# Enneagonal-truncated icosahedral duoprism

Enneagonal-truncated icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEti
Coxeter diagramx9o o5x3x ()
Elements
Tera12 pentagonal-enneagonal duoprisms, 20 hexagonal-enneagonal duoprisms, 9 truncated icosahedral prisms
Cells108 pentagonal prisms, 180 hexagonal prisms, 30+60 enneagonal prisms, 9 truncated icosahedra
Faces270+540 squares, 108 pentagons, 180 hexagons, 60 enneagons
Edges270+540+540
Vertices540
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2, 3, 3 (base triangle), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {29+9{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{9}}}}}{8}}}\approx 2.87711}$
Hypervolume${\displaystyle 9{\frac {125+43{\sqrt {5}}}{16\tan {\frac {\pi }{9}}}}\approx 341.77903}$
Tipe–ti–tipe: 140°
Hendip–ep–hendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}$
Peendip–pip–tipe: 90°
Hendip–hip–tipe: 90°
Central density1
Number of external pieces41
Level of complexity30
Related polytopes
ArmyEti
RegimentEti
DualEnneagonal-pentakis dodecahedral duotegum
ConjugatesEnneagrammic-truncated icosahedral duoprism, Great enneagrammic-truncated icosahedral duoprism, Enneagonal-truncated great icosahedral duoprism, Enneagrammic-truncated great icosahedral duoprism, Great enneagrammic-truncated great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(9), order 2160
ConvexYes
NatureTame

The enneagonal-truncated icosahedral duoprism or eti is a convex uniform duoprism that consists of 9 truncated icosahedral prisms, 20 hexagonal-enneagonal duoprisms, and 12 pentagonal-enneagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 pentagonal-enneagonal duoprism, and 2 hexagonal-enneagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 8.