Square-octagonal duoprism

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.

Square-octagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sodip
Coxeter diagram	x4o x8o ()
Elements
Cells	8 cubes, 4 octagonal prisms
Faces	8+32 squares, 4 octagons
Edges	32+32
Vertices	32
Vertex figure	Digonal disphenoid, edge lengths √2+√2 (base 1) and √2 (base 2 and sides)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{3+\sqrt2}{2}} ≈ 1.48563$
Hypervolume	$2(1+\sqrt2) ≈ 4.82843$
Dichoral angles	Cube–4–cube: 135°
	Cube–4–op: 90°
	Op–8–op: 90°
Height	1
Central density	1
Number of external pieces	12
Level of complexity	6
Related polytopes
Army	Sodip
Regiment	Sodip
Dual	Square-octagonal duotegum
Conjugate	Square-octagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B2×I2(8), order 128
Convex	Yes
Nature	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)

External links

• Bowers, Jonathan. "Category A: duoprisms".

• Klitzing, Richard. "sodip".

• Wikipedia Contributors. "4-8 duoprism".