Square-octagonal duoprism

Square-octagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSodip
Coxeter diagramx4o x8o ()
Elements
Cells8 cubes, 4 octagonal prisms
Faces8+32 squares, 4 octagons
Edges32+32
Vertices32
Vertex figureDigonal disphenoid, edge lengths 2+2 (base 1) and 2 (base 2 and sides)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{3+\sqrt2}{2}} ≈ 1.48563}$
Hypervolume${\displaystyle 2(1+\sqrt2) ≈ 4.82843}$
Dichoral anglesCube–4–cube: 135°
Cube–4–op: 90°
Op–8–op: 90°
Height1
Central density1
Number of pieces12
Level of complexity6
Related polytopes
ArmySodip
RegimentSodip
DualSquare-octagonal duotegum
ConjugateSquare-octagrammic duoprism
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryB2×I2(8), order 128
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

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.

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:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{1+\sqrt2}{2},\,±\frac12\right).}$

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)