Pentagonal duoprismatic prism

Pentagonal duoprismatic prism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Pepip
Coxeter diagram	x x5o x5o
Elements
Tera	10 square-pentagonal duoprisms, 2 pentagonal duoprisms
Cells	25 cubes, 10+20 pentagonal prisms
Faces	50+50 squares, 20 pentagons
Edges	25+100
Vertices	50
Vertex figure	Tetragonal disphenoidal pyramid, edge lengths (1+√5)/2 (disphenoid bases) and √2 (remaining edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {25+4{\sqrt {5}}}{20}}}\approx 1.30277$
Hypervolume	${\frac {5(5+2{\sqrt {5}})}{16}}\approx 2.96004$
Diteral angles	Squipdip–pip–squipdip: 108°
	Squipdip–cube–squipdip: 90°
	Pedip–pip–squipdip: 90°
Height	1
Central density	1
Number of external pieces	12
Level of complexity	15
Related polytopes
Army	Pepip
Regiment	Pepip
Dual	Pentagonal duotegmatic tegum
Conjugate	Pentagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H2≀S2×A1, order 400
Convex	Yes
Nature	Tame

The pentagonal duoprismatic prism or pepip, also known as the pentagonal-pentagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal duoprisms and 10 square-pentagonal duoprisms. Each vertex joins 4 square-pentagonal duoprisms and 1 pentagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates

The vertices of a pentagonal duoprismatic prism of edge length 1 are given by:

• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}}\right).$

Representations

A pentagonal duoprismatic prism has the following Coxeter diagrams:

• x x5o x5o (full symmetry)
• xx5oo xx5oo&#x (pentagonal duoprism atop pentagonal duoprism)

External links

• Klitzing, Richard. "pepip".