Square-octagonal duoprism

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Square-octagonal duoprism
4-8 duoprism.png
Bowers style acronymSodip
Coxeter diagramx4o x8o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel node.png)
Cells8 cubes, 4 octagonal prisms
Faces8+32 squares, 4 octagons
Vertex figureDigonal disphenoid, edge lengths 2+2 (base 1) and 2 (base 2 and sides)
Measures (edge length 1)
Dichoral anglesCube–4–cube: 135°
 Cube–4–op: 90°
 Op–8–op: 90°
Central density1
Number of external pieces12
Level of complexity6
Related polytopes
DualSquare-octagonal duotegum
ConjugateSquare-octagrammic duoprism
Abstract & topological properties
Euler characteristic0
SymmetryB2×I2(8), order 128

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[edit | edit source]

The vertices of a square-octagonal duoprism of edge length 1, centered at the origin, are given by:

Representations[edit | edit source]

A square-octagonal duoprism as the following Coxeter diagrams:

  • x4o x8o (full symmety)
  • x x x8o (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel node.png) (I2(8)×A1×A1 symmetry, square as rectangle)
  • x4o x4x (CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png) (B2×B2 symmetry, octagon as ditetragon)
  • x x x4x (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png) (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)

External links[edit | edit source]