Great stellated hecatonicosachoron

Great stellated hecatonicosachoron
(OFF file)
Rank	4
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Gishi
Coxeter diagram	x5/2o3o5o ()
Schläfli symbol	{5/2,3,5}
Elements
Cells	120 great stellated dodecahedra
Faces	720 pentagrams
Edges	720
Vertices	120
Vertex figure	Icosahedron, edge length (√5–1)/2
Edge figure	gissid 5/2 gissid 5/2 gissid 5/2 gissid 5/2 gissid 5/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt5-1}{2} \approx 0.61803$
Edge radius	$\frac{\sqrt{5-2\sqrt5}}{2} \approx 0.36327$
Face radius	$\sqrt{\frac{5-2\sqrt5}{5}} \approx 0.32492$
Inradius	$\frac{\sqrt5-1}{4} \approx 0.30902$
Hypervolume	$15\frac{25-11\sqrt5}{4} \approx 1.51220$
Dichoral angle	144°
Central density	20
Number of pieces	2400
Level of complexity	8
Related polytopes
Army	Ex
Regiment	Gishi
Company	Gishi
Dual	Grand hecatonicosachoron
Conjugate	Grand hecatonicosachoron
Convex core	Hecatonicosachoron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

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.

Cross-sections

Vertex coordinates

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

• $\left(±\frac{\sqrt5-1}{2},\,0,\,0,\,0\right),$
• $\left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4}\right),$

and all even permutations of:

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

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.

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#12).

• Klitzing, Richard. "gishi".

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

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