# Octagonal-cuboctahedral duoprism

Octagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOco
Coxeter diagramx8o o4x3o
Elements
Tera8 cuboctahedral prisms, 8 triangular-octagonal duoprisms, 6 square-octagonal duoprisms
Cells64 triangular prisms, 48 cubes, 8 cuboctahedra, 24 octagonal prisms
Faces64 triangles, 48+192 squares, 12 octagons
Edges96+192
Vertices96
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {4+{\sqrt {2}}}{2}}}\approx 1.64533}$
Hypervolume${\displaystyle 10{\frac {2+{\sqrt {2}}}{3}}\approx 11.38071}$
Diteral anglesCope–co–cope: 135°
Todip–op–sodip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Todip–trip–cope: 90°
Sodip–cube–cope: 90°
Central density1
Number of external pieces22
Level of complexity20
Related polytopes
ArmyOco
RegimentOco
DualOctagonal-rhombic dodecahedral duotegum
ConjugateOctagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(8), order 768
ConvexYes
NatureTame

The octagonal-cuboctahedral duoprism or oco is a convex uniform duoprism that consists of 8 cuboctahedral prisms, 6 square-octagonal duoprisms, and 8 triangular-octagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-octagonal duoprisms, and 2 square-octagonal duoprisms.

## Vertex coordinates

The vertices of an octagonal-cuboctahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}}\right).}$

## Representations

An octagonal-cuboctahedral duoprism has the following Coxeter diagrams:

• x8o o4x3o (full symmetry)
• x4x o4x3o (octagons as ditetragons)
• x8o x3o3x
• x4x x3o3x