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:
 * $$\left(\pm\frac{1}{2},\pm\frac{1+\sqrt{2}}{2}, \sqrt{\frac{\sqrt{2+\sqrt{2}}+1}{\sqrt{2+\sqrt{2}}+2}}\right),$$
 * $$\left(\pm\frac{1+\sqrt{2}}{2},\pm\frac{1}{2},\sqrt{\frac{\sqrt{2+\sqrt{2}}+1}{\sqrt{2+\sqrt{2}}+2}}\right),$$
 * $$\left(0,\pm\sqrt{\frac{2+\sqrt{2}}{2}},-\sqrt{\frac{\sqrt{2+\sqrt{2}}+1}{\sqrt{2+\sqrt{2}}+2}}\right),$$
 * $$\left(\pm\sqrt{\frac{2+\sqrt{2}}{2}},0,-\sqrt{\frac{\sqrt{2+\sqrt{2}}+1}{\sqrt{2+\sqrt{2}}+2}}\right),$$
 * $$\left(\pm\frac{\sqrt{2+\sqrt{2}}}{2},\pm\frac{\sqrt{2+\sqrt{2}}}{2},-\sqrt{\frac{\sqrt{2+\sqrt{2}}+1}{\sqrt{2+\sqrt{2}}+2}}\right).$$

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.