# Hendecagonal-great rhombicuboctahedral duoprism

Hendecagonal-great rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHengirco
Coxeter diagramx11o x4x3x
Elements
Tera12 square-hendecagonal duoprisms, 8 hexagonal-hendecagonal duoprisms, 6 octagonal-hendecagonal duoprisms
Cells132 cubes, 88 hexagonal prisms, 66 octagonal prisms, 24+24+24 hendecagonal prisms, 11 great rhombicuboctahedra
Faces132+264+264+264 squares, 88 hexagons, 66 octagons, 48 hendecagons
Edges264+264+264+528
Vertices528
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, 2+2 (base triangle), 2cos(π/11) (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13+6{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.91907}$
Hypervolume${\displaystyle 11{\frac {11+7{\sqrt {2}}}{2\tan {\frac {\pi }{11}}}}\approx 391.47429}$
Diteral anglesGircope–girco–gircope: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Shendip–henp–hahendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Shendip–henp–ohendip: 135°
Hahendip–henp–ohendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Shendip–cube–gircope: 90°
Hahendip–hip–gircope: 90°
Ohendip–op–gircope: 90°
Central density1
Number of external pieces37
Level of complexity60
Related polytopes
ArmyHengirco
RegimentHengirco
DualHendecagonal-disdyakis dodecahedral duotegum
ConjugatesSmall hendecagrammic-great rhombicuboctahedral duoprism, Hendecagrammic-great rhombicuboctahedral duoprism, Great hendecagrammic-great rhombicuboctahedral duoprism, Grand hendecagrammic-great rhombicuboctahedral duoprism, Hendecagonal-quasitruncated cuboctahedral duoprism, Small hendecagrammic-quasitruncated cuboctahedral duoprism, Hendecagrammic-quasitruncated cuboctahedral duoprism, Great hendecagrammic-quasitruncated cuboctahedral duoprism, Grand hendecagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(11), order 1056
ConvexYes
NatureTame

The hendecagonal-great rhombicuboctahedral duoprism or hengirco is a convex uniform duoprism that consists of 11 great rhombicuboctahedral prisms, 6 octagonal-hendecagonal duoprisms, 8 hexagonal-hendecagonal duoprisms, and 12 square-hendecagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-hendecagonal duoprism, 1 hexagonal-hendecagonal duoprism, and 1 octagonal-hendecagonal duoprism.

## Vertex coordinates

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

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

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