Quasiprismatorhombated grand hecatonicosachoron

Quasiprismatorhombated grand hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Quippirghi
Coxeter diagram	x5/3x3o5x ()
Elements
Cells	720 pentagonal prisms, 720 decagrammic prisms, 120 quasitruncated great stellated dodecahedra, 120 small rhombicosidodecahedra
Faces	2400 triangles, 3600+3600 squares, 1440 pentagons, 1440 decagrams
Edges	3600+7200+7200
Vertices	7200
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths (1+√5)/2, √2, 1, √2; lateral edge lengths √2, √2, √(5–√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {8-{\sqrt {5}}}}\approx 2.40082$
Hypervolume	$5{\frac {1215+421{\sqrt {5}}}{4}}\approx 2695.48077$
Dichoral angles	Quit gissid–10/3–stiddip: 162°
	Pip–4–stiddip: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	Srid–5–pip: 126°
	Srid–3–quit gissid: 60°
	Srid–4–stiddip: $\arccos \left({\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 58.28253^{\circ }$
Related polytopes
Army	Semi-uniform Prahi, edge lengths ${\sqrt {5}}-2$ (pentagonal prisms), ${\frac {3-{\sqrt {5}}}{2}}$ (triangles)
Regiment	Quippirghi
Conjugate	Prismatoquasirhombated great stellated hecatonicosachoron
Abstract & topological properties
Flag count	230400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The quasiprismatorhombated grand hecatonicosachoron, or quippirghi, is a nonconvex uniform polychoron that consists of 720 pentagonal prisms, 720 decagrammic prisms, 120 quasitruncated great stellated dodecahedra, and 120 small rhombicosidodecahedra. 1 pentagonal prism, 2 decagrammic prisms, 1 small rhombicosidodecahedron, and 1 quasitruncated great stellated dodecahedron join at each vertex. It can be obtained by quasiruncitruncating the great stellated hecatonicosachoron.

Gallery

Vertex coordinates

The vertices of a quasiprismatorhombated grand hecatonicosachoron of edge length 1 are given by all permutations of:

• $\left(0,\,\pm {\sqrt {5}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {2{\sqrt {5}}-1}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 1,\,\pm 2\right),$
• $\left(\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2{\sqrt {5}}-1}{2}}\right),$
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {\sqrt {5}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}}\right),$

Plus all even permutations of:

• $\left(0,\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}}\right),$
• $\left(0,\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm 2,\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\sqrt {5}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2{\sqrt {5}}-1}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\sqrt {5}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm 2,\,\pm {\frac {\sqrt {5}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2{\sqrt {5}}-1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 1\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,\pm 2,\,\pm {\frac {7-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {5}}{2}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {7-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 2,\,\pm 3{\frac {{\sqrt {5}}-1}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {2{\sqrt {5}}-1}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right).$

Related polychora

The quasiprismatorhombated grand hecatonicosachoron is the colonel of a three-member ergiment that also includes the grand prismatotrishecatonicosachoron and the rhombiprismic hecatonicosihecatonicosachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#388).

• Klitzing, Richard. "quippirghi".