Octagonal prism

The octagonal prism, or op, is a prismatic uniform polyhedron. It consists of 2 octagons and 8 squares. Each vertex joins one octagon and two squares. As the name suggests, it is a prism based on an octagon.

It can also be obtained from the small rhombicuboctahedron by removing two opposing square cupolas. It can therefore also be thought of as a bidiminished small rhombicuboctahedron.

Vertex coordinates
An octagonal prism of edge length 1 has vertex coordinates given by:
 * (±1/2, ±(1+$\sqrt{(5+2√2)/2}$)/2, ±1/2),
 * (±(1+$\sqrt{2}$)/2, ±1/2, ±1/2).

Representations
An octagonal prism has the following Coxeter diagrams:


 * x x8o (full symmetry)
 * x x4x (generally a ditetragonal prism)
 * s2s8x (generally a ditetragonal trapezoprism)
 * xx8oo&#x (octagonal frustum)
 * xx4xx&#x (ditetragonal frustum)
 * xxxx xwwx&#xt (A1×A1 axial)
 * xx xw wx&#zx (A1×A1×A1 symmetry)

Variations
There are several isogonal lower-symmetry variants of the octagonal prism, all of which are listed below:

Ditetragonal prism
A ditetragonal prism is a prism based on a ditetragon. The two bases are ditetragons, while the lateral sides are 4+4 rectangles.

Ditetragonal trapezoprism
A ditetragonal trapezoprism is made out of two opposite ditetragons in parallel planes, connected by 8 isosceles trapezoids.

Related polyhedra
A square cupola can be attached to a base of the octagonal prism to form the elongated square cupola. If a second square cupola is attached to the other base in the same orientation, the result is the elongated square orthobicupola, better known as the small rhombicuboctahedron. If the second cupola is rotated 45º the result is the elongated square gyrobicupola.