Pentagonalenneagonal duoprismatic prism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Pep 

Coxeter diagram  x x5o x9o 

Elements 

Tera  9 squarepentagonal duoprisms, 5 squareenneagonal duoprisms, 2 pentagonalenneagonal duoprisms 

Cells  45 cubes, 5+10 enneagonal prisms, 9+18 pentagonal prisms 

Faces  45+45+90 squares, 18 pentagons, 10 enneagons 

Edges  45+90+90 

Vertices  90 

Vertex figure  Digonal disphenoidal pyramid, edge lengths (1+√5)/2 (disphenoid base 1), 2cos(π/9) (disphenoid base 2), √2 (remaining edges) 

Measures (edge length 1) 

Circumradius  ${\sqrt {\frac {15+2{\sqrt {5}}+{\frac {5}{\sin ^{2}{\frac {\pi }{9}}}}}{20}}}\approx 1.76374$ 

Hypervolume  ${\frac {9{\sqrt {25+10{\sqrt {5}}}}}{16\tan {\frac {\pi }{9}}}}\approx 10.63569$ 

Diteral angles  Squipdip–pip–squipdip: 140° 

 Sendip–ep–sendip: 108° 

 Sendip–cube–squipdip: 90° 

 Peendip–pip–squipdip: 90° 

 Sendip–ep–peendip: 90° 

Height  1 

Central density  1 

Number of external pieces  16 

Level of complexity  30 

Related polytopes 

Army  Pep 

Regiment  Pep 

Dual  Pentagonalenneagonal duotegmatic tegum 

Conjugates  Pentagonalenneagrammic duoprismatic prism, Pentagonalgreat enneagrammic duoprismatic prism, Pentagrammicenneagonal duoprismatic prism, Pentagrammicenneagrammic duoprismatic prism, Pentagrammicgreat enneagrammic duoprismatic prism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{2}×I_{2}(9)×A_{1}, order 360 

Convex  Yes 

Nature  Tame 

The pentagonalenneagonal duoprismatic prism or pep, also known as the pentagonalenneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonalenneagonal duoprisms, 5 squareenneagonal duoprisms, and 9 squarepentagonal duoprisms. Each vertex joins 2 squarepentagonal duoprisms, 2 squareenneagonal duoprisms, and 1 pentagonalenneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.
The vertices of a pentagonalenneagonal duoprismatic prism of edge length 2sin(π/9) are given by:
 $\left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,1,\,0,\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,{\sqrt {\frac {5{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,1,\,0,\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm \sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{9}},\,1,\,0,\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,{\sqrt {\frac {5{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm \sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{9}},\,\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,{\sqrt {\frac {5{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm \sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{9}},\,{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
where j = 2, 4, 8.
A pentagonalenneagonal duoprismatic prism has the following Coxeter diagrams:
 x x5o x9o (full symmetry)
 xx5oo xx9oo&#x (pentagonalenneagonal duoprism atop pentagonalenneagonal duoprism)