# Enneagonal duoprismatic prism

Enneagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymEep
Coxeter diagramx x9o x9o ()
Elements
Tera18 square-enneagonal duoprisms, 2 enneagonal duoprisms
Cells81 cubes, 18+36 enneagonal prisms
Faces162+162 squares, 36 enneagons
Edges81+324
Vertices162
Vertex figureTetragonal disphenoidal pyramid, edge lengths 2cos(π/9) (disphenoid bases) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {1+{\frac {2}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.12704}$
Hypervolume${\displaystyle {\frac {81}{16\tan ^{2}{\frac {\pi }{9}}}}\approx 38.21495}$
Diteral anglesSendip–ep–sendip: 140°
Sendip–cube–sendip: 90°
Edip–ep–sendip: 90°
Height1
Central density1
Number of external pieces20
Level of complexity15
Related polytopes
ArmyEep
RegimentEep
DualEnneagonal duotegmatic tegum
ConjugatesEnneagrammic duoprismatic prism, Great enneagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(9)≀S2×A1, order 1296
ConvexYes
NatureTame

The enneagonal duoprismatic prism or eep, also known as the enneagonal-enneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 enneagonal duoprisms and 18 square-enneagonal duoprisms. Each vertex joins 4 square-enneagonal duoprisms and 1 enneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of an enneagonal duoprismatic prism of edge length 2sin(π/9) are given by:

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

where j, k = 2, 4, 8.

## Representations

An enneagonal duoprismatic prism has the following Coxeter diagrams:

• x x9o x9o () (full symmetry)
• xx9oo xx9oo&#x (enneagonal duoprism atop enneagonal duoprism)