# Hendecagonal-square antiprismatic duoprism

Hendecagonal-square antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHensquap
Coxeter diagramx11o s2s8o
Elements
Tera11 square antiprismatic prisms, 8 triangular-hendecagonal duoprisms, 2 square-hendecagonal duoprisms
Cells88 triangular prisms, 22 cubes, 11 square antiprisms, 8+8 hendecagonal prisms
Faces88 triangles, 22+88+88 squares, 8 hendecagons
Edges88+88+88
Vertices88
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 2 (base trapezoid), 2cos(π/11) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {4+{\sqrt {2}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 1.95613}$
Hypervolume${\displaystyle {\frac {11{\sqrt {4+3{\sqrt {2}}}}}{12\tan {\frac {\pi }{11}}}}\approx 8.96292}$
Diteral anglesSquappip–squap–squappip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–thendip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
Thendip–henp–shendip: = ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Thendip–trip–squappip: 90°
Shendip–cube–squappip: 90°
Height${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces21
Level of complexity40
Related polytopes
ArmyHensquap
RegimentHensquap
DualHendecagonal-square antitegmatic duotegum
ConjugatesSmall hendecagrammic-square antiprismatic duoprism, Hendecagrammic-square antiprismatic duoprism, Great hendecagrammic-square antiprismatic duoprism, Grand hendecagrammic-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(11)×I2(8)×A1+, order 352
ConvexYes
NatureTame

The hendecagonal-square antiprismatic duoprism or hensquap is a convex uniform duoprism that consists of 11 square antiprismatic prisms, 2 square-hendecagonal duoprisms, and 8 triangular-hendecagonal duoprisms. Each vertex joins 2 square antiprismatic prisms, 3 triangular-hendecagonal duoprisms, and 1 square-hendecagonal duoprism.

## Vertex coordinates

The vertices of a hendecagonal-square antiprismatic duoprism of edge length 2sin(π/11) are given by:

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{11}},\,\pm \sin {\frac {j\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{11}},\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{11}},\,\pm \sin {\frac {j\pi }{11}},\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{11}},\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\sqrt {2}}\sin {\frac {\pi }{11}},\,0,\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{11}},\,\pm \sin {\frac {j\pi }{11}},\,\pm {\sqrt {2}}\sin {\frac {\pi }{11}},\,0,\,-{\frac {{\sqrt[{4}]{8}}\sin {\frac {\pi }{11}}}{2}}\right),}$

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

## Representations

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

• x11o s2s8o (full symmetry; square antiprisms as alternated octagonal prisms)
• x11o s2s4s (square antiprisms as alternated ditetragonal prisms)