Hendecagonalicosidodecahedral duoprism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Henid 

Coxeter diagram  x11o o5x3o 

Elements 

Tera  20 triangularhendecagonal duoprisms, 12 pentagonalhendecagonal duoprisms, 11 icosidodecahedral prisms 

Cells  220 triangular prisms, 132 pentagonal prisms, 60 hendecagonal prisms, 11 icosidodecahedra 

Faces  220 triangles, 660 squares, 132 pentagons, 30 hendecagons 

Edges  330+660 

Vertices  330 

Vertex figure  Rectangular scalene, edge lengths 1, (1+√5)/2, 1, (1+√5)/2 (base rectangle), 2cos(π/11) (top), √2 (side edges) 

Measures (edge length 1) 

Circumradius  ${\sqrt {\frac {3+{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}{2}}}\approx 2.40161$ 

Hypervolume  $11{\frac {45+17{\sqrt {5}}}{24\tan {\frac {\pi }{11}}}}\approx 129.57855$ 

Diteral angles  Iddip–id–iddip: ${\frac {9\pi }{11}}\approx 147.27273^{\circ }$ 

 Thendip–henp–pahendip: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$ 

 Thendip–trip–iddip: 90° 

 Pahendip–pip–iddip: 90° 

Central density  1 

Number of external pieces  43 

Level of complexity  20 

Related polytopes 

Army  Henid 

Regiment  Henid 

Dual  Hendecagonalrhombic triacontahedral duotegum 

Conjugates  Small hendecagrammicicosidodecahedral duoprism, Hendecagrammicicosidodecahedral duoprism, Great hendecagrammicicosidodecahedral duoprism, Grand hendecagrammicicosidodecahedral duoprism, Hendecagonalgreat icosidodecahedral duoprism, Small hendecagrammicgreat icosidodecahedral duoprism, Hendecagrammicgreat icosidodecahedral duoprism, Great hendecagrammicgreat icosidodecahedral duoprism, Grand hendecagrammicgreat icosidodecahedral duoprism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{3}×I2(11), order 2640 

Convex  Yes 

Nature  Tame 

The hendecagonalicosidodecahedral duoprism or henid is a convex uniform duoprism that consists of 11 icosidodecahedral prisms, 12 pentagonalhendecagonal duoprisms, and 20 triangularhendecagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangularhendecagonal duoprisms, and 2 pentagonalhendecagonal duoprisms.
The vertices of a hendecagonalicosidodecahedral duoprism of edge length 2sin(π/11) are given by all permutations of the last three coordinates of:
 $\left(1,\,0,\,0,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),$
 $\left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,0,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),$
as well as all even permutations of the last three coordinates of:
 $\left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),$
 $\left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),$
where j = 2, 4, 6, 8, 10.