# Enneagonal-hexagonal antiprismatic duoprism

Enneagonal-hexagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEhap
Coxeter diagramx9o s2s12o ()
Elements
Tera9 hexagonal antiprismatic prisms, 12 triangular-enneagonal duoprisms, 2 hexagonal-enneagonal duoprisms
Cells108 triangular prisms, 18 hexagonal prisms, 9 hexagonal antiprisms, 12+12 enneagonal prisms
Faces108 triangles, 108+108 squares, 18 hexagons, 12 enneagons
Edges108+108+108
Vertices108
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 3 (base trapezoid), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\sqrt {3}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.82213}$
Hypervolume${\displaystyle {\frac {9{\sqrt {2+2{\sqrt {3}}}}}{4\tan {\frac {\pi }{9}}}}\approx 14.45027}$
Diteral anglesTedip–ep–tedip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
Happip–hap–happip: 140°
Tedip–ep–hendip: = ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Tedip–trip–happip: 90°
Hendip–hip–happip: 90°
Height${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces23
Level of complexity40
Related polytopes
ArmyEhap
RegimentEhap
DualEnneagonal-hexagonal antitegmatic duotegum
ConjugatesEnneagrammic-hexagonal antiprismatic duoprism, Great enneagrammic-hexagonal antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(9)×I2(12)×A1+, order 432
ConvexYes
NatureTame

The enneagonal-hexagonal antiprismatic duoprism or ehap is a convex uniform duoprism that consists of 9 hexagonal antiprismatic prisms, 2 hexagonal-enneagonal duoprisms, and 12 triangular-enneagonal duoprisms. Each vertex joins 2 hexagonal antiprismatic prisms, 3 triangular-enneagonal duoprisms, and 1 hexagonal-enneagonal duoprism.

## Vertex coordinates

The vertices of an enneagonal-hexagonal antiprismatic duoprism of edge length 2sin(π/9) are given by:

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

where j = 2, 4, 8.

## Representations

An enneagonal-hexagonal antiprismatic duoprism has the following Coxeter diagrams:

• x9o s2s12o () (full symmetry; hexagonal antiprisms as alternated dodecagonal prisms)
• x9o s2s6s () (hexagonal antiprisms as alternated dihexagonal prisms)