Small rhombated great grand hecatonicosachoron

Small rhombated great grand hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Sirgaghi
Coxeter diagram	x5o5/2x3o ()
Elements
Cells	1200 triangular prisms 120 great icosidodecahedra 120 rhombidodecadodecahedra
Faces	1440 pentagrams 2400 triangles 3600 squares 720 pentagons
Edges	3600+7200
Vertices	3600
Vertex figure	Wedge, edge lengths (√5–1)/2 (two edges of base rectangle), 1 (other 2 base edges), √2 (side edges), and (1+√5)/2 top edge)
Measures (edge length 1)
Circumradius	${\frac {{\sqrt {30}}-{\sqrt {6}}}{2}}\approx 1.51387$
Hypervolume	$5{\frac {1585-502{\sqrt {5}}}{2}}\approx 1156.23469$
Dichoral angle	Gid–3–trip: 150° Raded–4–trip: $\arccos \left(-{\frac {{\sqrt {15}}-{\sqrt {3}}}{6}}\right)\approx 110.90516^{\circ }$ Raded–5/2–gid: 108° Raded–5–raded: 72°
Related polytopes
Army	Semi-uniform Xhi, edge lengths ${\sqrt {5}}-2$ (pentagons), ${\frac {7-3{\sqrt {5}}}{2}}$ (triangles)
Regiment	Sirgaghi
Conjugate	Small rhombated small stellated hecatonicosachoron
Abstract & topological properties
Flag count	129600
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Flag orbits	9
Convex	No
Nature	Tame

The small rhombated great grand hecatonicosachoron, or sirgaghi, is a nonconvex uniform polychoron that consists of 1200 triangular prisms, 120 great icosidodecahedra, and 120 rhombidodecadodecahedra. 1 great icosidodecahedron, 2 triangular prisms, and 2 rhombidodecadodecahedra join at each vertex. it can be obtained by cantellating the great grand hecatonicosachoron.

Vertex coordinates

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

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

together with all even permutations of:

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

Related polychora

The small rhombated great grand hecatonicosachoron is the colonel of a regiment of 7 members. Its other members include the great retrosphenoverted hecatonicosihexacosihecatonicosachoron, rhombic great hecatonicosihexacosichoron, pseudorhombic great hexacosihecatonicosachoron, grand rhombic great dishecatonicosachoron, great hexacosihecatonicosintercepted dishecatonicosachoron, and great hecatonicosintercepted prismatodishecatonicosachoron.

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#200).

• Klitzing, Richard. "sirgaghi".