Quasiprismatorhombated grand hexacosichoron

Quasiprismatorhombated grand hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Quippirgax
Coxeter diagram	x5/3x3o3x ()
Elements
Cells	1200 triangular prisms, 720 decagrammic prisms, 600 cuboctahedra, 120 quasitruncated great stellated dodecahedra
Faces	2400+2400 triangles, 3600+3600 squares, 1440 decagrams
Edges	3600+7200+7200
Vertices	7200
Vertex figure	Skewed rectangular pyramid, base edge lengths 1, √2, 1, √2; lateral edge lengths √2, √2, √(5–√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {48-21{\sqrt {5}}}}\approx 1.02106$
Hypervolume	$25{\frac {3367-1475{\sqrt {5}}}{4}}\approx 429.99833$
Dichoral angles	Stiddip–4–trip: $\arccos \left(-{\sqrt {\frac {10-2{\sqrt {5}}}{15}}}\right)\approx 127.37737^{\circ }$
	Quit gissid–10/3–stiddip: 126°
	Quit gissid–3–co: $\arccos \left({\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 82.23876^{\circ }$
	Co–4–stiddip: $\arccos \left({\sqrt {\frac {5-2{\sqrt {5}}}{10}}}\right)\approx 76.71747^{\circ }$
	Co–3–trip: $\arccos \left({\frac {{\sqrt {30}}-{\sqrt {6}}}{8}}\right)\approx 67.76124^{\circ }$
Related polytopes
Army	Semi-uniform Prahi, edge lengths $9-4{\sqrt {5}}$ (pentagons), ${\frac {7-3{\sqrt {5}}}{2}}$ (triangles), ${\sqrt {5}}-2$ (sides of pentagonal prisms)
Regiment	Quippirgax
Conjugate	Prismatorhombated hexacosichoron
Abstract & topological properties
Flag count	230400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The quasiprismatorhombated grand hexacosichoron, or quippirgax, is a nonconvex uniform polychoron that consists of 1200 triangular prisms, 720 decagrammic prisms, 600 cuboctahedra, and 120 quasitruncated great stellated dodecahedra. 1 triangular prism, 2 decagrammic prisms, 1 cuboctahedron, and 1 quasitruncated great stellated dodecahedron join at each vertex. It can be obtained by quasiruncitruncating the great grand stellated hecatonicosachoron.

Vertex coordinates

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

• $\left(0,\,\pm 1,\,\pm {\frac {7-3{\sqrt {5}}}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {8-3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {9-4{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{2}}\right),$

Plus all even permutations of:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {9{\sqrt {5}}-17}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}}\right),$
• $\left(0,\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {7{\sqrt {5}}-13}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}}\right),$
• $\left(0,\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {19-7{\sqrt {5}}}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {17-7{\sqrt {5}}}{4}},\,\pm {\frac {5-2{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm (3-{\sqrt {5}}),\,\pm {\frac {5{\sqrt {5}}-11}{4}},\,\pm {\frac {13-5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {7{\sqrt {5}}-15}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {19-7{\sqrt {5}}}{4}},\,\pm ({\sqrt {5}}-2)\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {9{\sqrt {5}}-17}{4}},\,\pm {\frac {3-{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {17-7{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-9}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {9-4{\sqrt {5}}}{2}},\,\pm {\frac {5-2{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {7{\sqrt {5}}-13}{4}},\,\pm {\frac {13-5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {7{\sqrt {5}}-15}{4}},\,\pm (3-{\sqrt {5}})\right),$
• $\left(\pm 1,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {9-4{\sqrt {5}}}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{4}}\right),$
• $\left(\pm 1,\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {7{\sqrt {5}}-15}{4}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-9}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm (3-{\sqrt {5}})\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {9{\sqrt {5}}-17}{4}},\,\pm {\frac {{\sqrt {5}}-2}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {9-4{\sqrt {5}}}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{2}},\,\pm {\frac {13-5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm (3-{\sqrt {5}})\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm {\frac {13-5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {9-4{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {13-5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm ({\sqrt {5}}-2),\,\pm {\frac {7-3{\sqrt {5}}}{2}},\,\pm (3-{\sqrt {5}})\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {5{\sqrt {5}}-9}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{2}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {19-7{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm {\frac {5-2{\sqrt {5}}}{2}}\right),$
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm ({\sqrt {5}}-2),\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {5{\sqrt {5}}-11}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {8-3{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-9}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {17-7{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {7{\sqrt {5}}-15}{4}},\,\pm 3{\frac {3-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {5-2{\sqrt {5}}}{2}},\,\pm ({\sqrt {5}}-2),\,\pm {\frac {7{\sqrt {5}}-15}{4}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{2}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {7-3{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-9}{4}}\right),$
• $\left(\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {7{\sqrt {5}}-13}{4}},\,\pm ({\sqrt {5}}-2)\right).$

Related polychora

The quasiprismatorhombated grand hexacosichoron is the colonel of a 3-member regiment that also includes the great prismatohexacosihecatonicosihecatonicosachoron and the great rhombiprismic hexacosihecatonicosachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#391).

• Klitzing, Richard. "quippirgax".