# Enneagonal-icosidodecahedral duoprism

Enneagonal-icosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEid
Coxeter diagramx9o o5x3o ()
Elements
Tera20 triangular-enneagonal duoprisms, 12 pentagonal-enneagonal duoprisms, 9 icosidodecahedral prisms
Cells180 triangular prisms, 108 pentagonal prisms, 60 enneagonal prisms, 9 icosidodecahedra
Faces180 triangles, 540 squares, 108 pentagons, 30 enneagons
Edges270+540
Vertices270
Vertex figureRectangular scalene, edge lengths 1, (1+5)/2, 1, (1+5)/2 (base rectangle), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {3+{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}{2}}}\approx 2.18064}$
Hypervolume${\displaystyle 3{\frac {45+17{\sqrt {5}}}{8\tan {\frac {\pi }{9}}}}\approx 85.52879}$
Diteral anglesTedip–ep–peendip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Iddip–id–iddip: 140°
Tedip–trip–iddip: 90°
Peendip–pip–iddip: 90°
Central density1
Number of external pieces41
Level of complexity20
Related polytopes
ArmyEid
RegimentEid
DualEnneagonal-rhombic triacontahedral duotegum
ConjugatesEnneagrammic-icosidodecahedral duoprism, Great enneagrammic-icosidodecahedral duoprism, Enneagonal-great icosidodecahedral duoprism, Enneagrammic-great icosidodecahedral duoprism, Great enneagrammic-great icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(9), order 2160
ConvexYes
NatureTame

The enneagonal-icosidodecahedral duoprism or eid is a convex uniform duoprism that consists of 9 icosidodecahedral prisms, 12 pentagonal-enneagonal duoprisms, and 20 triangular-enneagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-enneagonal duoprisms, and 2 pentagonal-enneagonal duoprisms.

## Vertex coordinates

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

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

as well as all even permutations of the last three coordinates of:

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

where j = 2, 4, 8.