# Enneagonal-truncated dodecahedral duoprism

Enneagonal-truncated dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEtid
Coxeter diagramx9o x5x3o ()
Elements
Tera20 triangular-enneagonal duoprisms, 12 enneagonal-decagonal duoprisms
Cells180 triangular prisms, 30+60 enneagonal prisms, 108 decagonal prisms, 9 truncated dodecahedra
Faces180 triangles, 270+540 squares, 60 enneagons, 108 decagons
Edges270+540+540
Vertices540
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, (5+5)/2, (5+5)/2 (base triangle), cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {37+15{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{9}}}}}{8}}}\approx 3.30980}$
Hypervolume${\displaystyle 15{\frac {99+47{\sqrt {5}}}{16\tan {\frac {\pi }{9}}}}\approx 525.70026}$
Diteral anglesTedip–ep–edidip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Tiddip–tid–tiddip: 60°
Edidip–ep–edidip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Tedip–trip–tiddip: 90°
Edidip–dip–tiddip: 90°
Central density1
Number of external pieces41
Level of complexity30
Related polytopes
ArmyEtid
RegimentEtid
DualEnneagonal-triakis icosahedral duotegum
ConjugatesEnneagrammic-truncated dodecahedral duoprism, Great enneagrammic-truncated dodecahedral duoprism, Enneagonal-quasitruncated great stellated dodecahedral duoprism, Enneagrammic-quasitruncated great stellated dodecahedral duoprism, Great enneagrammic-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(9), order 2160
ConvexYes
NatureTame

The enneagonal-truncated dodecahedral duoprism or etid is a convex uniform duoprism that consists of 9 truncated dodecahedral prisms, 12 enneagonal-decagonal duoprisms, and 20 triangular-enneagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-enneagonal duoprism, and 2 enneagonal-decagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 8.