Octagonal antiprism

{{Infobox polytope The octagonal antiprism, or oap, is a prismatic uniform polyhedron. It consists of 16 triangles and 2 octagons. Each vertex joins one octagon and three triangles. As the name suggests, it is an antiprism based on an octagon.
 * dim = 3
 * obsa = Oap
 * img=Octagonal antiprism.png
 * 3d=Octagonal antiprism.stl
 * off = Octagonal antiprism.off
 * type=Uniform
 * coxeter = s2s16o
 * symmetry = I{{sub|2}}(16)×A{{sub|1}}+, order 32
 * army=Oap
 * reg=Oap
 * faces = 16 triangles, 2 octagons
 * edges = 16+16
 * vertices = 16
 * circum = $$\sqrt{\frac{6+2\sqrt2+\sqrt{20+14\sqrt2}}{8} ≈ 1.37555$$
 * height = $$\sqrt{\frac{-2-2\sqrt2+\sqrt{20+14\sqrt2}}{2}} ≈ 0.86030$$
 * volume = $$\frac{2\sqrt{4+2\sqrt2+2\sqrt{146+103\sqrt2}}}{3} ≈ 4.26796$$
 * verf = Isosceles trapezoid, edge lengths 1, 1, 1, $\sqrt{2+√2}$
 * dih= 3–3: $$\arccos\left(\frac{1-2\sqrt{2+\sqrt2}}{3}\right) ≈ 153.96238°$$
 * dih2=8–3: $$\arccos\left(-\sqrt{\frac{7+4\sqrt2-2\sqrt{20+14\sqrt2}}{3}}\right) ≈ 96.59451°$$
 * pieces = 18
 * loc = 4
 * dual = Octagonal antitegum
 * conjugate = Octagrammic antiprism, Octagrammic retroprism
 * conv=Yes
 * orient=Yes
 * nat=Tame}}

Vertex coordinates
An octagonal antiprism of edge length 1 has vertex coordinates given by:
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}2,\,H\right),$$
 * $$\left(±\frac{1+\sqrt2}2,\,±\frac12,\,H\right),$$
 * $$\left(0,\,±\sqrt{\frac{2+\sqrt2}2},\,-H\right),$$
 * $$\left(±\sqrt{\frac{2+\sqrt2}2},\,0,\,-H\right),$$
 * $$\left(±\frac{\sqrt{2+\sqrt2}}2,\,±\frac{\sqrt{2+\sqrt2}}2,\,-H\right),$$

where $$H=\sqrt{\frac{-2-2\sqrt2+\sqrt{20+14\sqrt2}}{8}}$$ is the distance between the antiprism's center and the center of one of its bases.

Representations
An octagonal prism can be represented by the following Coxeter diagrams:


 * s2s16o (alternated hexadecagonal prism)
 * s2s8s (alternated dioctagonal prism)
 * xo8ox&#x (bases considered separately)

Related polyhedra
A square cupola can be attached to a base of the octagonal antiprism to form the gyroelongated square cupola. If a second square cupola is attached to the other base, the result is the gyroelongated square bicupola.