Enneagonal-truncated icosahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Eti Coxeter diagram x9o o5x3x ( ) Elements Tera 12 pentagonal-enneagonal duoprisms , 20 hexagonal-enneagonal duoprisms , 9 truncated icosahedral prisms Cells 108 pentagonal prisms , 180 hexagonal prisms , 30+60 enneagonal prisms , 9 truncated icosahedra Faces 270+540 squares , 108 pentagons , 180 hexagons , 60 enneagons Edges 270+540+540 Vertices 540 Vertex figure Digonal disphenoidal pyramid , edge lengths (1+√5 )/2, √3 , √3 (base triangle), 2cos(π/9) (top), √2 (side edges)Measures (edge length 1) Circumradius
29
+
9
5
+
2
sin
2
π
9
8
≈
2.87711
{\displaystyle {\sqrt {\frac {29+9{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{9}}}}}{8}}}\approx 2.87711}
Hypervolume
9
125
+
43
5
16
tan
π
9
≈
341.77903
{\displaystyle 9{\frac {125+43{\sqrt {5}}}{16\tan {\frac {\pi }{9}}}}\approx 341.77903}
Tipe–ti–tipe: 140° Hendip–ep–hendip:
arccos
(
−
5
3
)
≈
138.18968
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}
Peendip–pip–tipe: 90° Hendip–hip–tipe: 90° Central density 1 Number of external pieces 41 Level of complexity 30 Related polytopes Army Eti Regiment Eti Dual Enneagonal-pentakis dodecahedral duotegum Conjugates Enneagrammic-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 characteristic 2 Orientable Yes Properties Symmetry H3 ×I2(9) , order 2160Convex Yes Nature Tame
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.
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:
(
1
,
0
,
0
,
±
sin
π
9
,
±
3
(
1
+
5
)
sin
π
9
2
)
,
{\displaystyle \left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{9}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}}\right),}
(
1
,
0
,
±
sin
π
9
,
±
(
5
+
5
)
sin
π
9
2
,
±
(
1
+
5
)
sin
π
9
)
,
{\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),}
(
1
,
0
,
±
(
1
+
5
)
sin
π
9
2
,
±
2
sin
π
9
,
±
(
2
+
5
)
sin
π
9
)
,
{\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),}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
0
,
±
sin
π
9
,
±
3
(
1
+
5
)
sin
π
9
2
)
,
{\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),}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
±
sin
π
9
,
±
(
5
+
5
)
sin
π
9
2
,
±
(
1
+
5
)
sin
π
9
)
,
{\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),}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
±
(
1
+
5
)
sin
π
9
2
,
±
2
sin
π
9
,
±
(
2
+
5
)
sin
π
9
)
,
{\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),}
(
−
1
2
,
±
3
2
,
0
,
±
sin
π
9
,
±
3
(
1
+
5
)
sin
π
9
2
)
,
{\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),}
(
−
1
2
,
±
3
2
,
±
sin
π
9
,
±
(
5
+
5
)
sin
π
9
2
,
±
(
1
+
5
)
sin
π
9
)
,
{\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),}
(
−
1
2
,
±
3
2
,
±
(
1
+
5
)
sin
π
9
2
,
±
2
sin
π
9
,
±
(
2
+
5
)
sin
π
9
)
,
{\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.