# Heptagonal-hendecagonal duoprismatic prism

Heptagonal-hendecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHehenip
Coxeter diagramx x7o x11o
Elements
Tera11 square-heptagonal duoprisms, 7 square-hendecagonal duoprisms, 2 heptagonal-hendecagonal duoprisms
Cells77 cube]]s, 7+14 hendecagonal prisms, 11+22 heptagonal prisms
Faces77+77+154 squares, 22 heptagons, 14 hendecagons
Edges77+154+154
Vertices154
Vertex figureDigonal disphenoidal pyramid, edge lengths 2cos(π/7) (disphenoid base 1), 2cos(π/11) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {1+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.17432}$
Hypervolume${\displaystyle {\frac {77}{16\tan {\frac {\pi }{7}}\tan {\frac {\pi }{11}}}}\approx 34.03392}$
Diteral anglesSquahedip–hep–squahedip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Shendip–henp–shendip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Shendip–cube–squahedip: 90°
Hehendip–hep–squahedip: 90°
Shendip–henp–hehendip: 90°
Height1
Central density1
Number of external pieces20
Level of complexity30
Related polytopes
ArmyHehenip
RegimentHehenip
DualHeptagonal-hendecagonal duotegmatic tegum
ConjugatesHeptagonal-small hendecagrammic duoprismatic prism, Heptagonal-hendecagrammic duoprismatic prism, Heptagonal-great hendecagrammic duoprismatic prism, Heptagonal-grand hendecagrammic duoprismatic prism, Heptagrammic-hendecagonal duoprismatic prism, Heptagrammic-small hendecagrammic duoprismatic prism, Heptagrammic-hendecagrammic duoprismatic prism, Heptagrammic-great hendecagrammic duoprismatic prism, Heptagrammic-grand hendecagrammic duoprismatic prism, Great heptagrammic-hendecagonal duoprismatic prism, Great heptagrammic-small hendecagrammic duoprismatic prism, Great heptagrammic-hendecagrammic duoprismatic prism, Great heptagrammic-great hendecagrammic duoprismatic prism, Great heptagrammic-grand hendecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)×I2(11)×A1, order 616
ConvexYes
NatureTame

The heptagonal-hendecagonal duoprismatic prism or hehenip, also known as the heptagonal-hendecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonal-hendecagonal duoprisms, 7 square-hendecagonal duoprisms, and 11 square-heptagonal duoprisms. Each vertex joins 2 square-heptagonal duoprisms, 2 square-hendecagonal duoprisms, and 1 heptagonal-hendecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a heptagonal-hendecagonal duoprismatic prism of edge length 4sin(π/7)sin(π/11) are given by:

• ${\displaystyle \left(2\sin {\frac {\pi }{11}},\,0,\,2\sin {\frac {\pi }{7}},\,0,\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(2\sin {\frac {\pi }{11}},\,0,\,2\cos \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{11}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{11}},\,2\sin {\frac {\pi }{7}},\,0,\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{11}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{11}},\,2\cos \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin \left({\frac {k\pi }{11}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{11}}\right),}$

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

## Representations

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

• x x7o x11o (full symmetry)
• xx7oo xx11oo&#x (heptagonal-hendecagonal duoprism atop heptagonal-hendecagonal duoprism)