# Octagonal-cubic duoprism

Octagonal-cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOcube
Coxeter diagramx8o x4o3o
Elements
Tera8 tesseracts, 6 square-octagonal duoprisms
Cells8+48 cubes, 12 octagonal prisms
Faces48+96 squares, 8 octagons
Edges64+96
Vertices64
Vertex figureTriangular scalene, edge lengths 2+2 (top), 2 (base triangle and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+2{\sqrt {2}}}}{2}}\approx 1.56752}$
Hypervolume${\displaystyle 2(1+{\sqrt {2}})\approx 4.82843}$
Diteral anglesTes–cube–tes: 135°
Tes–cube–sodip: 90°
Sodip–op–sodip: 90°
Height1
Central density1
Number of external pieces14
Level of complexity10
Related polytopes
ArmyOcube
RegimentOcube
DualOctagonal-octahedral duotegum
ConjugateOctagrammic-cubic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(8), order 768
ConvexYes
NatureTame

The octagonal-cubic duoprism or ocube, also known as a square-octagonal duoprismatic prism, is a convex uniform duoprism that consists of 8 tesseracts and 6 square-octagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-octagonal duoprisms. It is a duoprism based on a square and an octagonal prism, which makes it a convex segmentoteron.

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

The octagonal-cubic duoprism can be vertex-inscribed into the small cellated penteractitriacontaditeron.

## Vertex coordinates

The vertices of an octagonal-cubic duoprism of edge length 1 are given by:

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

## Representations

An octagonal-cubic duoprism has the following Coxeter diagrams:

• x8o x4o3o (full symmetry)
• x4x x4o3o (octagons as ditetragons)
• x x4o x8o (square-octagonal duoprismatic prism)
• x x4o x4x
• x x x x8o (octagonal prismatic prismatic prism)
• x x x x4x