Prismatorhombated great hecatonicosachoron

Prismatorhombated great hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Pirghi
Coxeter diagram	x5x5/2o5x ()
Elements
Cells	720 pentagonal prisms 720 decagonal prisms 120 truncated great dodecahedra 120 rhombidodecadodecahedra
Faces	1440 pentagrams 3600+3600 squares 1440 pentagons 1440 decagons
Edges	3600+7200+7200
Vertices	7200
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths (1+√5)/2, √2, (√5–1)/2, √2; lateral edge lengths √2, √2, √(5+√5)/2, √(5+√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {13+5{\sqrt {5}}}}\approx 4.91735$
Hypervolume	$45{\frac {825+359{\sqrt {5}}}{4}}\approx 18312.1695$
Dichoral angle	Tigid–10–dip: 162° Raded–5–pip: 162° Pip–4–dip: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$ Raded–4–dip: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$ Raded–5/2–tigid: 144°
Related polytopes
Army	Semi-uniform Grix, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (pentagonal prisms), ${\frac {3-{\sqrt {5}}}{2}}$ (remaining edges)
Regiment	Pirghi
Conjugate	Quasiprismatorhombated grand stellated hecatonicosachoron
Abstract & topological properties
Flag count	230400
Euler characteristic	–960
Orientable	Yes
Properties
Symmetry	H4, order 14400
Flag orbits	16
Convex	No
Nature	Tame

The prismatorhombated great hecatonicosachoron, or pirghi, is a nonconvex uniform polychoron that consists of 720 pentagonal prisms, 720 decagonal prisms, 120 truncated great dodecahedra, and 120 rhombidodecadodecahedra. 1 pentagonal prism, 2 decagonal prisms, 1 truncated great dodecahedron, and 1 rhombidodecadodecahedron join at each vertex. It can be obtained by runcitruncating the great hecatonicosachoron.

Vertex coordinates

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

• $\left(0,\,\pm (1+{\sqrt {5}}),\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 1,\,\pm (2+{\sqrt {5}})\right)$,
• $\left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)$,

plus all even permutations of:

• $\left(0,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)$,
• $\left(0,\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right)$,
• $\left(0,\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,\pm (2+{\sqrt {5}}),\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm 1\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {7+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm (1+{\sqrt {5}})\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm (1+{\sqrt {5}})\right)$,
• $\left(\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm 1,\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}})\right)$,
• $\left(\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}}\right)$.

Related polychora

The prismatorhombated great hecatonicosachoron is the colonel of a three-member regiment that also includes the small prismatohecatonicosidishecatonicosachoron and the small rhombiprismic hecatonicosihecatonicosachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#373).

• Klitzing, Richard. "pirghi".