# Hendecagonal-pentagonal antiprismatic duoprism

Hendecagonal-pentagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHenpap
Coxeter diagramx11o s2s10o
Elements
Tera11 pentagonal antiprismatic prisms, 10 triangular-hendecagonal duoprisms, 2 pentagonal-hendecagonal duoprisms
Cells110 triangular prisms, 22 pentagonal prisms, 11 pentagonal antiprisms, 10+10 hendecagonal prisms
Faces110 triangles, 110+110 squares, 22 pentagons, 10 hendecagons
Edges110+110+110
Vertices110
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+5)/2 (base trapezoid), 2cos(π/11) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 2.01350}$
Hypervolume${\displaystyle 11{\frac {5+2{\sqrt {5}}}{24\tan {\frac {\pi }{11}}}}\approx 14.78544}$
Diteral anglesPappip–pap–pappip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–thendip: = ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Thendip–henp–pahendip: = ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Thendip–trip–pappip: 90°
Pahendip–pip–pappip: 90°
${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces23
Level of complexity40
Related polytopes
ArmyHenpap
RegimentHenpap
DualHendecagonal-pentagonal antitegmatic duotegum
ConjugatesSmall hendecagrammic-pentagonal antiprismatic duoprism, Hendecagrammic-pentagonal antiprismatic duoprism, Great hendecagrammic-pentagonal antiprismatic duoprism, Grand hendecagrammic-pentagonal antiprismatic duoprism, hendecagonal-pentagrammic retroprismatic duoprism, small hendecagrammic-pentagrammic retroprismatic duoprism, Hendecagrammic-pentagrammic retroprismatic duoprism, Great hendecagrammic-pentagrammic retroprismatic duoprism, Grand hendecagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(11)×I2(10)×A1+, order 440
ConvexYes
NatureTame

The hendecagonal-pentagonal antiprismatic duoprism or henpap is a convex uniform duoprism that consists of 11 pentagonal antiprismatic prisms, 2 pentagonal-hendecagonal duoprisms, and 10 triangular-hendecagonal duoprisms. Each vertex joins 2 pentagonal antiprismatic prisms, 3 triangular-hendecagonal duoprisms, and 1 pentagonal-hendecagonal duoprism.

## Vertex coordinates

The vertices of a hendecagonal-pentagonal antiprismatic duoprism of edge length 2sin(π/11) are given by all central inversions of the last three coordinates of:

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

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

## Representations

A hendecagonal-pentagonal antiprismatic duoprism has the following Coxeter diagrams:

• x11o s2s10o (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x11o s2s5s (pentagonal antiprisms as alternated dipentagonal prisms)