# Hendecagonal-cubic duoprism

Hendecagonal-cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHencube
Coxeter diagramx11o x4o3o
Elements
Tera11 tesseracts, 6 square-hendecagonal duoprisms
Cells11+66 cubes, 12 hendecagonal prisms
Faces66+132 squares, 8 hendecagons
Edges88+132
Vertices88
Vertex figureTriangular scalene, edge lengths 2cos(π/11) (top), 2 (base triangle and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 1.97476}$
Hypervolume${\displaystyle {\frac {11}{4\tan \left({\frac {\pi }{11}}\right)}}\approx 9.36564}$
Diteral anglesTes–cube–tes: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Tes–cube–shendip: 90°
Shendip–henp–shendip: 90°
Height1
Central density1
Number of external pieces17
Level of complexity10
Related polytopes
ArmyHencube
RegimentHencube
DualHendecagonal-octahedral duotegum
ConjugatesSmall hendecagrammic-cubic duoprism, Hendecagrammic-cubic duoprism, Great hendecagrammic-cubic duoprism, Grand hendecagrammic-cubic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(11), order 1056
ConvexYes
NatureTame

The hendecagonal-cubic duoprism or hencube, also known as the square-hendecagonal duoprismatic prism, is a convex uniform duoprism that consists of 11 tesseracts and 6 square-hendecagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-hendecagonal duoprisms. It is a duoprism based on a square and a hendecagonal prism, which makes it a convex segmentoteron.

## Vertex coordinates

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

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

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

## Representations

A hendecagonal-cubic duoprism has the following Coxeter diagrams:

• x11o x4o3o (full symmetry)
• x x4o x11o (square-hendecagonal duoprismatic prism)
• x x x x11o (hendecagonal prismatic prismatic prism)