Hecatonicosihecatonicosaquasitruncated dishecatonicosachoron

Hecatonicosihecatonicosaquasitruncated dishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Hihiquatady
Coxeter diagram	o3x5x3x5/3*b ()
Elements
Cells	120 truncated dodecahedra, 120 quasitruncated great stellated dodecahedra, 120 icosidodecatruncated icosidodecahedra
Faces	2400 triangles, 1200 hexagons, 1440 decagons, 1440 decagrams
Edges	3600+3600+7200
Vertices	7200
Vertex figure	Sphenoid, edge lengths 1 (1), √3 (1), √(5+√5)/2 (2), and √(5–√5)/2 (2)
Measures (edge length 1)
Circumradius	${\sqrt {13}}\approx 3.60555$
Hypervolume	${\frac {25375}{2}}=12687.5$
Dichoral angles	Idtid–10/3–quit gissid: 144°
	Idtid–6–idtid: 120°
	Tid–3–quit gissid: 120°
	Idtid–10–tid: 108°
Number of external pieces	8520
Level of complexity	57
Related polytopes
Army	Semi-uniform Prahi, edge lengths ${\sqrt {5}}-2$ (pentagons), ${\frac {3-{\sqrt {5}}}{2}}$ (triangles), 1 (sides of pentagonal prisms)
Regiment	Hihiquatady
Conjugate	Hecatonicosihecatonciosiquasitruncated dishecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	172800
Euler characteristic	–1080
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The hecatonicosihecatonicosiquasitruncated dishecatonicosachoron, or hihiquatady, is a nonconvex uniform polychoron that consists of 120 truncated dodecahedra, 120 quasitruncated great stellated dodecahedra, and 120 icosidodecatruncated icosidodecahedra. 1 truncated dodecahedron, 1 quasitruncated great stellated dodecahedron, and 2 icosidodecatruncated icosidodecahedra join at each vertex.

Cross-sections

Vertex coordinates

The vertices of a hecatonicosihecatonicosiquasitruncated dishecatonicosachoron of edge length 1 are given by all even permutations of:

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

External links

• Bowers, Jonathan. "Category 8: Great Rhombates" (#327).

• Klitzing, Richard. "hihiquatady".