# Quasitruncated great stellated hecatonicosachoron

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Quasitruncated great stellated hecatonicosachoron
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymQuit gishi
Coxeter diagramx5/3x3o5o ()
Elements
Cells120 icosahedra, 120 quasitruncated great stellated dodecahedra
Faces2400 triangles, 720 decagrams
Edges720+3600
Vertices1440
Vertex figurePentagonal pyramid, edge lengths 1 (base) and (5–5)/2 (side)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{13-5\sqrt5}{2}} ≈ 0.95385}$
Hypervolume${\displaystyle 50(21\sqrt5-46) ≈ 47.87318}$
Dichoral anglesQuit gissid–10/3–quit gissid: 144°
Quit gissid–3–ike: 60°
Central density—20
Number of external pieces35040
Level of complexity120
Related polytopes
ArmySemi-uniform Tex
RegimentQuit gishi
ConjugateTruncated grand hecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The quasitruncated great stellated hecatonicosachoron, or quit gishi, is a nonconvex uniform polychoron that consists of 120 regular icosahedra and 120 quasitruncated great stellated dodecahedra. One icosahedron and five quasitruncated great stellated dodecahedra join at each vertex. As the name suggests, it can be obtained by quasitruncating the great stellated hecatonicosachoron.

## Vertex coordinates

The vertices of a quasitruncated great stellated hecatonicosachoron of edge length 1 are given by all permutations of:

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

plus all even permutations of:

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

## Related polychora

The quasitruncated great stellated hecatonicosachoron is the colonel of a two-member regiment that also includes the quasitruncated grand stellated hecatonicosachoron.