Hendecagonal-icosahedral duoprism

Hendecagonal-icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHenike
Coxeter diagramx11o o5o3x
Elements
Tera20 triangular-hendecagonal duoprisms, 11 icosahedral prisms
Cells220 triangular prisms, 30 hendecagonal prisms, 11 icosahedra
Faces220 triangles, 330 squares, 12 hendecagons
Edges132+330
Vertices132
Vertex figurePentagonal scalene, edge lengths 1 (base pentagon), 2cos(π/11) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 2.01350}$
Hypervolume${\displaystyle {\frac {55(3+{\sqrt {5}})}{48\tan {\frac {\pi }{11}}}}\approx 20.43297}$
Diteral anglesIpe–ike–ipe: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–thendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Thendip–trip–ipe: 90°
Central density1
Number of external pieces31
Level of complexity10
Related polytopes
ArmyHenike
RegimentHenike
DualHendecagonal-dodecahedral duotegum
ConjugatesSmall hendecagrammic-icosahedral duoprism, Hendecagrammic-icosahedral duoprism, Great hendecagrammic-icosahedral duoprism, Grand hendecagrammic-icosahedral duoprism, Hendecagonal-great icosahedral duoprism, Small hendecagrammic-great icosahedral duoprism, Hendecagrammic-great icosahedral duoprism, Great hendecagrammic-great icosahedral duoprism, Grand hendecagrammic-great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(11), order 2640
ConvexYes
NatureTame

The hendecagonal-icosahedral duoprism or henike is a convex uniform duoprism that consists of 11 icosahedral prisms and 20 triangular-hendecagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-hendecagonal duoprisms.

Vertex coordinates

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

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

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