# Octagonal-hendecagonal duoprismatic prism

Octagonal-hendecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymOhenip
Coxeter diagramx x8o x11o
Elements
Tera11 square-octagonal duoprisms, 8 square-hendecagonal duoprisms, 2 octagonal-hendecagonal duoprisms
Cells88 cubes, 8+16 hendecagonal prisms, 11+22 octagonal prisms
Faces88+88+176 squares, 22 octagons, 16 hendecagons
Edges88+176+176
Vertices176
Vertex figureDigonal disphenoidal pyramid, edge lengths 2+2 (disphenoid base 1), 2cos(π/11) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.25982}$
Hypervolume${\displaystyle 11{\frac {1+{\sqrt {2}}}{2\tan {\frac {\pi }{11}}}}\approx 45.22131}$
Diteral anglesSodip–op–sodip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Shendip–henp–shendip: 135°
Shendip–cube–sodip: 90°
Ohendip–op–sodip: 90°
Shendip–henp–ohendip: 90°
Height1
Central density1
Number of external pieces21
Level of complexity30
Related polytopes
ArmyOhenip
RegimentOhenip
DualOctagonal-hendecagonal duotegmatic tegum
ConjugatesOctagonal-small hendecagrammic duoprismatic prism, Octagonal-hendecagrammic duoprismatic prism, Octagonal-great hendecagrammic duoprismatic prism, Octagonal-grand hendecagrammic duoprismatic prism, Octagrammic-hendecagonal duoprismatic prism, Octagrammic-small hendecagrammic duoprismatic prism, Octagrammic-hendecagrammic duoprismatic prism, Octagrammic-great hendecagrammic duoprismatic prism, Octagrammic-grand hendecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(8)×I2(11)×A1, order 704
ConvexYes
NatureTame

The octagonal-hendecagonal duoprismatic prism or ohenip, also known as the octagonal-hendecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 octagonal-hendecagonal duoprisms, 8 square-hendecagonal duoprisms, and 11 square-octagonal duoprisms. Each vertex joins 2 square-octagonal duoprisms, 2 square-hendecagonal duoprisms, and 1 octagonal-hendecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of an octagonal-hendecagonal duoprismatic prism of edge length 2sin(π/11) are given by all permutations of the first two coordinates of:

• ${\displaystyle \left(\pm \sin {\frac {\pi }{11}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{11}},\,1,\,0,\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm \sin {\frac {\pi }{11}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{11}},\,\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}}\right),}$

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

## Representations

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

• x x8o x11o (full symmetry)
• x x4x x11o (octagons as ditetragons)
• xx8oo xx11oo&#x (octagonal-hendecagonal duoprism atop octagonal-hendecagonal duoprism)
• xx4xx xx11oo&#x