# Pentagonal-hendecagonal duoprismatic prism

Pentagonal-hendecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymPehenip
Coxeter diagramx x5o x11o
Elements
Tera11 square-pentagonal duoprisms, 5 square-hendecagonal duoprisms, 2 pentagonal-hendecagonal duoprisms
Cells55 cubes, 5+10 hendecagonal prisms, 11+22 pentagonal prisms
Faces55+55+110 squares, 22 pentagons, 10 hendecagons
Edges55+110+110
Vertices110
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2 (disphenoid base 1), 2cos(π/11) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+2{\sqrt {5}}+{\frac {5}{\sin ^{2}{\frac {\pi }{11}}}}}{20}}}\approx 2.03059}$
Hypervolume${\displaystyle {\frac {11{\sqrt {25+10{\sqrt {5}}}}}{16\tan {\frac {\pi }{11}}}}\approx 16.11337}$
Diteral anglesSquipdip–pip–squipdip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Shendip–henp–shendip: 108°
Shendip–cube–squipdip: 90°
Pahendip–pip–squipdip: 90°
Shendip–henp–pahendip: 90°
Height1
Central density1
Number of external pieces18
Level of complexity30
Related polytopes
ArmyPehenip
RegimentPehenip
DualPentagonal-hendecagonal duotegmatic tegum
ConjugatesPentagonal-small hendecagrammic duoprismatic prism, Pentagonal-hendecagrammic duoprismatic prism, Pentagonal-great hendecagrammic duoprismatic prism, Pentagonal-grand hendecagrammic duoprismatic prism, Pentagrammic-hendecagonal duoprismatic prism, Pentagrammic-small hendecagrammic duoprismatic prism, Pentagrammic-hendecagrammic duoprismatic prism, Pentagrammic-great hendecagrammic duoprismatic prism, Pentagrammic-grand hendecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×I2(11)×A1, order 440
ConvexYes
NatureTame

The pentagonal-hendecagonal duoprismatic prism or pehenip, also known as the pentagonal-hendecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal-hendecagonal duoprisms, 5 square-hendecagonal duoprisms, and 11 square-pentagonal duoprisms. Each vertex joins 2 square-pentagonal duoprisms, 2 square-hendecagonal duoprisms, and 1 pentagonal-hendecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{11}},\,1,\,0,\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{11}},\,1,\,0,\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm \sin {\frac {\pi }{11}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{11}},\,1,\,0,\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{11}},\,\cos {\frac {j\pi }{11}},\,\pm \sin {\frac {j\pi }{11}},\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{11}},\,\cos {\frac {j\pi }{11}},\,\pm \sin {\frac {j\pi }{11}},\,\pm \sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\pm \sin {\frac {\pi }{11}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{11}},\,\cos {\frac {j\pi }{11}},\,\pm \sin {\frac {j\pi }{11}},\,\pm \sin {\frac {\pi }{11}}\right),}$

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

## Representations

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

• x x5o x11o (full symmetry)
• xx5oo xx11oo&#x (pentagonal-hendecagonal duoprism atop pentagonal-hendecagonal duoprism)