# Great stellated hecatonicosachoron

Great stellated hecatonicosachoron
Rank4
TypeRegular
Notation
Bowers style acronymGishi
Coxeter diagramx5/2o3o5o ()
Schläfli symbol{5/2,3,5}
${\displaystyle \{5,3,5\mid *3\}}$
Elements
Cells120 great stellated dodecahedra
Faces720 pentagrams
Edges720
Vertices120
Vertex figureIcosahedron, edge length (5–1)/2
Edge figuregissid 5/2 gissid 5/2 gissid 5/2 gissid 5/2 gissid 5/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {5}}-1}{2}}\approx 0.61803}$
Edge radius${\displaystyle {\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\approx 0.36327}$
Face radius${\displaystyle {\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\approx 0.32492}$
Inradius${\displaystyle {\frac {{\sqrt {5}}-1}{4}}\approx 0.30902}$
Hypervolume${\displaystyle 15{\frac {25-11{\sqrt {5}}}{4}}\approx 1.51220}$
Dichoral angle144°
Central density20
Number of external pieces2400
Level of complexity8
Related polytopes
ArmyEx, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
RegimentGishi
CompanyGishi
DualGrand hecatonicosachoron
ConjugateGrand hecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count14400
Euler characteristic0
Schläfli type{5,3,5}
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The great stellated hecatonicosachoron, or gishi, also commonly called the great stellated 120-cell, is one of the 10 regular Schläfli–Hess polychora. It has 120 great stellated dodecahedra as cells, joining 5 to an edge and 20 to a vertex in an icosahedral arrangement.

## Vertex coordinates

The vertices of a great stellated hecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of:

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

and all even permutations of:

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

The first two sets of vertices form an icositetrachoron that can be inscribed into the great stellated hecatonicosachoron.

## Related polychora

The great stellated hecatonicosachoron is the colonel of a five-member regiment that includes three other regular polychora, namely the grand stellated hecatonicosachoron, the great faceted hexacosichoron, and the grand hexacosichoron. Of these, the grand stellated hecatonicosachoron also shares the same faces, so the great stellated hecatonicosachoron is the captain of a two-member company.

Uniform polychoron compounds composed of great stellated hecatonicosachora include: