# Enneagonal-great rhombicosidodecahedral duoprism

Enneagonal-great rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEgrid
Coxeter diagramx9o x5x3x ()
Elements
Tera30 square-enneagonal duoprisms, 20 hexagonal-enneagonal duoprisms, 12 enneagonal-decagonal duoprisms, 9 great rhombicosidodecahedral prisms
Cells270 cubes, 180 hexagonal prisms, 60+60+60 enneagonal prisms, 108 decagonal prisms, 9 great rhombicosidodecahedra
Faces270+540+540+540 squares, 180 hexagons, 120 enneagons, 108 decagons
Edges540+540+540+1080
Vertices1080
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, (5+5)/2 (base triangle), 2cos(π/9) (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {31+12{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 4.07374}$
Hypervolume${\displaystyle 45{\frac {19+10{\sqrt {5}}}{4\tan {\frac {\pi }{9}}}}\approx 1278.42225}$
Diteral anglesSendip–ep–hendip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Sendip–ep–edidip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Hendip–ep–edidip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Griddip–grid–griddip: 140°
Sendip–cube–griddip: 90°
Hendip–hip–griddip: 90°
Edidip–dip–griddip: 90°
Central density1
Number of external pieces71
Level of complexity60
Related polytopes
ArmyEgrid
RegimentEgrid
DualEnneagonal-disdyakis triacontahedral duotegum
ConjugatesEnneagrammic-great rhombicosidodecahedral duoprism, Great enneagrammic-great rhombicosidodecahedral duoprism, Enneagonal-great quasitruncated icosidodecahedral duoprism, Enneagrammic-great quasitruncated icosidodecahedral duoprism, Great enneagrammic-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(9), order 2160
ConvexYes
NatureTame

The enneagonal-great rhombicosidodecahedral duoprism or egrid is a convex uniform duoprism that consists of 9 great rhombicosidodecahedral prisms, 12 enneagonal-decagonal duoprisms, 20 hexagonal-enneagonal duoprisms, and 30 square-enneagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-enneagonal duoprism, 1 hexagonal-enneagonal duoprism, and 1 enneagonal-decagonal duoprism.

## Vertex coordinates

The vertices of an enneagonal-great rhombicosidodecahedral duoprism of edge length 2sin(π/9) are given by all permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$

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

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{9}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm \sin {\frac {\pi }{9}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm 2\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm 2\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm 2\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}}\right),}$

where j = 2, 4, 8.