# Hendecagonal-dodecahedral duoprism

Hendecagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHendoe
Coxeter diagramx11o x5o3o
Elements
Tera12 pentagonal-hendecagonal duoprisms, 11 dodecahedral prisms
Cells132 pentagonal prisms, 30 hendecagonal prisms, 11 dodecahedra
Faces330 squares, 132 pentagons, 20 hendecagons
Edges220+330
Vertices220
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle), 2cos(π/11) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {9+3{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 2.26124}$
Hypervolume${\displaystyle {\frac {11(15+7{\sqrt {5}})}{16\tan {\frac {\pi }{1}}1}}\approx 71.77001}$
Diteral anglesDope–doe–dope: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Pahendip–henp–pahendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Pahendip–pip–dope: 90°
Central density1
Number of external pieces23
Level of complexity10
Related polytopes
ArmyHendoe
RegimentHendoe
DualHendecagonal-icosahedral duotegum
ConjugatesSmall hendecagrammic-dodecahedral duoprism, Hendecagrammic-dodecahedral duoprism, Great hendecagrammic-dodecahedral duoprism, Grand hendecagrammic-dodecahedral duoprism, Hendecagonal-great stellated dodecahedral duorprism, Small hendecagrammic-great stellated dodecahedral duoprism, Hendecagrammic-great stellated dodecahedral duoprism, Great hendecagrammic-great stellated dodecahedral duoprism, Grand hendecagrammic-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(11), order 2640
ConvexYes
NatureTame

The hendecagonal-dodecahedral duoprism or hendoe is a convex uniform duoprism that consists of 11 dodecahedral prisms and 12 pentagonal-hendecagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal-hendecagonal duoprisms.

## Vertex coordinates

The vertices of a hendecagonal-dodecahedral duoprism of edge length 2sin(π/11) are given by:

• ${\displaystyle \left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$

as well as all even permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$

where j = 2, 4, 6, 8, 10.