# Enneagonal-truncated tetrahedral duoprism

Enneagonal-truncated tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEtut
Coxeter diagramx9o x3x3o ()
Elements
Tera4 triangular-enneagonal duoprisms, 9 truncated tetrahedral prisms, 4 hexagonal-enneagonal duoprisms
Cells36 triangular prisms, 36 hexagonal prisms, 9 truncated tetrahedra, 6+12 enneagonal prisms
Faces36 triangles, 54+108 squares, 36 hexagons, 12 enneagons
Edges54+108+108
Vertices108
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 3, 3 (base triangle), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {{\frac {11}{8}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{9}}}}}}\approx 1.87408}$
Hypervolume${\displaystyle {\frac {69{\sqrt {2}}}{16\tan {\frac {\pi }{9}}}}\approx 16.75630}$
Diteral anglesTuttip–tut–tuttip: 140°
Tedip-ep-hendip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Tedip–trip–tuttip: 90°
Hendip-hip-tuttip: 90°
Hendip–ep–hendip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density1
Number of external pieces17
Level of complexity30
Related polytopes
ArmyEtut
RegimentEtut
DualEnneagonal-triakis tetrahedral duotegum
ConjugatesEnneagrammic-truncated tetrahedral duoprism, Great enneagrammic-truncated tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(9), order 432
ConvexYes
NatureTame

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.

## Vertex coordinates

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:

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

where j = 2, 4, 8.