# Enneagonal-dodecahedral duoprism

Enneagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEdoe
Coxeter diagramx9o x5o3o ()
Elements
Tera12 pentagonal-enneagonal duoprisms, 9 dodecahedral prisms
Cells108 pentagonal prisms, 30 enneagonal prisms, 9 dodecahedra
Faces270 squares, 108 pentagons, 20 enneagons
Edges180+270
Vertices180
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle), 2cos(π/9) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {9+3{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{9}}}}}{8}}}\approx 2.02501}$
Hypervolume${\displaystyle {\frac {9(15+7{\sqrt {5}})}{16\tan {\frac {\pi }{9}}}}\approx 47.37205}$
Diteral anglesDope–doe–dope: 140°
Peendip–ep–peendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Peendip–pip–dope: 90°
Central density1
Number of external pieces21
Level of complexity10
Related polytopes
ArmyEdoe
RegimentEdoe
DualEnneagonal-icosahedral duotegum
ConjugatesEnneagrammic-dodecahedral duoprism, Great enneagrammic-dodecahedral duoprism, Enneagonal-great stellated dodecahedral duoprism, Enneagrammic-great stellated dodecahedral duoprism, Great enneagrammic-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(9), order 2160
ConvexYes
NatureTame

The enneagonal-dodecahedral duoprism or edoe is a convex uniform duoprism that consists of 9 dodecahedral prisms and 12 pentagonal-enneagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal-enneagonal duoprisms.

## Vertex coordinates

The vertices of an enneagonal-dodecahedral duoprism of edge length 2sin(π/9) are given by:

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

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

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

where j = 2, 4, 8.