Enneagonal antiprism

Enneagonal antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Eap
Coxeter diagram	s2s18o
Elements
Faces	18 triangles, 2 enneagons
Edges	18+18
Vertices	18
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, 2cos(π/9)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{3-2\cos\frac\pi9}{8-8\cos\frac\pi9}} ≈ 1.52405$
Volume	$\frac{3\sqrt{4\cos^2\frac\pi{18}-1}}{8\sin^2\frac\pi9} ≈ 5.43974$
Dihedral angles	3–3: $\arccos\left(\frac{1-4\cos\frac\pi9}3\right) ≈ 156.86624°$
	9–3: $\arccos\left(-\frac{\sqrt3\tan\frac\pi{18}}3\right) ≈ 95.84297°$
Height	$\sqrt{\frac{1+2\cos\frac\pi9}{2+2\cos\frac\pi9}} ≈ 0.86153$
Central density	1
Number of pieces	20
Level of complexity	4
Related polytopes
Army	Eap
Regiment	Eap
Dual	Enneagonal antitegum
Conjugate	Great enneagrammic retroprism
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	I2(18)×A1/2, order 36
Convex	Yes
Nature	Tame

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:

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

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

External links

• Wikipedia Contributors. "Enneagonal antiprism".