Enneagonal-hendecagonal duoprismatic prism

Enneagonal-hendecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymEhenip
Coxeter diagramx x9o x11o ()
Elements
Tera11 square-enneagonal duoprisms, 9 square-hendecagonal duoprisms, 2 enneagonal-hendecagonal duoprisms
Cells99 cubes, 9+18 hendecagonal prisms, 11+22 enneagonal prisms
Faces99+99+198 squares, 22 enneagons, 18 hendecagons
Edges99+198+198
Vertices198
Vertex figureDigonal disphenoidal pyramid, edge lengths 2cos(π/9) (disphenoid base 1), 2cos(π/11) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {1+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.35305}$
Hypervolume${\displaystyle {\frac {99}{16\tan {\frac {\pi }{9}}\tan {\frac {\pi }{11}}}}\approx 57.89674}$
Diteral anglesSendip–ep–sendip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Shendip–henp–shendip: 140°
Shendip–cube–sendip: 90°
Ehendip–ep–sendip: 90°
Shendip–henp–ehendip: 90°
Height1
Central density1
Number of external pieces22
Level of complexity30
Related polytopes
ArmyEhenip
RegimentEhenip
DualEnneagonal-hendecagonal duotegmatic tegum
ConjugatesEnneagonal-small hendecagrammic duoprismatic prism, Enneagonal-hendecagrammic duoprismatic prism, Enneagonal-great hendecagrammic duoprismatic prism, Enneagonal-grand hendecagrammic duoprismatic prism, Enneagrammic-hendecagonal duoprismatic prism, Enneagrammic-small hendecagrammic duoprismatic prism, Enneagrammic-hendecagrammic duoprismatic prism, Enneagrammic-great hendecagrammic duoprismatic prism, Enneagrammic-grand hendecagrammic duoprismatic prism, Great enneagrammic-hendecagonal duoprismatic prism, Great enneagrammic-small hendecagrammic duoprismatic prism, Great enneagrammic-hendecagrammic duoprismatic prism, Great enneagrammic-great hendecagrammic duoprismatic prism, Great enneagrammic-grand hendecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(9)×I2(11)×A1, order 792
ConvexYes
NatureTame

The enneagonal-hendecagonal duoprismatic prism or ehenip, also known as the enneagonal-hendecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 enneagonal-hendecagonal duoprisms, 9 square-hendecagonal duoprisms, and 11 square-enneagonal duoprisms. Each vertex joins 2 square-enneagonal duoprisms, 2 square-hendecagonal duoprisms, and 1 enneagonal-hendecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates

The vertices of an enneagonal-hendecagonal duoprismatic prism of edge length 4sin(π/9)sin(π/11) are given by:

• ${\displaystyle \left(2\sin {\frac {\pi }{11}},\,0,\,2\sin {\frac {\pi }{9}},\,0,\,\pm 2\sin {\frac {\pi }{9}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(2\sin {\frac {\pi }{11}},\,0,\,2\cos \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin {\frac {\pi }{9}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(2\cos \left({\frac {j\pi }{9}}\right)\sin {\frac {\pi }{11}},\,\pm 2\sin \left({\frac {j\pi }{9}}\right)\sin {\frac {\pi }{11}},\,2\sin {\frac {\pi }{9}},\,0,\,\pm 2\sin {\frac {\pi }{9}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(2\cos \left({\frac {j\pi }{9}}\right)\sin {\frac {\pi }{11}},\,\pm 2\sin \left({\frac {j\pi }{9}}\right)\sin {\frac {\pi }{11}},\,2\cos \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin {\frac {\pi }{9}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(-\sin {\frac {\pi }{11}},\,\pm {\sqrt {3}}\sin {\frac {\pi }{11}},\,2\sin {\frac {\pi }{9}},\,0,\,\pm 2\sin {\frac {\pi }{9}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(-\sin {\frac {\pi }{11}},\,\pm {\sqrt {3}}\sin {\frac {\pi }{11}},\,2\cos \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin {\frac {\pi }{9}}\sin {\frac {\pi }{11}}\right),}$

where j = 2, 4, 8 and k = 2, 4, 6, 8, 10.

Representations

An enneagonal-hendecagonal duoprismatic prism has the following Coxeter diagrams:

• x x9o x11o () (full symmetry)
• xx9oo xx11oo&#x (enneagonal-hendecagonal duoprism atop enneagonal-hendecagonal duoprism)