Square-octagonal duoprism

{{Infobox polytope The square-octagonal duoprism or sodip, also known as the 4-8 duoprism, is a uniform duoprism that consists of 4 octagonal prisms and 8 cubes, with two of each joining at each vertex.
 * type=Uniform
 * dim = 4
 * img= 4-8 duoprism.png
 * off=Square-octagonal duoprism.off
 * obsa = Sodip
 * cells = 8 cubes, 4 octagonal prisms
 * faces = 8+32 squares, 4 octagons
 * edges = 32+32
 * vertices = 32
 * verf = Digonal disphenoid, edge lengths $\sqrt{2+√2}$ (base 1) and $\sqrt{2}$ (base 2 and sides)
 * symmetry = B{{sub|2}}×I{{sub|2}}(8), order 128
 * coxeter = x4o x8o ({{CDD|node_1|4|node|2|node_1|8|node}})
 * army=Sodip
 * reg=Sodip
 * circum = $$\sqrt{\frac{3+\sqrt2}{2}} ≈ 1.48563$$
 * height = 1
 * hypervolume= $$2(1+\sqrt2) ≈ 4.82843$$
 * dich = Cube–4–cube: 135°
 * dich2 = Cube–4–op: 90°
 * dich3 = Op–8–op: 90°
 * pieces = 12
 * loc = 6
 * dual=Square-octagonal duotegum
 * conjugate = Square-octagrammic duoprism
 * conv = Yes
 * orientable=Yes
 * nat=Tame]}

The square-octagonal duoprism, being the prism of the octagonal prism, is also the central part of the small rhombicuboctahedral prism, which can in turn be constructed as part of the small disprismatotesseractihexadecachoron.

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

It is also a CRF segmentochoron, designated K-4.70 on Richard Klitzing's list. As such it can also be viewed as a prism based on the octagonal prism.

The convex hull of two orthogonal square-octagonal duoprisms is either the small disprismatotesseractihexadecachoron or the square duotruncatoprism.

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

Representations
A square-octagonal duoprism as the following Coxeter diagrams:


 * x4o x8o (full symmety)
 * x x x8o (I2(8)×A1×A1 symmetry, square as rectangle)
 * x4o x4x (B2×B2 symmetry, octagon as ditetragon)
 * x x x4x (B2×A1×A1 symmetry, both of the above)
 * xx xx8oo&#x (I2(8)×A1 octagon prism prism)
 * xx xx4xx&#x (B2×A1 symmetry, as above)
 * xxx8ooo oqo&#xt (I2(8)×A1 symmetry, octagon-first)
 * oqo xxx4xxx&#xt (B2×A1 axial, octagon-first)
 * xwwx xxxx4oooo&#xt (B2×A1 axial, cube-first)
 * xxxx xxxx xwwx&#xt (A1×A1×A1 axial, cube-first)