Small rhombated grand stellated hecatonicosachoron

Small rhombated grand stellated hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Sirgashi
Coxeter diagram	x5/2o5x5/2o ()
Elements
Cells	720 pentagrammic prisms, 120 dodecadodecahedra, 120 rhombidodecadodecahedra
Faces	3600 squares, 1440 pentagons, 720+1440 pentagrams
Edges	3600+7200
Vertices	3600
Vertex figure	Wedge, edge lengths (√5–1)/2 (two edges of base rectangle and top edge), (1+√5)/2 (other 2 base edges), and √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {7-2{\sqrt {5}}}}\approx 1.58993$
Hypervolume	$75{\frac {41{\sqrt {5}}-54}{2}}\approx 1412.95451$
Dichoral angles	Did–5/2–stip: 126°
	Raded–4–stip: $\arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }$
	Raded–5–did: 108°
	Raded–5/2–raded: 72°
Related polytopes
Army	Srahi, edge length ${\sqrt {5}}-2$
Regiment	Sirgashi
Conjugate	Small rhombated great hecatonicosachoron
Abstract & topological properties
Flag count	129600
Euler characteristic	–960
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small rhombated grand stellated hecatonicosachoron, or sirgashi, is a nonconvex uniform polychoron that consists of 720 pentagrammic prisms, 120 dodecadodecahedra, and 120 rhombidodecadodecahedra. 1 dodecadodecahedron, 2 pentagrammic prisms, and 2 rhombidodecadodecahedra join at each vertex. it can be obtained by cantellating the grand stellated hecatonicosachoron.

Vertex coordinates

Coordinates for the vertices of a small rhombated grand stellated hecatonicosachoron of edge length 1 are given by all permutations of:

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

together with all even permutations of:

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

Related polychora

The small rhombated grand stellated hecatonicosachoron is the colonel of a regiment with 7 members. Its other members include the great retrosphenoverted hecatonicosidishecatonicosachoron, rhombic great dishecatonicosachoron, pseudorhombic great hecatonicosihexacosichoron, grand rhombic great hecatonicosihexacosichoron, great hecatonicosihexacosintercepted dishecatonicosachoron, and great hexacosintercepted prismatodishecatonicosachoron.

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#207).

• Klitzing, Richard. "sirgashi".