# Hendecagonal-great rhombicosidodecahedral duoprism

Hendecagonal-great rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHengrid
Coxeter diagramx11o x5x3x
Elements
Tera30 square-hendecagonal duoprisms, 20 hexagonal-hendecagonal duoprisms, 12 decagonal-hendecagonal duoprisms, 11 great rhombicosidodecahedral prisms
Cells330 cubes, 220 hexagonal prisms, 132 decagonal prisms, 60+60+60 hendecagonal prisms, 11 great rhombicosidodecahedra
Faces330+660+660+660 squares, 220 hexagons, 132 decagons, 120 hendecagons
Edges660+660+660+1320
Vertices1320
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, (5+5)/2 (base triangle), 2cos(π/11) (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {31+12{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 4.19617}$
Hypervolume${\displaystyle 55{\frac {19+10{\sqrt {5}}}{4\tan {\frac {\pi }{11}}}}\approx 1936.84617}$
Diteral anglesShendip–henp–hahendip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Shendip–henp–dahendip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Griddip–grid–griddip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Hahendip–henp–dahendip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Shendip–cube–griddip: 90°
Hahendip–hip–griddip: 90°
Dahendip–dip–griddip: 90°
Central density1
Number of external pieces73
Level of complexity60
Related polytopes
ArmyHengrid
RegimentHengrid
DualHendecagonal-disdyakis triacontahedral duotegum
ConjugatesSmall hendecagrammic-great rhombicosidodecahedral duoprism, Hendecagrammic-great rhombicosidodecahedral duoprism, Great hendecagrammic-great rhombicosidodecahedral duoprism, Grand hendecagrammic-great rhombicosidodecahedral duoprism, Hendecagonal-great quasitruncated icosidodecahedral duoprism, Small hendecagrammic-great quasitruncated icosidodecahedral duoprism, Hendecagrammic-great quasitruncated icosidodecahedral duoprism, Great hendecagrammic-great quasitruncated icosidodecahedral duoprism, Grand hendecagrammic-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(11), order 2640
ConvexYes
NatureTame

The hendecagonal-great rhombicosidodecahedral duoprism or hengrid is a convex uniform duoprism that consists of 11 great rhombicosidodecahedral prisms, 12 decagonal-hendecagonal duoprisms, 20 hexagonal-hendecagonal duoprisms, and 30 square-hendecagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-hendecagonal duoprism, 1 hexagonal-hendecagonal duoprism, and 1 decagonal-hendecagonal duoprism.

## Vertex coordinates

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

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$

along with all even permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm 2\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm 2\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$

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