# Octagonal-truncated octahedral duoprism

Octagonal-truncated octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOtoe
Coxeter diagramx8o o4x3x ()
Elements
Tera6 square-octagonal duoprisms, 8 truncated octahedral prisms, 8 hexagonal-octagonal duoprisms
Cells48 cubes, 64 hexagonal prisms, 12+24 octagonal prisms, 8 truncated octahedra
Faces48+96+192 squares, 64 hexagons, 24 octagons
Edges96+192+192
Vertices192
Vertex figureDigonal disphenoidal pyramid, edge lengths 2, 3, 3 (base triangle), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {7+{\sqrt {2}}}{2}}}\approx 2.05112}$
Hypervolume${\displaystyle 32+16{\sqrt {2}}\approx 54.62742}$
Diteral anglesTope–toe–tope: 135°
Sodip–op–hodip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Hodip–op–hodip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Sodip–cube–tope: 90°
Hodip–hip–tope: 90°
Central density1
Number of external pieces22
Level of complexity30
Related polytopes
ArmyOtoe
RegimentOtoe
DualOctagonal-tetrakis hexahedral duotegum
ConjugateOctagrammic-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(8), order 768
ConvexYes
NatureTame

The octagonal-truncated octahedral duoprism or otoe is a convex uniform duoprism that consists of 8 truncated octahedral prisms, 8 hexagonal-octagonal duoprisms, and 6 square-octagonal duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 square-octagonal duoprism, and 2 hexagonal-octagonal duoprisms.

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

## Vertex coordinates

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

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

## Representations

An octagonal-truncated octahedral duoprism has the following Coxeter diagrams:

• x8o o4x3x (full symmetry)
• x8o x3x3x ()
• x4x o4x3x () (octagons as ditetragons)
• x4x x3x3x ()