Great quasidisprismatohecatonicosihecatonicosachoron

Great quasidisprismatohecatonicosihecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Gaquidphihi
Coxeter diagram	x5/3x3x5x ()
Elements
Cells	720 decagonal prisms, 720 decagrammic prisms, 120 great rhombicosidodecahedra, 120 great quasitruncated icosidodecahedra
Faces	3600+3600+3600 squares, 2400 hexagons, 1440 decagons, 1440 decagrams
Edges	7200+7200+7200+7200
Vertices	14400
Vertex figure	Irregular tetrahedron, edge lengths √2 (3), √3 (1), √(5+√5)/2 (1), and √(5–√5)/2 (1)
Measures (edge length 1)
Circumradius	$3{\sqrt {2}}\approx 4.24264$
Hypervolume	2850
Dichoral angles	Gaquatid–10/3–stiddip: 162°
	Dip–4–stiddip: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	Grid–10–dip: 126°
	Gaquatid–6–grid: 60°
	Grid–4–stiddip: $\arccos \left({\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 58.28253^{\circ }$
	Gaquatid–4–dip: $\arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }$
Number of external pieces	105240
Level of complexity	363
Related polytopes
Army	Semi-uniform Gidpixhi, edge lengths ${\frac {3-{\sqrt {5}}}{2}}$ (sides of ditrigonal prisms), ${\sqrt {5}}-2$ (remaining edges of dipentagons), ${\frac {7-3{\sqrt {5}}}{2}}$ (non-dipentagonal edges of great rhombicosidodecahedra)), ${\frac {3{\sqrt {5}}-5}{2}}$ (edges not in great rhombicosidodecahedra)
Regiment	Gaquidphihi
Conjugate	Great quasidisprismatohecatonicosihecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	345600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The great quasidisprismatohecatonicosihecatonicosachoron, or gaquidphihi, is a nonconvex uniform polychoron that consists of 720 decagonal prisms, 720 decagrammic prisms, 120 great rhombicosidodecahedron, and 120 great quasitruncated icosidodecahedra. 1 of each type of cell join at each vertex. It is the quasiomnitruncate of the grand hecatonicosachoron and the great stellated hecatonicosachoron.

Cross-sections

Vertex coordinates

Vertex coordinates for a great quasidisprismatohecatonicosihecatonicosachoron of edge length 1 are given by all permutations of:

• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {6+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {6-{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {5}{2}},\,\pm {\frac {3{\sqrt {5}}}{2}}\right),$
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {11+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\frac {11-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}}\right),$
• $\left(\pm {\frac {9+{\sqrt {5}}}{4}},\,\pm {\frac {9+{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}}\right),$

Plus all even permutations of:

• $\left(\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {6+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {9+{\sqrt {5}}5}{4}}\right),$
• $\left(\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {7-3{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {3+5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {11+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {6+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm ({\sqrt {5}}-1),\,\pm {\frac {9+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {6+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {13+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{2}},\,\pm {\frac {9-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm (1+{\sqrt {5}}),\,\pm {\frac {13-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {13+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {5{\sqrt {5}}-3}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-3}{4}},\,\pm {\frac {9+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {9-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {11-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {9+{\sqrt {5}}}{4}},\,\pm {\frac {5}{2}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {1}{2}},\,\pm {\sqrt {5}},\,\pm {\frac {3+5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}-5}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {6+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-5}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm (1+{\sqrt {5}})\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {11-{\sqrt {5}}}{4}},\,\pm {\frac {3+5{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {13+{\sqrt {5}}}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm ({\sqrt {5}}-1),\,\pm {\frac {13+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{2}},\,\pm {\frac {9+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5{\sqrt {5}}-3}{4}},\,\pm {\frac {11+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {9+{\sqrt {5}}}{4}},\,\pm {\frac {13-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {6-{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}-1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5{\sqrt {5}}-3}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\sqrt {5}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {13+{\sqrt {5}}}{4}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {5{\sqrt {5}}-1}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm ({\sqrt {5}}-1),\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{4}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {13-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {11+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {3}{2}}\right),$
• $\left(\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {2{\sqrt {5}}-3}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm (1+{\sqrt {5}})\right),$
• $\left(\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm (1+{\sqrt {5}}),\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {5}{2}}\right),$
• $\left(\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {3+5{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {4-{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}}\right),$
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm ({\sqrt {5}}-1),\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+5{\sqrt {5}}}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}}\right),$
• $\left(\pm 3{\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm 1,\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {11+{\sqrt {5}}}{4}},\,\pm {\frac {11-{\sqrt {5}}}{4}}\right),$
• $\left(\pm 1,\,\pm ({\sqrt {5}}-1),\,\pm (1+{\sqrt {5}}),\,\pm {\sqrt {5}}\right),$
• $\left(\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {6-{\sqrt {5}}}{2}}\right),$
• $\left(\pm 1,\,\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {3{\sqrt {5}}}{2}},\,\pm {\frac {1+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {6-{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {7-{\sqrt {5}}}{4}},\,\pm {\sqrt {5}},\,\pm {\frac {9+{\sqrt {5}}}{4}},\,\pm {\frac {5}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+3{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {13-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\frac {5}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {3{\sqrt {5}}-1}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {11-{\sqrt {5}}}{4}},\,\pm {\frac {9+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {13-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {9-{\sqrt {5}}}{4}},\,\pm {\sqrt {5}},\,\pm {\frac {9+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {6-{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right).$

External links

• Bowers, Jonathan. "Category 9: Omnitruncates" (#338).

• Klitzing, Richard. "gaquidphihi".