Small rhombated grand hexacosichoron

Small rhombated grand hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Sirgax
Coxeter diagram	o5/2x3o3x ()
Elements
Cells	600 cuboctahedra, 720 pentagrammic prisms, 120 great icosidodecahedra
Faces	1200+2400 triangles, 3600 squares, 1440 pentagrams
Edges	3600+7200
Vertices	3600
Vertex figure	Wedge, edge lengths 1 (two edges of base rectangle and top edge), √2 (side edges), and (√5–1)/2 (remaining base edges)
Measures (edge length 1)
Circumradius	${\sqrt {19-8{\sqrt {5}}}}\approx 1.05425$
Hypervolume	$5{\frac {591{\sqrt {5}}-1240}{2}}\approx 203.79044$
Dichoral angles	Gid–5/2–stip: 126°
	Co–4–stip: $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{10}}}\right)\approx 103.28253^{\circ }$
	Gid–3–co: $\arccos \left(-{\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 97.76124^{\circ }$
	Co–3–co: $\arccos \left({\frac {3{\sqrt {5}}-1}{8}}\right)\approx 44.47751^{\circ }$
Related polytopes
Army	Semi-uniform Srahi, edge lengths ${\frac {5{\sqrt {5}}-11}{2}}$ (pentagons), ${\sqrt {5}}-2$ (octahedra)
Regiment	Sirgax
Conjugate	Small rhombated hexacosichoron
Abstract & topological properties
Flag count	129600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small rhombated grand hexacosichoron, or sirgax, is a nonconvex uniform polychoron that consists of 600 cuboctahedra, 720 pentagrammic prisms, and 120 great icosidodecahedra. 1 great icosidodecahedron, 2 pentagrammic prisms, and 2 cuboctahedra join at each vertex. It can be obtained by cantellating the grand hexacosichoron.

Vertex coordinates

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

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

together with all even permutations of:

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

Related polychora

The small rhombated grand hexacosichoron is the colonel of a seven-member regiment. Its other members include the grand retrosphenoverted hecatonicosihexacosihecatonicosachoron, rhombic great hexacosihecatonicosachoron, pseudorhombic great dishecatonicosachoron, grand rhombic great hexacosihecatonicosachoron, great dishecatonicosintercepted hexacosihecatonicosachoron, and hecatonicosintercepted prismatohecatonicosihexacosichoron.

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#214).

• Klitzing, Richard. "sirgax".