# Hendecagonal-small rhombicuboctahedral duoprism

Hendecagonal-small rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHensirco
Coxeter diagramx11o x4o3x
Elements
Tera8 triangular-hendecagonal duoprisms, 6+12 square-hendecagonal duoprisms
Cells88 triangular prisms, 66+132 cubes, 24+24 hendecagonal prisms, 11 small rhombicuboctahedra
Faces88 triangles, 66+132+264+264 squares, 24 hendecagon
Edges264+264+264
Vertices264
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, 2, 2 (base trapezoid), 2cos(π/11) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.25982}$
Hypervolume${\displaystyle 11{\frac {6+5{\sqrt {2}}}{6\tan {\frac {\pi }{11}}}}\approx 81.61261}$
Diteral anglesSircope–sirco–sircope: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–shendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Shendip–henp–shendip: 135°
Thendip–trip–sircope: 90°
Shendip–cube–sircope: 90°
Central density1
Number of external pieces37
Level of complexity40
Related polytopes
ArmyHensirco
RegimentHensirco
DualHendecagonal-deltoidal icositetrahedral duotegum
ConjugatesSmall hendecagrammic-small rhombicuboctahedral duoprism, Hendecagrammic-small rhombicuboctahedral duoprism, Great hendecagrammic-small rhombicuboctahedral duoprism, Grand hendecagrammic-small rhombicuboctahedral duoprism, Hendecagonal-quasirhombicuboctahedral duoprism, Small hendecagrammic-quasirhombicuboctahedral duoprism, Hendecagrammic-quasirhombicuboctahedral duoprism, Great hendecagrammic-quasirhombicuboctahedral duoprism, Grand hendecagrammic-quasirhombicuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(11), order 1056
ConvexYes
NatureTame

The hendecagonal-small rhombicuboctahedral duoprism or hensirco is a convex uniform duoprism that consists of 11 small rhombicuboctahedral prisms, 18 square-hendecagonal duoprisms of two kinds, and 8 triangular-hendecagonal duoprisms. Each vertex joins 2 small rhombicuboctahedral prisms, 1 triangular-hendecagonal duoprism, and 3 square-hendecagonal duoprisms.

## Vertex coordinates

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

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

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