# Great grand stellated hecatonicosachoron

Great grand stellated hecatonicosachoron Rank4
TypeRegular
SpaceSpherical
Notation
Bowers style acronymGogishi
Coxeter diagramx5/2o3o3o (         )
Schläfli symbol{5/2,3,3}
Elements
Cells120 great stellated dodecahedra
Faces720 pentagrams
Edges1200
Vertices600
Vertex figureTetrahedron, edge length (5–1)/2
Edge figuregissid 5/2 gissid 5/2 gissid 5/2
Measures (edge length 1)
Circumradius$\frac{3\sqrt2-\sqrt{10}}{2} ≈ 0.54018$ Edge radius$\frac{\sqrt{15}-2\sqrt3}{2} ≈ 0.20444$ Face radius$\sqrt{\frac{65-29\sqrt5}{10}} ≈ 0.12411$ Inradius$\frac{7-3\sqrt5}{4} ≈ 0.072949$ Hypervolume$15\frac{47\sqrt5-105}{4} ≈ 0.35698$ Dichoral angle72°
Central density191
Number of pieces9600
Level of complexity30
Related polytopes
ArmyHi
RegimentGogishi
DualGrand hexacosichoron
ConjugateHecatonicosachoron
Convex coreHecatonicosachoron
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The great grand stellated hecatonicosachoron, or gogishi, also commonly called the great grand stellated 120-cell, is one of the 10 regular Schläfli–Hess polychora. It has 120 great stellated dodecahedra as cells, joining 3 to an edge and 4 to a vertex.

## Vertex coordinates

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

• $\left(±\frac{3-\sqrt5}{2},\,±\frac{3-\sqrt5}{2},\,0,\,0\right),$ • $\left(±\frac{3\sqrt5-5}{4},\,±\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{4}\right),$ • $\left(±\frac{\sqrt5-2}{2},\,±\frac{\sqrt5-2}{2},\,±\frac{\sqrt5-2}{2},\,±\frac12\right),$ • $\left(±\frac{7-3\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4}\right),$ together with all the even permutations of:

• $\left(±\frac{7-3\sqrt5}{4},\,±\frac{3-\sqrt5}{4},\,±\frac12,\,0\right),$ • $\left(±\frac{\sqrt5-2}{2},\,±\frac{3\sqrt5-5}{4},\,0,\,±\frac{\sqrt5-1}{4}\right),$ • $\left(±\frac{\sqrt5-2}{2},\,±\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{2},\,±\frac{\sqrt5-1}{4}\right).$ 