# Enneagonal-small rhombicosidodecahedral duoprism

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Enneagonal-small rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEsrid
Coxeter diagramx9o x5o3x ()
Elements
Tera20 triangular-enneagonal duoprisms, 30 square-enneagonal duoprisms, 12 pentagonal-enneagonal duoprisms, 9 small rhombicosidodecahedral prisms
Cells180 triangular prisms, 270 cubes, 108 pentagonal prisms, 60+60 enneagonal prisms, 9 small rhombicosidodecahedra
Faces180 triangles, 270+540+540 squares, 108 pentagons, 60 enneagons
Edges540+540+540
Vertices540
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, (1+5)/2, 2 (base trapezoid), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.66894}$
Hypervolume${\displaystyle 3{\frac {60+29{\sqrt {5}}}{4\tan {\frac {\pi }{9}}}}\approx 257.25862}$
Diteral anglesTedip–ep–sendip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Sendip–ep–peendip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Sriddip–srid–sriddip: 140°
Tedip–trip–sriddip: 90°
Sendip–cube–sriddip: 90°
Peendip–pip–sriddip: 90°
Central density1
Number of external pieces71
Level of complexity40
Related polytopes
ArmyEsrid
RegimentEsrid
DualEnneagonal-deltoidal hexecontahedral duotegum
ConjugatesEnneagrammic-small rhombicosidodecahedral duoprism, Great enneagrammic-small rhombicosidodecahedral duoprism, Enneagonal-quasirhombicosidodecahedral duoprism, Enneagrammic-quasirhombicosidodecahedral duoprism, Great enneagrammic-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(9), order 2160
ConvexYes
NatureTame

The enneagonal-small rhombicosidodecahedral duoprism or esrid is a convex uniform duoprism that consists of 9 small rhombicosidodecahedral prisms, 12 pentagonal-enneagonal duoprisms, 30 square-enneagonal duoprisms, and 20 triangular-enneagonal duoprisms. Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-enneagonal duoprism, 2 square-enneagonal duoprisms, and 1 pentagonal-enneagonal duoprism.

## Vertex coordinates

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

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

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

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

where j = 2, 4, 8.