# Enneagonal-tetrahedral duoprism

Enneagonal-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEtet
Coxeter diagramx9o x3o3o ()
Elements
Tera9 tetrahedral prisms, 4 triangular-enneagonal duoprisms
Cells9 tetrahedra, 36 triangular prisms, 6 enneagonal prisms
Faces36 triangles, 54 squares, 4 enneagons
Edges36+54
Vertices36
Vertex figureTriangular scalene, edge lengths 1 (base triangle), 2cos(π/9) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {{\frac {3}{8}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{9}}}}}}\approx 1.58498}$
Hypervolume${\displaystyle {\frac {3{\sqrt {2}}}{16\tan {\frac {\pi }{9}}}}\approx 0.72853}$
Diteral anglesTepe–tet–tepe: 140°
Tepe–trip–tedip: 90°
Tedip–ep–tedip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
HeightsEn atop tedip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Ep atop perp ep: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces13
Level of complexity10
Related polytopes
ArmyEtet
RegimentEtet
DualEnneagonal-tetrahedral duotegum
ConjugatesEnneagrammic-tetrahedral duoprism, Great enneagrammic-tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(9), order 432
ConvexYes
NatureTame

The enneagonal-tetrahedral duoprism or etet is a convex uniform duoprism that consists of 9 tetrahedral prisms and 4 triangular-enneagonal duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular-enneagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 8.