# Enneagonal antiprism

Enneagonal antiprism
Rank3
TypeUniform
Notation
Bowers style acronymEap
Coxeter diagrams2s18o
Conway notationA9
Elements
Faces18 triangles, 2 enneagons
Edges18+18
Vertices18
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 2cos(π/9)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {3-2\cos {\frac {\pi }{9}}}{8-8\cos {\frac {\pi }{9}}}}}\approx 1.52405}$
Volume${\displaystyle {\frac {3{\sqrt {4\cos ^{2}{\frac {\pi }{18}}-1}}}{8\sin ^{2}{\frac {\pi }{9}}}}\approx 5.43974}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {1-4\cos {\frac {\pi }{9}}}{3}}\right)\approx 156.86624^{\circ }}$
9–3: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}\tan {\frac {\pi }{18}}}{3}}\right)\approx 95.84297^{\circ }}$
Height${\displaystyle {\sqrt {\frac {1+2\cos {\frac {\pi }{9}}}{2+2\cos {\frac {\pi }{9}}}}}\approx 0.86153}$
Central density1
Number of external pieces20
Level of complexity4
Related polytopes
ArmyEap
RegimentEap
DualEnneagonal antitegum
ConjugateGreat enneagrammic retroprism
Abstract & topological properties
Flag count144
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(18)×A1)/2, order 36
ConvexYes
NatureTame

The enneagonal antiprism, or eap, is a prismatic uniform polyhedron. It consists of 18 triangles and 2 enneagons. Each vertex joins one enneagon and three triangles. As the name suggests, it is an antiprism based on an enneagon.

## Vertex coordinates

The vertices of an enneagonal antiprism, centered at the origin and with edge length 2sin(π/9), are given by the following points, as well as their central inversions:

• ${\displaystyle \left(1,\,0,\,H\right),}$
• ${\displaystyle \left(\cos \left({\frac {2\pi }{9}}\right),\,\pm \sin \left({\frac {2\pi }{9}}\right),\,H\right),}$
• ${\displaystyle \left(\cos \left({\frac {4\pi }{9}}\right),\,\pm \sin \left({\frac {4\pi }{9}}\right),\,H\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,H\right),}$
• ${\displaystyle \left(\cos \left({\frac {8\pi }{9}}\right),\,\pm \sin \left({\frac {8\pi }{9}}\right),\,H\right),}$

where ${\displaystyle H={\sqrt {\frac {1+2\cos {\frac {\pi }{9}}}{2+2\cos {\frac {\pi }{9}}}}}\sin {\frac {\pi }{9}}.}$