Octagonal antiprism

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.

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.