Octagonalenneagonal duoprismatic prism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Oep 

Coxeter diagram  x x8o x9o 

Elements 

Tera  9 squareoctagonal duoprisms, 8 squareenneagonal duoprisms, 2 octagonalenneagonal duoprisms 

Cells  72 cubes, 8+16 enneagonal prisms, 9+18 octagonal prisms 

Faces  72+72+144 squares, 18 octagons, 16 enneagons 

Edges  72+144+144 

Vertices  144 

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

Measures (edge length 1) 

Circumradius  ${\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.02343$ 

Hypervolume  $9{\frac {1+{\sqrt {2}}}{2\tan {\frac {\pi }{9}}}}\approx 29.84849$ 

Diteral angles  Sodip–op–sodip: 140° 

 Sendip–ep–sendip: 135° 

 Sendip–cube–sodip: 90° 

 Oedip–op–sodip: 90° 

 Sendip–ep–oedip: 90° 

Height  1 

Central density  1 

Number of external pieces  19 

Level of complexity  30 

Related polytopes 

Army  Oep 

Regiment  Oep 

Dual  Octagonalenneagonal duotegmatic tegum 

Conjugates  Octagonalenneagrammic duoprismatic prism, Octagonalgreat enneagrammic duoprismatic prism, Octagrammicenneagonal duoprismatic prism, Octagrammicenneagrammic duoprismatic prism, Octagrammicgreat enneagrammic duoprismatic prism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  I_{2}(8)×I_{2}(9)×A_{1}, order 576 

Convex  Yes 

Nature  Tame 

The octagonalenneagonal duoprismatic prism or oep, also known as the octagonalenneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 octagonalenneagonal duoprisms, 8 squareenneagonal duoprisms, and 9 squareoctagonal duoprisms. Each vertex joins 2 squareoctagonal duoprisms, 2 squareenneagonal duoprisms, and 1 octagonalenneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.
The vertices of an octagonalenneagonal duoprismatic prism of edge length 2sin(π/9) are given by all permutations of the first two coordinates of:
 $\left(\pm \sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,1,\,0,\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm \sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\pm \sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}},\,{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
where j = 2, 4, 8.
An octagonalenneagonal duoprismatic prism has the following Coxeter diagrams:
 x x8o x9o (full symmetry)
 x x4x x9o (octagons as ditetragons)
 xx8oo xx9oo&#x (octagonalenneagonal duoprism atop octagonalenneagonal duoprism)
 xx4xx xx9oo&#x