# Triangular-truncated octahedral duoprism

Triangular-truncated octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTratoe
Coxeter diagramx3o o4x3x ()
Elements
Tera6 triangular-square duoprisms, 8 triangular-hexagonal duoprisms, 3 truncated octahedral prisms
Cells12+24 triangular prisms, 18 cubes, 24 hexagonal prisms, 3 truncated octahedra
Faces24 triangles, 18+36+72 squares, 24 hexagons
Edges36+72+72
Vertices72
Vertex figureDigonal disphenoidal pyramid, edge lengths 2, 3, 3 (base triangle), 1 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {102}}{6}}\approx 1.68325}$
Hypervolume${\displaystyle 2{\sqrt {6}}\approx 4.89898}$
Diteral anglesTisdip–trip–thiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Thiddip–trip–thiddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Tisdip–cube–tope: 90°
Thiddip–hip–tope: 90°
Tope–toe–tope: 60°
Height${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Central density1
Number of external pieces17
Level of complexity30
Related polytopes
ArmyTratoe
RegimentTratoe
DualTriangular-tetrakis hexahedral duotegum
ConjugateNone
Abstract & topological properties
Flag count8640
Euler characteristic2
OrientableYes
Properties
SymmetryB3×A2, order 288
ConvexYes
NatureTame

The triangular-truncated octahedral duoprism or tratoe is a convex uniform duoprism that consists of 3 truncated octahedral prisms, 8 triangular-hexagonal duoprisms, and 6 triangular-square duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 triangular-square duoprism, and 2 triangular-hexagonal duoprisms. It is a duoprism based on a triangle and a truncated octahedron, which makes it a convex segmentoteron.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\sqrt {2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\sqrt {2}}\right).}$

## Representations

A triangular-truncated octahedral duoprism has the following Coxeter diagrams:

• x3o o4x3x (full symmetry)
• x3o x3x3x
• ox xx3xx3xx&#x (truncated octahedron atop truncated octahedral prism)
• xxx3xxx3xxx&#x