# Hendecagonal-cuboctahedral duoprism

Hendecagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHenco
Coxeter diagramx11o o4x3o
Elements
Tera11 cuboctahedral prisms, 8 triangular-hendecagonal duoprisms, 6 square-hendecagonal duoprisms
Cells88 triangular prisms, 66 cubes, 11 cuboctahedra, 24 hendecagonal prisms
Faces88 triangles, 66+264 squares, 24 hendecagons
Edges132+264
Vertices132
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), 2cos(π/11) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {4+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.03708}$
Hypervolume${\displaystyle {\frac {55{\sqrt {2}}}{12\tan {\frac {\pi }{11}}}}\approx 22.07502}$
Diteral anglesCope–co–cope: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–shendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Thendip–trip–cope: 90°
Shendip–cube–cope: 90°
Central density1
Number of external pieces25
Level of complexity20
Related polytopes
ArmyHenco
RegimentHenco
DualHendecagonal-rhombic dodecahedral duotegum
ConjugatesSmall hendecagrammic-cuboctahedral duoprism, Hendecagrammic-cuboctahedral duoprism, Great hendecagrammic-cuboctahedral duoprism, Grand hendecagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(11), order 1056
ConvexYes
NatureTame

The hendecagonal-cuboctahedral duoprism or henco is a convex uniform duoprism that consists of 11 cuboctahedral prisms, 6 square-hendecagonal duoprisms, and 8 triangular-hendecagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-hendecagonal duoprisms, and 2 square-hendecagonal duoprisms.

## Vertex coordinates

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

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

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

## Representations

A hendecagonal-cuboctahedral duoprism has the following Coxeter diagrams:

• x11o o4x3o (full symmetry)
• x11o x3o3x