Enneagonaldecagonal duoprismatic prism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Eddip 

Coxeter diagram  x x9o x10o () 

Elements 

Tera  10 squareenneagonal duoprisms, 9 squaredecagonal duoprisms, 2 enneagonaldecagonal duoprisms 

Cells  90 cubes, 9+18 decagonal prisms, 10+20 enneagonal prisms 

Faces  90+90+180 squares, 20 enneagons, 18 decagons 

Edges  90+180+180 

Vertices  180 

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

Measures (edge length 1) 

Circumradius  ${\frac {\sqrt {7+2{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.23723$ 

Hypervolume  ${\frac {45{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{9}}}}\approx 47.56425$ 

Diteral angles  Sendip–ep–sendip: 144° 

 Squadedip–dip–squadedip: 140° 

 Squadedip–cube–sendip: 90° 

 Edidip–ep–sendip: 90° 

 Squadedip–dip–edidip: 90° 

Height  1 

Central density  1 

Number of external pieces  21 

Level of complexity  30 

Related polytopes 

Army  Eddip 

Regiment  Eddip 

Dual  Enneagonaldecagonal duotegmatic tegum 

Conjugates  Enneagonaldecagrammic duoprismatic prism, Enneagrammicdecagonal duoprismatic prism, Enneagrammicdecagrammic duoprismatic prism, Great enneagrammicdecagonal duoprismatic prism, Great enneagrammicdecagrammic duoprismatic prism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  I_{2}(9)×I_{2}(10)×A_{1}, order 720 

Convex  Yes 

Nature  Tame 

The enneagonaldecagonal duoprismatic prism or eddip, also known as the enneagonaldecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 enneagonaldecagonal duoprisms, 9 squaredecagonal duoprisms, and 10 squareenneagonal duoprisms. Each vertex joins 2 squareenneagonal duoprisms, 2 squaredecagonal duoprisms, and 1 enneagonaldecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.
The vertices of an enneagonaldecagonal duoprismatic prism of edge length 2sin(π/9) are given by:
 $\left(0,\,1,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(0,\,1,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(0,\,1,\,\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left({\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left({\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
 $\left({\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
where j = 2, 4, 8.
An enneagonaldecagonal duoprismatic prism has the following Coxeter diagrams:
 x x9o x10o () (full symmetry)
 x x9o x5x () (decagons as dipentagons)
 xx9oo xx10oo&#x (enneagonaldecagonal duoprism atop enneagonaldecagonal duoprism)
 xx9oo xx5xx&#x