Hendecagonaldodecahedral duoprism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Hendoe 

Coxeter diagram  x11o x5o3o 

Elements 

Tera  12 pentagonalhendecagonal duoprisms, 11 dodecahedral prisms 

Cells  132 pentagonal prisms, 30 hendecagonal prisms, 11 dodecahedra 

Faces  330 squares, 132 pentagons, 20 hendecagons 

Edges  220+330 

Vertices  220 

Vertex figure  Triangular scalene, edge lengths (1+√5)/2 (base triangle), 2cos(π/11) (top), √2 (sides) 

Measures (edge length 1) 

Circumradius  ${\sqrt {\frac {9+3{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 2.26124$ 

Hypervolume  ${\frac {11(15+7{\sqrt {5}})}{16\tan {\frac {\pi }{1}}1}}\approx 71.77001$ 

Diteral angles  Dope–doe–dope: ${\frac {9\pi }{11}}\approx 147.27273^{\circ }$ 

 Pahendip–henp–pahendip: $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$ 

 Pahendip–pip–dope: 90° 

Central density  1 

Number of external pieces  23 

Level of complexity  10 

Related polytopes 

Army  Hendoe 

Regiment  Hendoe 

Dual  Hendecagonalicosahedral duotegum 

Conjugates  Small hendecagrammicdodecahedral duoprism, Hendecagrammicdodecahedral duoprism, Great hendecagrammicdodecahedral duoprism, Grand hendecagrammicdodecahedral duoprism, Hendecagonalgreat stellated dodecahedral duorprism, Small hendecagrammicgreat stellated dodecahedral duoprism, Hendecagrammicgreat stellated dodecahedral duoprism, Great hendecagrammicgreat stellated dodecahedral duoprism, Grand hendecagrammicgreat stellated dodecahedral duoprism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{3}×I2(11), order 2640 

Convex  Yes 

Nature  Tame 

The hendecagonaldodecahedral duoprism or hendoe is a convex uniform duoprism that consists of 11 dodecahedral prisms and 12 pentagonalhendecagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonalhendecagonal duoprisms.
The vertices of a hendecagonaldodecahedral duoprism of edge length 2sin(π/11) are given by:
 $\left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),$
 $\left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),$
as well as all even permutations of the last three coordinates of:
 $\left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),$
 $\left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),$
where j = 2, 4, 6, 8, 10.