# Octagonal duoprismatic prism

Octagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymOop
Coxeter diagramx x8o x8o ()
Elements
Tera16 square-octagonal duoprisms, 2 octagonal duoprisms
Cells64 cubes, 16+32 octagonal prisms
Faces128+128 squares, 32 octagons
Edges64+256
Vertices128
Vertex figureTetragonal disphenoidal pyramid, edge lengths 2+2 (disphenoid bases) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+2{\sqrt {2}}}{2}}\approx 1.91421}$
Hypervolume${\displaystyle 4(3+2{\sqrt {2}})\approx 23.31371}$
Diteral anglesSodip–op–sodip: 135°
Sodip–cube–sodip: 90°
Odip–op–sodip: 90°
Height1
Central density1
Number of external pieces18
Level of complexity15
Related polytopes
ArmyOop
RegimentOop
DualOctagonal duotegmatic tegum
ConjugateOctagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(8)≀S2×A1, order 1024
ConvexYes
NatureTame

The octagonal duoprismatic prism or oop, also known as the octagonal-octagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 octagonal duoprisms and 16 square-octagonal duoprisms. Each vertex joins 4 square-octagonal duoprisms and 1 octagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

The octagonal duoprismatic prism can be vertex-inscribed into a small prismated penteract.

This polyteron can be alternated into a square duoantiprismatic antiprism, although it cannot be made uniform. Half of the octagons can also be alternated into long rectangles to create a square-square prismatic prismantiprismoid, which is also nonuniform.

## Vertex coordinates

The vertices of a octagonal duoprismatic prism of edge length 1 are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

An octagonal duoprismatic prism has the following Coxeter diagrams:

• x x8o x8o () (full symmetry)
• x x4x x4x () (octagons as ditetragons)
• xx8oo xx8oo&#x (octagonal duoprism atop octagonal duoprism)
• xx4xx xx4xx&#x