Octagonal-truncated octahedral duoprism

{{Infobox polytope The octagonal-truncated octahedral duoprism or otoe is a convex uniform duoprism that consists of 8 truncated octahedral prisms, 8 hexagonal-octagonal duoprisms, and 6 square-octagonal duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 square-octagonal duoprism, and 2 hexagonal-octagonal duoprisms.
 * type=Uniform
 * dim = 5
 * img=
 * off = auto
 * obsa = Otoe
 * coxeter = x8o o4x3x
 * army = Otoe
 * reg = Otoe
 * terons = 6 square-octagonal duoprisms, 8 truncated octahedral prisms, 8 hexagonal-octagonal duoprisms
 * cells = 48 cubes, 64 hexagonal prisms, 12+24 octagonal prisms, 8 truncated octahedra
 * faces = 48+96+192 squares, 64 hexagons, 24 octagons
 * edges = 96+192+192
 * vertices = 192
 * circum = $$\sqrt{\frac{7+\sqrt2}{2} ≈ 2.05112$$
 * hypervol = $$32+16\sqrt2 ≈ 54.62742$$
 * dit = Tope–toe–tope: 135°
 * dit2 = Sodip–op–hodip: $$\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$$
 * dit3 = Hodip–op–hodip: $$\arccos\left(-\frac13\right) ≈ 109.47122°$$
 * dit4 = Sodip–cube–tope: 90°
 * dit5 = Hodip–hip–tope: 90°
 * verf = Digonal disphenoidal pyramid, edge lengths $\sqrt{2}$, $\sqrt{3}$, $\sqrt{3}$ (base triangle), $\sqrt{2+√2}$ (top), $\sqrt{2}$ (side edges)
 * symmetry = BC{{sub|3}}×I2(8), order 768
 * pieces = 22
 * loc = 30
 * dual=Octagonal-tetrakis hexahedral duotegum
 * conjugate = Octagrammic-truncated octahedral duoprism
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

This polyteron can be alternated into a square-pyritohedral icosahedral duoantiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a pyritohedral icosahedral-square prismantiprismoid, which is also nonuniform.

Vertex coordinates
The vertices of an octagonal-truncated octahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}2,\,0,\,±\frac{\sqrt2}2,\,±\sqrt2\right),$$
 * $$\left(±\frac{1+\sqrt2}2,\,±\frac12,\,0,\,±\frac{\sqrt2}2,\,±\sqrt2\right).$$

Representations
An octagonal-truncated octahedral duoprism has the following Coxeter diagrams:
 * x8o o4x3x (full symmetry)
 * x8o x3x3x
 * x4x o4x3x (octagons as ditetragons)
 * x4x x3x3x