Quasitruncated great stellated hecatonicosachoron

Quasitruncated great stellated hecatonicosachoron
	(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Quit gishi
Coxeter diagram	x5/3x3o5o ()
Elements
Cells	120 icosahedra, 120 quasitruncated great stellated dodecahedra
Faces	2400 triangles, 720 decagrams
Edges	720+3600
Vertices	1440
Vertex figure	Pentagonal pyramid, edge lengths 1 (base) and √(5–√5)/2 (side)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral angles	Quit gissid–10/3–quit gissid: 144°
	Quit gissid–3–ike: 60°
Central density	—20
Number of external pieces	35040
Level of complexity	120
Related polytopes
Army	Semi-uniform Tex, edge lengths (icosahedra), (surrounded by truncated tetrahedra)
Regiment	Quit gishi
Conjugate	Truncated grand hecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	57600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

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.

Gallery[edit | edit source]

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

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

plus all even permutations of:

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

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