Octagonal duoprism

The octagonal duoprism or odip, also known as the octagonal-octagonal duoprism, the 8 duoprism or the 8-8 duoprism, is a noble uniform duoprism that consists of 16 octagonal prisms, with 4 joining at each vertex. It is also the digonal double gyrotrapezohedroid and the 16-7 gyrochoron. It is the first in an infinite family of isogonal octagonal dihedral swirlchora, the first in an infinite family of isochoric octagonal hosohedral swirlchora and also the first in an infinite family of isochoric digonal bipyramidal swirlchora.

The octagonal duoprism can be vertex-inscribed into a small rhombated tesseract or small prismatotetracontoctachoron.

This polychoron can be alternated into a square duoantiprism, although it cannot be made uniform. Eight of the octagons can also be alternated into long rectangles to create a square-square prismantiprismoid, which is also nonuniform.

It can form a non-Wythoffian uniform tiling with 288 octagonal duoprisms at each vertex with a bitetracontoctachoron as the vertex figure, called an octagonal duoprismatic honeycomb.

Vertex coordinates
Coordinates for the vertices of an octagonal duoprism of edge length 1, centered at the origin, are given by:
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$

Representations
An octagonal duoprism has the following Coxeter diagrams:


 * x8o x8o (full symmetry)
 * x4x x8o (BC2×I2(8) symmetry)
 * x4x x4x (BC2×BC2 symmetry, both octagons as ditetragons)
 * xwwx xxxx4xxxx&#xt (BC2×A1 axial, octagonal prism-first)

Related polychora
Non-adjacent cells of the octagonal duoprism can be augmented with square pucofastegiums. If 8 cells are augmented in this way, so that all the cupolas blend with the prisms into small rhombicuboctahedra, the result is the uniform small rhombated tesseract.

An octagonal duoprism of edge length 1 contains the vertices of a hexadecachoron of edge length $$\sqrt{4+2\sqrt2}$$, since the hexadecachoron is the 8-3 step prism.