Hecatonicosihexacosidishecatonicosachoron

Hecatonicosihexacosidishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Hixady
Coxeter diagram	x3o3x5x5/3*a ()
Elements
Cells	600 cuboctahedra, 120 truncated dodecahedra, 120 quasitruncated great stellated dodecahedra, 120 quasitruncated dodecadodecahedra
Faces	2400+2400 triangles, 3600 squares, 1440 decagons, 1440 decagrams
Edges	3600+7200+7200
Vertices	7200
Vertex figure	Skewed Rectangular pyramid, base edge lengths 1, √2, 1, √2; lateral edge lengths √(5+√5)/2, √(5+√5)/2, √(5–√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	3
Hypervolume	9625
Dichoral angles	Co–3–quit gissid: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Quit gissid–10/3–quitdid: 144°
	Co–4–quitdid: 135°
	Tid–10–quitdid: 108°
	Co–3–tid: $\arccos \left({\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 82.23876^{\circ }$
Related polytopes
Army	Semi-uniform Prahi, edge lengths ${\sqrt {5}}-2$ (pentagons), ${\frac {3-{\sqrt {5}}}{2}}$ (triangles), ${\frac {{\sqrt {5}}-1}{2}}$ (sides of pentagonal prisms)
Regiment	Hixady
Conjugate	Hecatonicosihexacosidishecatonicosachoron
Abstract & topological properties
Flag count	230400
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The hecatonicosihexacosidishecatonicosachoron, or hixady, is a nonconvex uniform polychoron that consists of 600 cuboctahedra, 120 truncated dodecahedra, 120 quasitruncated great stellated dodecahedra, and 120 quasitruncated dodecadodecahedra. 1 truncated dodecahedron, 1 quasitruncated great stellated dodecahedron, 1 cuboctahedron, and 2 quasitruncated dodecadodecahedra join at each vertex.

Vertex coordinates

The vertices of a hecatonicosihexacosidishecatonicosachoron of edge length 1 are given by all permutations of:

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

Plus all even permutations of:

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

Related polychora

The hecatonicosihexacosidishecatonicosachoron is the colonel of a 3-member regiment that also includes the hexacosihecatonicosidishecatonicosachoron and rhombic hexacosidishecatonicosachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#418).

• Klitzing, Richard. "hixady".