Hendecagonal-icosahedral duoprism

Hendecagonal-icosahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Henike
Coxeter diagram	x11o o5o3x
Elements
Tera	20 triangular-hendecagonal duoprisms, 11 icosahedral prisms
Cells	220 triangular prisms, 30 hendecagonal prisms, 11 icosahedra
Faces	220 triangles, 330 squares, 12 hendecagons
Edges	132+330
Vertices	132
Vertex figure	Pentagonal scalene, edge lengths 1 (base pentagon), 2cos(π/11) (top), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5+{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 2.01350$
Hypervolume	${\frac {55(3+{\sqrt {5}})}{48\tan {\frac {\pi }{11}}}}\approx 20.43297$
Diteral angles	Ipe–ike–ipe: ${\frac {9\pi }{11}}\approx 147.27273^{\circ }$
	Thendip–henp–thendip: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	Thendip–trip–ipe: 90°
Central density	1
Number of external pieces	31
Level of complexity	10
Related polytopes
Army	Henike
Regiment	Henike
Dual	Hendecagonal-dodecahedral duotegum
Conjugates	Small 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 characteristic	2
Orientable	Yes
Properties
Symmetry	H3×I2(11), order 2640
Convex	Yes
Nature	Tame

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:

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

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