# Decagonal-hendecagonal duoprismatic prism

Decagonal-hendecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymDahenip
Coxeter diagramx x10o x11o ()
Elements
Tera11 square-decagonal duoprisms, 10 square-hendecagonal duoprisms, 2 decagonal-hendecagonal duoprisms
Cells110 cubes, 10+20 hendecagonal prisms, 11+22 decagonal prisms
Faces110+110+220 squares, 22 decagons, 20 hendecagons
Edges110+220+220
Vertices220
Vertex figureDigonal disphenoidal pyramid, edge lengths (5+5)/2 (disphenoid base 1), 2cos(π/11) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+2{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.45310}$
Hypervolume${\displaystyle {\frac {55{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{11}}}}\approx 72.06119}$
Diteral anglesSquadedip–dip–squadedip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Shendip–henp–shendip: 144°
Shendip–henp–dahendip: 90°
Height1
Central density1
Number of external pieces23
Level of complexity30
Related polytopes
ArmyDahenip
RegimentDahenip
DualDecagonal-hendecagonal duotegmatic tegum
ConjugatesDecagonal-small hendecagrammic duoprismatic prism, Decagonal-hendecagrammic duoprismatic prism, Decagonal-great hendecagrammic duoprismatic prism, Decagonal-grand hendecagrammic duoprismatic prism, Decagrammic-hendecagonal duoprismatic prism, Decagrammic-small hendecagrammic duoprismatic prism, Decagrammic-hendecagrammic duoprismatic prism, Decagrammic-great hendecagrammic duoprismatic prism, Decagrammic-grand hendecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(10)×I2(11)×A1, order 880
ConvexYes
NatureTame

The decagonal-hendecagonal duoprismatic prism or dahenip, also known as the decagonal-hendecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 decagonal-hendecagonal duoprisms, 10 square-hendecagonal duoprisms, and 11 square-decagonal duoprisms. Each vertex joins 2 square-decagonal duoprisms, 2 square-hendecagonal duoprisms, and 1 decagonal-hendecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of an decagonal-hendecagonal duoprismatic prism of edge length 2sin(π/11) are given by:

• ${\displaystyle \left(0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,0,\,1,\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,0,\,1,\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,0,\,1,\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{11}}\right),}$

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

## Representations

A decagonal-hendecagonal duoprismatic prism has the following Coxeter diagrams:

• x x10o x11o () (full symmetry)
• x x5x x11o () (decagons as dipentagons)
• xx10oo xx11oo&#x (decagonal-hendecagonal duoprism atop decagonal-hendecagonal duoprism)
• xx5xx xx11oo&#x