# Hexagonal-enneagonal duoprismatic prism

Hexagonal-enneagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHaep
Coxeter diagramx x6o x9o
Elements
Tera9 square-hexagonal duoprisms, 6 square-enneagonal duoprisms, 2 hexagonal-enneagonal duoprisms
Cells54 cubes, 6+12 enneagonal prisms, 9+18 hexagonal prisms
Faces54+54+108 squares, 18 hexagons, 12 enneagons
Edges54+108+108
Vertices108
Vertex figureDigonal disphenoidal pyramid, edge lengths 3 (disphenoid base 1), 2cos(π/9) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.84042}$
Hypervolume${\displaystyle {\frac {27{\sqrt {3}}}{8\tan {\frac {\pi }{9}}}}\approx 16.06085}$
Diteral anglesShiddip–hip–shiddip: 140°
Sendip–ep–sendip: 120°
Sendip–cube–shiddip: 90°
Hendip–hip–shiddip: 90°
Sendip–ep–hendip: 90°
Height1
Central density1
Number of external pieces17
Level of complexity30
Related polytopes
ArmyHaep
RegimentHaep
DualHexagonal-enneagonal duotegmatic tegum
ConjugatesHexagonal-enneagrammic duoprismatic prism, Hexagonal-great enneagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(9)×A1, order 432
ConvexYes
NatureTame

The hexagonal-enneagonal duoprismatic prism or haep, also known as the hexagonal-enneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal-enneagonal duoprisms, 6 square-enneagonal duoprisms, and 9 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-enneagonal duoprisms, and 1 hexagonal-enneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a hexagonal-enneagonal duoprismatic prism of edge length 2sin(π/9) are given by:

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

where j = 2, 4, 8.

## Representations

A hexagonal-enneagonal duoprismatic prism has the following Coxeter diagrams:

• x x6o x9o (full symmetry)
• x x3x x9o (hexagons as ditrigons)
• xx6oo xx9oo&#x (hexagonal-enneagonal duoprism atop hexagonal-enneagonal duoprism)
• xx3xx xx9oo&#x