Prismatorhombated pentachoric prism

Prismatorhombated pentachoric prism
File:Prismatorhombated pentachoric prism.png
Rank	5
Type	Uniform
Notation
Bowers style acronym	Prippip
Coxeter diagram	x x3x3o3x ()
Elements
Tera	10 triangular-square duoprisms, 10 square-hexagonal duoprisms, 5 truncated tetrahedral prisms, 5 cuboctahedral prisms, 2 prismatorhombated pentachora
Cells	20+20+20 triangular prisms, 30+30 cubes, 20+20 hexagonal prisms, 10 truncated tetrahedra, 10 cuboctahedra
Faces	40+40 triangles, 30+60+60+60+60 squares, 40 hexagons
Edges	60+60+120+120
Vertices	120
Vertex figure	Skew rectangular pyramidal pyramid, edge lengths 1, √2, √3 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {285}}{10}}\approx 1.68819$
Hypervolume	${\frac {237{\sqrt {5}}}{32}}\approx 16.56088$
Diteral angles	Shiddip–cube–tisdip: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Cope–trip–tisdip: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76125^{\circ }$
	Tuttip–hip–shiddip: $\arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76125^{\circ }$
	Cope–cube–shiddip: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
	Cope–trip–tuttip: $\arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }$
	Prip–co–cope: 90°
	Prip–tut–tuttip: 90°
	Prip–hip–shiddip: 90°
	Prip–trip–tisdip: 90°
Height	1
Central density	1
Number of external pieces	32
Level of complexity	80
Related polytopes
Army	Prippip
Regiment	Prippip
Dual	Rhombipyramidal hexecontachoric tegum
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A4×A1, order 240
Convex	Yes
Nature	Tame

The prismatorhombated pentachoric prism or prippip is a prismatic uniform polyteron. It consists of 2 prismatorhombated pentachora, 5 cuboctahedral prisms, 5 truncated tetrahedral prisms, 10 square-hexagonal duoprisms, and 10 triangular-square duoprisms. 1 prismatorhombated pentachoron, 1 cuboctahedral prism, 1 truncated tetrahedral prism, 2 square-hexagonal duoprisms, and 1 triangular-square duoprism join at each vertex. As the name suggests, it is a prism based on the prismatorhombated pentachoron, which also makes it a convex segmentoteron.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "Prippip".