Enneagonal-truncated tetrahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Etut Coxeter diagram x9o x3x3o ( ) Elements Tera 4 triangular-enneagonal duoprisms , 9 truncated tetrahedral prisms , 4 hexagonal-enneagonal duoprisms Cells 36 triangular prisms , 36 hexagonal prisms , 9 truncated tetrahedra , 6+12 enneagonal prisms Faces 36 triangles , 54+108 squares , 36 hexagons , 12 enneagons Edges 54+108+108 Vertices 108 Vertex figure Digonal disphenoidal pyramid , edge lengths 1, √3 , √3 (base triangle), 2cos(π/9) (top), √2 (side edges)Measures (edge length 1) Circumradius ${\sqrt {{\frac {11}{8}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{9}}}}}}\approx 1.87408$ Hypervolume ${\frac {69{\sqrt {2}}}{16\tan {\frac {\pi }{9}}}}\approx 16.75630$ Diteral angles Tuttip–tut–tuttip: 140° Tedip-ep-hendip: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$ Tedip–trip–tuttip: 90° Hendip-hip-tuttip: 90° Hendip–ep–hendip: $\arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }$ Central density 1 Number of external pieces 17 Level of complexity 30 Related polytopes Army Etut Regiment Etut Dual Enneagonal-triakis tetrahedral duotegum Conjugates Enneagrammic-truncated tetrahedral duoprism , Great enneagrammic-truncated tetrahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry A_{3} ×I2(9) , order 432Convex Yes Nature Tame

The enneagonal-truncated tetrahedral duoprism or etut is a convex uniform duoprism that consists of 9 truncated tetrahedral prisms , 4 hexagonal-enneagonal duoprisms , and 4 triangular-enneagonal duoprisms . Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-enneagonal duoprism, and 2 hexagonal-enneagonal duoprisms.

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

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