# Great hexacosihecatonicosachoron

Great hexacosihecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymGixhi
Coxeter diagramo5/2x3x3o ()
Elements
Cells
Faces
Edges3600+3600
Vertices3600
Vertex figureDigonal disphenoid, edge lengths 1 (base 1), (5–1)/2 (base 2) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {59-25{\sqrt {5}}}{2}}}\approx 1.24465}$
Hypervolume${\displaystyle 5{\frac {2726{\sqrt {5}}-5845}{4}}\approx 313.15163}$
Dichoral anglesTiggy–6–tut: ${\displaystyle \arccos \left(-{\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 97.76124^{\circ }}$
Tiggy–5/2–tiggy: 72°
Tut–3–tut: ${\displaystyle \arccos \left({\frac {3{\sqrt {5}}-1}{8}}\right)\approx 44.47751^{\circ }}$
Central density191
Number of external pieces122640
Level of complexity494
Related polytopes
ArmySemi-uniform Srix, edge lengths ${\displaystyle {\frac {5-{\sqrt {5}}-11}{2}}}$ (icosidodecahedra), ${\displaystyle 5-2{\sqrt {5}}}$ (main triangles)
RegimentGixhi
ConjugateHexacosihecatonicosachoron
Convex coreHexacosichoron
Abstract & topological properties
Flag count86400
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits6
ConvexNo
NatureTame

The great hexacosihecatonicosachoron, or gixhi, is a nonconvex uniform polychoron that consists of 600 truncated tetrahedra and 120 truncated great icosahedra. 2 of each join at each vertex.

It is the medial stage of the truncation series between a great grand stellated hecatonicosachoron and its dual grand hexacosichoron, which makes it the bitruncation of either of these polychora.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,0,\,\pm ({\sqrt {5}}-1),\,\pm {\frac {7-3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-9}{4}},\,\pm {\frac {5{\sqrt {5}}-9}{4}}\right)}$,

together with all even permutations of:

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