# Hendecagonal-icosidodecahedral duoprism

Hendecagonal-icosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHenid
Coxeter diagramx11o o5x3o
Elements
Tera20 triangular-hendecagonal duoprisms, 12 pentagonal-hendecagonal duoprisms, 11 icosidodecahedral prisms
Cells220 triangular prisms, 132 pentagonal prisms, 60 hendecagonal prisms, 11 icosidodecahedra
Faces220 triangles, 660 squares, 132 pentagons, 30 hendecagons
Edges330+660
Vertices330
Vertex figureRectangular 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${\displaystyle {\sqrt {\frac {3+{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}{2}}}\approx 2.40161}$
Hypervolume${\displaystyle 11{\frac {45+17{\sqrt {5}}}{24\tan {\frac {\pi }{11}}}}\approx 129.57855}$
Diteral anglesIddip–id–iddip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–pahendip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Thendip–trip–iddip: 90°
Pahendip–pip–iddip: 90°
Central density1
Number of external pieces43
Level of complexity20
Related polytopes
ArmyHenid
RegimentHenid
DualHendecagonal-rhombic triacontahedral duotegum
ConjugatesSmall hendecagrammic-icosidodecahedral duoprism, Hendecagrammic-icosidodecahedral duoprism, Great hendecagrammic-icosidodecahedral duoprism, Grand hendecagrammic-icosidodecahedral duoprism, Hendecagonal-great icosidodecahedral duoprism, Small hendecagrammic-great icosidodecahedral duoprism, Hendecagrammic-great icosidodecahedral duoprism, Great hendecagrammic-great icosidodecahedral duoprism, Grand hendecagrammic-great icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(11), order 2640
ConvexYes
NatureTame

The hendecagonal-icosidodecahedral duoprism or henid is a convex uniform duoprism that consists of 11 icosidodecahedral prisms, 12 pentagonal-hendecagonal duoprisms, and 20 triangular-hendecagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-hendecagonal duoprisms, and 2 pentagonal-hendecagonal duoprisms.

## Vertex coordinates

The vertices of a hendecagonal-icosidodecahedral duoprism of edge length 2sin(π/11) are given by all permutations of the last three coordinates of:

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

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

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

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