# Hendecagonal-hexagonal antiprismatic duoprism

Hendecagonal-hexagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHenhap
Coxeter diagramx11o s2s12o ()
Elements
Tera11 hexagonal antiprismatic prisms, 12 triangular-hendecagonal duoprisms, 2 hexagonal-hendecagonal duoprisms
Cells132 triangular prisms, 22 hexagonal prisms, 11 hexagonal antiprisms, 12+12 hendecagonal prisms
Faces132 triangles, 132+132 squares, 22 hexagons, 12 hendecagons
Edges132+132+132
Vertices132
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 3 (base trapezoid), 2cos(π/11) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\sqrt {3}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.08151}$
Hypervolume${\displaystyle {\frac {11{\sqrt {2+2{\sqrt {3}}}}}{4\tan {\frac {\pi }{11}}}}\approx 21.89257}$
Diteral anglesHappip–hap–happip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–thendip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
Thendip–henp–hahendip: = ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Thendip–trip–happip: 90°
Hahendip–hip–happip: 90°
Height${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces25
Level of complexity40
Related polytopes
ArmyHenhap
RegimentHenhap
DualHendecagonal-hexagonal antitegmatic duotegum
ConjugatesSmall hendecagrammic-hexagonal antiprismatic duoprism, Hendecagrammic-hexagonal antiprismatic duoprism, Great hendecagrammic-hexagonal antiprismatic duoprism, Grand hendecagrammic-hexagonal antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(11)×I2(12)×A1+, order 528
ConvexYes
NatureTame

The hendecagonal-hexagonal antiprismatic duoprism or henhap is a convex uniform duoprism that consists of 11 hexagonal antiprismatic prisms, 2 hexagonal-hendecagonal duoprisms, and 12 triangular-hendecagonal duoprisms. Each vertex joins 2 hexagonal antiprismatic prisms, 3 triangular-hendecagonal duoprisms, and 1 hexagonal-hendecagonal duoprism.

## Vertex coordinates

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

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

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

## Representations

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

• x11o s2s12o (full symmetry; hexagonal antiprisms as alternated dodecagonal prisms)
• x11o s2s6s () (hexagonal antiprisms as alternated dihexagonal prisms)