Great grand stellated hecatonicosachoron

Great grand stellated hecatonicosachoron
(OFF file)
Rank	4
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Gogishi
Coxeter diagram	x5/2o3o3o ()
Schläfli symbol	{5/2,3,3}
Elements
Cells	120 great stellated dodecahedra
Faces	720 pentagrams
Edges	1200
Vertices	600
Vertex figure	Tetrahedron, edge length (√5–1)/2
Edge figure	gissid 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 angle	72°
Central density	191
Number of pieces	9600
Level of complexity	30
Related polytopes
Army	Hi
Regiment	Gogishi
Dual	Grand hexacosichoron
Conjugate	Hecatonicosachoron
Convex core	Hecatonicosachoron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

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.

Cross-sections

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:

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

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#16).

• Bowers, Jonathan. "How to Make Gogishi".

• Klitzing, Richard. "gogishi".

• Nan Ma. "Great grand stellated 120-cell {5/2, 3, 3}".

• Wikipedia Contributors. "Great grand stellated 120-cell".