# Hexagonal-enneagonal duoprism

Hexagonal-enneagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymHendip
Coxeter diagramx6o x9o ()
Elements
Cells9 hexagonal prisms, 6 enneagonal prisms
Faces54 squares, 9 hexagons, 6 enneagons
Edges54+54
Vertices54
Vertex figureDigonal disphenoid, edge lengths 3 (base 1), 2cos(π/9) (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {4+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.77120}$
Hypervolume${\displaystyle {\frac {27{\sqrt {3}}}{8\tan {\frac {\pi }{9}}}}\approx 16.06085}$
Dichoral anglesHip–6–hip: 140°
Ep–9–ep: 120°
Hip–4–ep: 90°
Central density1
Number of external pieces15
Level of complexity6
Related polytopes
ArmyHendip
RegimentHendip
DualHexagonal-enneagonal duotegum
ConjugatesHexagonal-enneagrammic duoprism,
Hexagonal-great enneagrammic duoprism
Abstract & topological properties
Flag count1296
Euler characteristic0
OrientableYes
Properties
SymmetryG2×I2(9), order 216
Flag orbits6
ConvexYes
NatureTame

The hexagonal-enneagonal duoprism or hendip, also known as the 6-9 duoprism, is a uniform duoprism that consists of 6 enneagonal prisms and 9 hexagonal prisms, with two of each joining at each vertex.

This polychoron can be subsymmetrically faceted into a digonal-triangular triswirlprism, although it cannot be made uniform.

## Vertex coordinates

The coordinates of a hexagonal-enneagonal duoprism, centered at the origin and with edge length 2sin(π/9), are given by:

• ${\displaystyle \left(\pm 2\sin {\frac {\pi }{9}},0,1,0\right)}$,
• ${\displaystyle \left(\pm 2\sin {\frac {\pi }{9}},0,\cos \left({\frac {j\pi }{9}}\right),\pm \sin \left({\frac {j\pi }{9}}\right)\right)}$,
• ${\displaystyle \left(\pm 2\sin {\frac {\pi }{9}},0,-{\frac {1}{2}},\pm {\frac {\sqrt {3}}{2}}\right)}$,
• ${\displaystyle \left(\pm \sin {\frac {\pi }{9}},\pm {\sqrt {3}}\sin {\frac {\pi }{9}},1,0\right)}$,
• ${\displaystyle \left(\pm \sin {\frac {\pi }{9}},\pm {\sqrt {3}}\sin {\frac {\pi }{9}},\cos \left({\frac {j\pi }{9}}\right),\pm \sin \left({\frac {j\pi }{9}}\right)\right)}$,
• ${\displaystyle \left(\pm \sin {\frac {\pi }{9}},\pm {\sqrt {3}}\sin {\frac {\pi }{9}},-{\frac {1}{2}},\pm {\frac {\sqrt {3}}{2}}\right)}$,

where j = 2, 4, 8.

## Representations

A hexagonal-enneagonal duoprism has the following Coxeter diagrams:

• x6o x9o () (full symmetry)
• x3x x9o () (A2×I2(9) symmetry, hexagons as ditrigons)