# Enneagonal antiprism

Enneagonal antiprism
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymEap
Coxeter diagrams2s18o
Elements
Faces18 triangles, 2 enneagons
Edges18+18
Vertices18
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 2cos(π/9)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{3-2\cos\frac\pi9}{8-8\cos\frac\pi9}} ≈ 1.52405}$
Volume${\displaystyle \frac{3\sqrt{4\cos^2\frac\pi{18}-1}}{8\sin^2\frac\pi9} ≈ 5.43974}$
Dihedral angles3–3: ${\displaystyle \arccos\left(\frac{1-4\cos\frac\pi9}3\right) ≈ 156.86624°}$
9–3: ${\displaystyle \arccos\left(-\frac{\sqrt3\tan\frac\pi{18}}3\right) ≈ 95.84297°}$
Height${\displaystyle \sqrt{\frac{1+2\cos\frac\pi9}{2+2\cos\frac\pi9}} ≈ 0.86153}$
Central density1
Number of pieces20
Level of complexity4
Related polytopes
ArmyEap
RegimentEap
DualEnneagonal antitegum
ConjugateGreat enneagrammic retroprism
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryI2(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:

• ${\displaystyle \left(1,\,0,\,H\right),}$
• ${\displaystyle \left(\cos\left(\frac{2\pi}9\right),\,±\sin\left(\frac{2\pi}9\right),\,H\right),}$
• ${\displaystyle \left(\cos\left(\frac{4\pi}9\right),\,±\sin\left(\frac{4\pi}9\right),\,H\right),}$
• ${\displaystyle \left(-\frac12,\,±\frac{\sqrt3}2,\,H\right),}$
• ${\displaystyle \left(\cos\left(\frac{8\pi}9\right),\,±\sin\left(\frac{8\pi}9\right),\,H\right),}$

where ${\displaystyle H=\sqrt{\frac{1+2\cos\frac\pi9}{2+2\cos\frac\pi9}}\sin\frac\pi9.}$