Small hexacositrishecatonicosachoron

Small hexacositrishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Sixathi
Coxeter diagram	x3x3o5x5/3*a ()
Elements
Cells	600 truncated tetrahedra, 120 quasitruncated small stellated dodecahedra, 120 small rhombicosidodecahedra, 120 great quasitruncated icosidodecahedra
Faces	2400 triangles, 3600 squares, 1440 pentagons, 2400 hexagons, 1440 decagrams
Edges	3600+7200+7200
Vertices	7200
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths 1, √2, (1+√5)/2, √2; lateral edge lengths √3, √3, √(5–√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {10-2{\sqrt {5}}}}\approx 2.35114$
Hypervolume	$5{\frac {925+704{\sqrt {5}}}{4}}\approx 3123.98982$
Dichoral angles	Tut–6–gaquatid: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Quit sissid–10/3–gaquatid: 144°
	Srid–5–quit sissid: 108°
	Srid–3–tut: $\arccos \left(-{\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 97.76124^{\circ }$
	Srid–4–gaquatid: 90°
Number of external pieces	94080
Related polytopes
Army	Semi-uniform Prahi, edge lengths ${\frac {3-{\sqrt {5}}}{2}}$ (pentagons), ${\sqrt {5}}-2$ (triangles), ${\frac {7-3{\sqrt {5}}}{2}}$ (sides of pentagonal prisms)
Regiment	Sixathi
Conjugate	Great hexacositrishecatonicosachoron
Abstract & topological properties
Flag count	230400
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small hexacositrishecatonicosachoron, or sixathi, is a nonconvex uniform polychoron that consists of 600 truncated tetrahedra, 120 quasitruncated small stellated dodecahedra, 120 small rhombicosidodecahedra, and 120 great quasitruncated icosidodecahedra. 1 truncated tetrahedron, 1 quasitruncated small stellated dodecahedron, 1 small rhombicosidodecahedron, and 2 great quasitruncated icosidodecahedra join at each vertex.

Vertex coordinates

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

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

plus all even permutations of:

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

Related polychora

The small hexacositrishecatonicosachoron is the colonel of a 3-member regiment that also includes the hecatonicosihexacosiquasitruncated hexacosihecatonicosachoron and small rhombic trishecatonicosachoron.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#421).

• Bowers, Jonathan. "How to Make Sixathi".

• Klitzing, Richard. "sixathi".