Enneagonal-truncated octahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Etoe Coxeter diagram x9o o4x3x ( ) Elements Tera 6 square-enneagonal duoprisms , 8 truncated octahedral prisms , 8 hexagonal-enneagonal duoprisms Cells 54 cubes , 72 hexagonal prisms , 12+24 enneagonal prisms , 9 truncated octahedra Faces 54+108+216 squares , 72 hexagons , 24 enneagons Edges 108+216+216 Vertices 216 Vertex figure Digonal disphenoidal pyramid , edge lengths √2 , √3 , √3 (base triangle), cos(π/9) (top), √2 (side edges)Measures (edge length 1) Circumradius ${\frac {\sqrt {10+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.15341$ Hypervolume ${\frac {18{\sqrt {2}}}{\tan {\frac {\pi }{9}}}}\approx 69.93936$ Diteral angles Tope–toe–tope: 140° Sendip–ep–hendip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$ Hendip–ep–hendip: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$ Sendip–cube–tope: 90° Hendip–hip–tope: 90° Central density 1 Number of external pieces 23 Level of complexity 30 Related polytopes Army Etoe Regiment Etoe Dual Enneagonal-tetrakis hexahedral duotegum Conjugates Enneagrammic-truncated octahedral duoprism , Great enneagrammic-truncated octahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry B_{3} ×I_{2} (9) , order 864Convex Yes Nature Tame

The enneagonal-truncated octahedral duoprism or etoe is a convex uniform duoprism that consists of 9 truncated octahedral prisms , 8 hexagonal-enneagonal duoprisms , and 6 square-enneagonal duoprisms . Each vertex joins 2 truncated octahedral prisms, 1 square-enneagonal duoprism, and 2 hexagonal-enneagonal duoprisms.

The vertices of an enneagonal-truncated octahedral duoprism of edge length $2\sin(\pi /9)$ are given by all permutations of the last three coordinates of:

$\left(1,\,0,\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,\pm 2{\sqrt {2}}\sin {\frac {\pi }{9}}\right)$ ,
$\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,\pm 2{\sqrt {2}}\sin {\frac {\pi }{9}}\right)$ ,
$\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{9}},\,\pm 2{\sqrt {2}}\sin {\frac {\pi }{9}}\right)$ ,
where j = 2, 4, 8.