystyle 5{\frac {591{\sqrt {5}}-1240}{2}}\approx 203.79044}$
Dichoral anglesGid–5/2–stip: 126°
Co–4–stip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{10}}}\right)\approx 103.28253^{\circ }}$
Gid–3–co: ${\displaystyle \arccos \left(-{\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 97.76124^{\circ }}$
Co–3–co: ${\displaystyle \arccos \left({\frac {3{\sqrt {5}}-1}{8}}\right)\approx 44.47751^{\circ }}$
Related polytopes
ArmySemi-uniform Srahi, edge lengths ${\displaystyle {\frac {5{\sqrt {5}}-11}{2}}}$ (pentagons), ${\displaystyle {\sqrt {5}}-2}$ (octahedra)
RegimentSirgax
ConjugateSmall rhombated hexacosichoron
Abstract & topological properties
Flag count129600
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The small rhombated grand hexacosichoron, or sirgax, is a nonconvex uniform polychoron that consists of 600 cuboctahedra, 720 pentagrammic prisms, and 120 great icosidodecahedra. 1 great icosidodecahedron, 2 pentagrammic prisms, and 2 cuboctahedra join at each vertex. It can be obtained by cantellating the grand hexacosichoron.

## Vertex coordinates

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

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

together with all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm (3-{\sqrt {5}}),\,\pm {\frac {3-{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm (3-{\sqrt {5}}),\,\pm {\frac {3{\sqrt {5}}-5}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {5{\sqrt {5}}-11}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm ({\sqrt {5}}-2),\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {4-{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 1,\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm ({\sqrt {5}}-2),\,\pm {\frac {4-{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm 1,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm (3-{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm ({\sqrt {5}}-2)\right)}$,
• ${\displaystyle \left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {5-2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm ({\sqrt {5}}-2),\,\pm {\frac {3{\sqrt {5}}-5}{4}}\right)}$.

## Related polychora

The small rhombated grand hexacosichoron is the colonel of a seven-member regiment. Its other members include the grand retrosphenoverted hecatonicosihexacosihecatonicosachoron, rhombic great hexacosihecatonicosachoron, pseudorhombic great dishecatonicosachoron, grand rhombic great hexacosihecatonicosachoron, great dishecatonicosintercepted hexacosihecatonicosachoron, and hecatonicosintercepted prismatohecatonicosihexacosichoron.