Great rhombated pentachoric prism

Great rhombated pentachoric prism
File:Great rhombated pentachoric prism.png
Rank	5
Type	Uniform
Notation
Bowers style acronym	Grippip
Coxeter diagram	x x3x3x3o
Elements
Tera	10 triangular-square duoprisms, 5 truncated tetrahedral prisms, 5 truncated octahedral prisms, 2 great rhombated pentachora
Cells	20+20 triangular prisms, 30 cubes, 10+20 hexagonal prisms, 10 truncated tetrahedra, 10 truncated octahedra
Faces	40 triangles, 30+30+60+60 squares, 20+40 hexagons
Edges	60+60+60+120
Vertices	120
Vertex figure	Sphenoidal pyramid, edge lengths 1, √2, √3 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {365}}{10}}\approx 1.91050$
Hypervolume	${\frac {287{\sqrt {5}}}{32}}\approx 20.05473$
Diteral angles	Tuttip–trip–tisdip: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }$
	Tope–cube–tisdip: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Tope–hip–tuttip: $\arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }$
	Grip–toe–tope: 90°
	Grip–tut–tuttip: 90°
	Grip–trip–tisdip: 90°
	Tope–hip–tope: $\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$
Height	1
Central density	1
Number of external pieces	22
Level of complexity	60
Related polytopes
Army	Grippip
Regiment	Grippip
Dual	Sphenoidal hexecontachoric tegum
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A4×A1, order 240
Convex	Yes
Nature	Tame

The great rhombated pentachoric prism or grippip is a prismatic uniform polyteron that consists of 2 great rhombated pentachora, 5 truncated octahedral prisms, 5 truncated tetrahedral prisms, and 10 triangular-square duoprisms. 1 great rhombated pentachoron, 2 truncated octahedral prisms, 1 truncated tetrahedral prism, and 1 triangular-square duoprism join at each vertex. As the name suggests, it can be obtained as a prism based on the great rhombated pentachoron, which also makes it a convex segmentoteron.

Vertex coordinates

The vertices of a great rhombated pentachoric prism of edge length 1 are given by:

• $\left({\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {6}}{4}},\,{\frac {\sqrt {3}}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {6}}{4}},\,-{\sqrt {3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{20}},\,{\frac {5{\sqrt {6}}}{12}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {3}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{20}},\,{\frac {5{\sqrt {6}}}{12}},\,-{\frac {2{\sqrt {3}}}{3}},\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{20}},\,{\frac {5{\sqrt {6}}}{12}},\,{\frac {5{\sqrt {3}}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{20}},\,-{\frac {7{\sqrt {6}}}{12}},\,{\frac {\sqrt {3}}{3}},\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {\sqrt {10}}{20}},\,-{\frac {7{\sqrt {6}}}{12}},\,-{\frac {2{\sqrt {3}}}{3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {\sqrt {10}}{5}},\,0,\,{\frac {\sqrt {3}}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {\sqrt {10}}{5}},\,0,\,-{\sqrt {3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {\sqrt {10}}{5}},\,{\frac {\sqrt {6}}{3}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {3}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {\sqrt {10}}{5}},\,{\frac {\sqrt {6}}{3}},\,{\frac {2{\sqrt {3}}}{3}},\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {\sqrt {10}}{5}},\,{\frac {\sqrt {6}}{3}},\,-{\frac {5{\sqrt {3}}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {\sqrt {10}}{5}},\,-{\frac {2{\sqrt {6}}}{3}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {\sqrt {10}}{5}},\,-{\frac {2{\sqrt {6}}}{3}},\,-{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,-{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {3}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {3}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,-{\frac {\sqrt {6}}{6}},\,{\frac {2{\sqrt {3}}}{3}},\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {2{\sqrt {3}}}{3}},\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,-{\frac {\sqrt {6}}{6}},\,-{\frac {5{\sqrt {3}}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,{\frac {\sqrt {6}}{6}},\,{\frac {5{\sqrt {3}}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,\pm {\frac {\sqrt {6}}{2}},\,0,\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left({\frac {3{\sqrt {10}}}{10}},\,\pm {\frac {\sqrt {6}}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {9{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{3}},\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {9{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {2{\sqrt {3}}}{3}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {9{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{4}},\,0,\,\pm 1,\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {9{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {9{\sqrt {10}}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(-{\frac {9{\sqrt {10}}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right).$

External links

• Klitzing, Richard. "Grippip".