Hendecagonal-great rhombicuboctahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hengirco Coxeter diagram x11o x4x3x Elements Tera 12 square-hendecagonal duoprisms , 8 hexagonal-hendecagonal duoprisms , 6 octagonal-hendecagonal duoprisms Cells 132 cubes , 88 hexagonal prisms , 66 octagonal prisms , 24+24+24 hendecagonal prisms , 11 great rhombicuboctahedra Faces 132+264+264+264 squares , 88 hexagons , 66 octagons , 48 hendecagons Edges 264+264+264+528 Vertices 528 Vertex figure Mirror-symmetric pentachoron , edge lengths √2 , √3 , √2+√2 (base triangle), 2cos(π/11) (top edge), √2 (side edges) Measures (edge length 1) Circumradius ${\frac {\sqrt {13+6{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.91907$ Hypervolume $11{\frac {11+7{\sqrt {2}}}{2\tan {\frac {\pi }{11}}}}\approx 391.47429$ Diteral angles Gircope–girco–gircope: ${\frac {9\pi }{11}}\approx 147.27273^{\circ }$ Shendip–henp–hahendip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$ Shendip–henp–ohendip: 135° Hahendip–henp–ohendip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$ Shendip–cube–gircope: 90° Hahendip–hip–gircope: 90° Ohendip–op–gircope: 90° Central density 1 Number of external pieces 37 Level of complexity 60 Related polytopes Army Hengirco Regiment Hengirco Dual Hendecagonal-disdyakis dodecahedral duotegum Conjugates Small 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 characteristic 2 Orientable Yes Properties Symmetry B_{3} ×I2(11) , order 1056Convex Yes Nature Tame

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.

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:

$\left(1,\,0,\,\pm (1+2{\sqrt {2}})\sin {\frac {\pi }{11}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}}\right),$
$\left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm (1+2{\sqrt {2}})\sin {\frac {\pi }{11}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}}\right),$
where j = 2, 4, 6, 8, 10.