Great dishecatonicosaquasitruncated prismatodishecatonicosachoron

Great dishecatonicosaquasitruncated prismatodishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Gidhiquit paddy
Coxeter diagram	x5/3x3x5/3x5*b ()
Elements
Cells	720 decagrammic prisms, 120 great quasitruncated icosidodecahedra, 120 quasitruncated dodecadodecahedra, 120 icosidodecatruncated icosidodecahedra
Faces	3600+3600 squares, 2400 hexagons, 1440 decagons, 1440+1440 decagrams
Edges	7200+7200+7200+7200
Vertices	14400
Vertex figure	Irregular tetrahedron, edge lengths √2 (2), √3 (1), √(5+√5)/2 (1), and √(5–√5)/2 (2)
Measures (edge length 1)
Circumradius	${\sqrt {15}}-{\sqrt {3}}\approx 2.14093$
Hypervolume	$75(144{\sqrt {5}}-317)\approx 374.53416$
Dichoral angles	Gaquatid–4–stiddip: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	Quitdid–4–stiddip: $\arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }$
	Gaquatid–10/3–quitdid: 108°
	Idtid–10–quitdid: 72°
	Idtid–6–gaquatid: 60°
	Idtid–10/3–stiddip: 54°
Number of external pieces	250920
Level of complexity	931
Related polytopes
Army	Semi-uniform Gidpixhi, edge lengths ${\frac {7-3{\sqrt {5}}}{2}}$ (decagonal prisms), ${\sqrt {5}}-2$ (remaining edges)
Regiment	Gidhiquit paddy
Conjugate	Small dishecatonicosiquasitruncated prismatodishecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	345600
Euler characteristic	–1560
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The great dishecatonicosiquasitruncated prismatodishecatonicosachoron, or gidhiquit paddy, is a nonconvex uniform polychoron that consists of 720 decagrammic prisms, 120 great quasitruncated icosidodecahedra, 120 quasitruncated dodecadodecahedra, and 120 icosidodecatruncated icosidodecahedra. 1 of each type of cell join at each vertex.

Gallery

Vertex coordinates

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

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

plus all even permutations of:

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

External links

• Bowers, Jonathan. "Category 9: Omnitruncates" (#347).

• Bowers, Jonathan. "How to Make Gidhiquit Paddy".

• Klitzing, Richard. "gidhiquit paddy".