Great rhombated grand hexacosichoron

Great rhombated grand hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Graggix
Coxeter diagram	o5/2x3x3x ()
Elements
Cells	720 pentagrammic prisms, 600 truncated octahedra, 120 truncated great icosahedra
Faces	3600 squares, 1440 pentagrams, 1200+2400 hexagons
Edges	3600+3600+7200
Vertices	7200
Vertex figure	Sphenoid, edge lengths √2 (1), (√5–1)/2 (2), and √3 (3)
Measures (edge length 1)
Circumradius	${\sqrt {43-18{\sqrt {5}}}}\approx 1.65855$
Hypervolume	$45{\frac {673{\sqrt {5}}-1395}{4}}\approx 1236.07967$
Dichoral angles	Tiggy–5/2–stip: 126°
	Toe–4–stip: $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{10}}}\right)\approx 103.28253^{\circ }$
	Tiggy–6–toe: $\arccos \left(-{\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 97.76124^{\circ }$
	Toe–6–toe: $\arccos \left({\frac {3{\sqrt {5}}-1}{8}}\right)\approx 44.47751^{\circ }$
Number of external pieces	294480
Level of complexity	1052
Related polytopes
Army	Semi-uniform Prahi, edge lengths $7-3{\sqrt {5}}$ (pentagons), ${\frac {7-3{\sqrt {5}}}{2}}$ (triangles), ${\frac {5{\sqrt {5}}-11}{2}}$ (sides of pentagonal prisms)
Regiment	Graggix
Conjugate	Great rhombated hexacosichoron
Convex core	Hexacosichoron
Abstract & topological properties
Flag count	172800
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The great rhombated grand hexacosichoron, or graggix, is a nonconvex uniform polychoron that consists of 720 pentagrammic prisms, 600 truncated octahedra, and 120 truncated great icosahedra. 1 pentagrammic prism, 1 truncated great icosahedron, and 2 truncated octahedra join at each vertex. As the name suggests, it can be obtained by cantitruncating the grand hexacosichoron.

Cross-sections

Vertex coordinates

The vertices of a great rhombated grand hexacosichoron of edge length 1 are given by all permutations of:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3{\sqrt {5}}-4}{2}},\,\pm {\frac {8-3{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}}\right)$,
• $\left(\pm 1,\,\pm 1,\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{2}}\right)$,

plus all even permutations of:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3{\sqrt {5}}-5}{4}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm 5{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {17-7{\sqrt {5}}}{4}}\right)$,
• $\left(0,\,\pm 1,\,\pm ({\sqrt {5}}-1),\,\pm 2({\sqrt {5}}-2)\right)$,
• $\left(0,\,\pm {\frac {11-3{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-11}{4}},\,\pm {\frac {7-2{\sqrt {5}}}{2}}\right)$,
• $\left(0,\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {13-5{\sqrt {5}}}{4}},\,\pm {\frac {13-3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-7}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 2({\sqrt {5}}-2)\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {7-2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{2}},\,\pm {\frac {13-3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 1,\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm 3{\frac {3{\sqrt {5}}-5}{4}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm 2({\sqrt {5}}-2)\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm {\frac {7-2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm ({\sqrt {5}}-2),\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {13-3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm 1,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{2}},\,\pm ({\sqrt {5}}-2)\right)$,
• $\left(\pm 1,\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {13-5{\sqrt {5}}}{4}}\right)$,
• $\left(\pm 1,\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {17-7{\sqrt {5}}}{4}},\,\pm {\frac {4-{\sqrt {5}}}{2}}\right)$,
• $\left(\pm 1,\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm {\frac {11-3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {17-7{\sqrt {5}}}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right)$,
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm {\frac {5-2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-4}{2}},\,\pm {\frac {13-5{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {11-3{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-7}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}}\right)$,
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}}\right)$,
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {4-{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm ({\sqrt {5}}-1),\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-4}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}},\,\pm 5{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm ({\sqrt {5}}-2)\right)$,
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-7}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{2}}\right)$,
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm ({\sqrt {5}}-2),\,\pm {\frac {3{\sqrt {5}}-4}{2}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right)$,
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {5{\sqrt {5}}-7}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm {\frac {5-2{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-4}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{2}}\right)$,
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm 5{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm ({\sqrt {5}}-2)\right)$,
• $\left(\pm ({\sqrt {5}}-1),\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-7}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}}\right)$.

External links

• Bowers, Jonathan. "Category 8: Great Rhombates" (#314).

• Klitzing, Richard. "graggix".