# Great rhombated grand hexacosichoron

Great rhombated grand hexacosichoron
Rank4
TypeUniform
Notation
Bowers style acronymGraggix
Coxeter diagramo5/2x3x3x ()
Elements
Cells720 pentagrammic prisms, 600 truncated octahedra, 120 truncated great icosahedra
Faces3600 squares, 1440 pentagrams, 1200+2400 hexagons
Edges3600+3600+7200
Vertices7200
Vertex figureSphenoid, edge lengths 2 (1), (5–1)/2 (2), and 3 (3)
Measures (edge length 1)
Circumradius$\displaystyle \sqrt{43-18\sqrt5} \approx 1.65855$
Hypervolume$\displaystyle 45\frac{673\sqrt5-1395}{4} \approx 1236.07967$
Dichoral anglesTiggy–5/2–stip: 126°
Toe–4–stip: $\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{10}}\right) \approx 103.28253^\circ$
Tiggy–6–toe: $\displaystyle \arccos\left(-\frac{\sqrt{7-3\sqrt5}}{4}\right) \approx 97.76124^\circ$
Toe–6–toe: $\displaystyle \arccos\left(\frac{3\sqrt5-1}{8}\right) \approx 44.47751^\circ$
Number of external pieces294480
Level of complexity1052
Related polytopes
ArmySemi-uniform Prahi, edge lengths $\displaystyle 7-3\sqrt5$ (pentagons), $\displaystyle \frac{7-3\sqrt5}{2}$ (triangles), ${\displaystyle {\frac {5{\sqrt {5}}-11}{2}}}$ (sides of pentagonal prisms)
RegimentGraggix
ConjugateGreat rhombated hexacosichoron
Convex coreHexacosichoron
Abstract & topological properties
Flag count172800
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

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.

## Vertex coordinates

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

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

plus all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3{\sqrt {5}}-5}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 5{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {17-7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm 1,\,\pm ({\sqrt {5}}-1),\,\pm 2({\sqrt {5}}-2)\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {11-3{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-11}{4}},\,\pm {\frac {7-2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {13-5{\sqrt {5}}}{4}},\,\pm {\frac {13-3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-7}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 2({\sqrt {5}}-2)\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {7-2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{2}},\,\pm {\frac {13-3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 1,\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm 3{\frac {3{\sqrt {5}}-5}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm 2({\sqrt {5}}-2)\right)}$,
• $\displaystyle \left(\pm \frac{\sqrt5-1}{4},\,\pm 3\frac{3-\sqrt5}{4},\,\pm \frac{3\sqrt5-5}{2},\,\pm \frac{7-2\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{\sqrt5-1}{4},\,\pm (\sqrt5-2),\,\pm 3\frac{\sqrt5-2}{2},\,\pm \frac{13-3\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm 1,\,\pm \frac{\sqrt5-1}{2},\,\pm 3\frac{3-\sqrt5}{2},\,\pm (\sqrt5-2)\right)$ ,
• $\displaystyle \left(\pm 1,\,\pm \frac{7-\sqrt5}{4},\,\pm 3\frac{\sqrt5-2}{2},\,\pm \frac{13-5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm 1,\,\pm 3\frac{\sqrt5-1}{4},\,\pm \frac{17-7\sqrt5}{4},\,\pm \frac{4-\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm 1,\,\pm \frac{3\sqrt5-5}{4},\,\pm \frac{8-3\sqrt5}{2},\,\pm \frac{11-3\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm \frac32,\,\pm \frac{\sqrt5-1}{2},\,\pm \frac{17-7\sqrt5}{4},\,\pm \frac{7-3\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm \frac32,\,\pm \frac{5-\sqrt5}{4},\,\pm \frac{7-3\sqrt5}{2},\,\pm \frac{5\sqrt5-11}{4}\right)$ ,
• $\displaystyle \left(\pm \frac32,\,\pm \frac{\sqrt5-2}{2},\,\pm \frac{8-3\sqrt5}{2},\,\pm \frac{5-2\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{\sqrt5-1}{2},\,\pm 3\frac{3-\sqrt5}{4},\,\pm \frac{3\sqrt5-4}{2},\,\pm \frac{13-5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm \frac{\sqrt5-1}{2},\,\pm \frac{11-3\sqrt5}{4},\,\pm \frac{5\sqrt5-7}{4},\,\pm 3\frac{\sqrt5-2}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{5-\sqrt5}{4},\,\pm \frac{7-\sqrt5}{4},\,\pm \frac{3\sqrt5-5}{2},\,\pm 3\frac{\sqrt5-2}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{5-\sqrt5}{4},\,\pm \frac{3\sqrt5-5}{4},\,\pm 3\frac{3-\sqrt5}{2},\,\pm \frac{4-\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{5-\sqrt5}{4},\,\pm (\sqrt5-1),\,\pm \frac{8-3\sqrt5}{2},\,\pm 3\frac{3-\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm \frac{\sqrt5-2}{2},\,\pm \frac{7-\sqrt5}{4},\,\pm \frac{7-3\sqrt5}{4},\,\pm 3\frac{3-\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{\sqrt5-2}{2},\,\pm \frac{4-\sqrt5}{2},\,\pm \frac{3\sqrt5-4}{2},\,\pm 3\frac{\sqrt5-2}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{\sqrt5-2}{2},\,\pm 3\frac{3-\sqrt5}{4},\,\pm 5\frac{\sqrt5-1}{4},\,\pm \frac{7-3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{7-\sqrt5}{4},\,\pm 3\frac{\sqrt5-1}{4},\,\pm \frac{8-3\sqrt5}{2},\,\pm (\sqrt5-2)\right)$ ,
• $\displaystyle \left(\pm 3\frac{\sqrt5-1}{4},\,\pm \frac{4-\sqrt5}{2},\,\pm \frac{5\sqrt5-7}{4},\,\pm \frac{7-3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm 3\frac{\sqrt5-1}{4},\,\pm (\sqrt5-2),\,\pm \frac{3\sqrt5-4}{2},\,\pm \frac{5\sqrt5-11}{4}\right)$ ,
• $\displaystyle \left(\pm 3\frac{\sqrt5-1}{4},\,\pm \frac{5\sqrt5-7}{4},\,\pm \frac{3\sqrt5-5}{2},\,\pm \frac{5-2\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{3\sqrt5-5}{4},\,\pm \frac{7-3\sqrt5}{4},\,\pm \frac{3\sqrt5-4}{2},\,\pm \frac{3\sqrt5-5}{2}\right)$ ,
• $\displaystyle \left(\pm \frac{3\sqrt5-5}{4},\,\pm 5\frac{\sqrt5-1}{4},\,\pm 3\frac{\sqrt5-2}{2},\,\pm (\sqrt5-2)\right)$ ,
• $\displaystyle \left(\pm (\sqrt5-1),\,\pm \frac{7-3\sqrt5}{4},\,\pm \frac{5\sqrt5-7}{4},\,\pm 3\frac{\sqrt5-2}{2}\right)$ .