# Triangular-truncated tetrahedral duoprism

Triangular-truncated tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTratut
Coxeter diagramx3o x3x3o ()
Elements
Tera4 triangular duoprisms, 4 triangular-hexagonal duoprisms, 3 truncated tetrahedral prisms
Cells6+12+12 triangular prisms, 12 hexagonal prisms, 3 truncated tetrahedra
Faces12+12 triangles, 18+36 squares, 12 hexagons
Edges18+36+36
Vertices36
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 3, 3 (base triangle), 1 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {246}}{12}}\approx 1.30703}$
Hypervolume${\displaystyle {\frac {23{\sqrt {6}}}{48}}\approx 1.17371}$
Diteral anglesTriddip–trip–thiddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Triddip–trip–tuttip: 90°
Thiddip–hip–tuttip: 90°
Thiddip–trip–thiddip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Tuttip–tut–tuttip: 60°
Height${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Central density1
Number of external pieces11
Level of complexity30
Related polytopes
ArmyTratut
RegimentTratut
DualTriangular-triakis tetrahedral duotegum
ConjugateNone
Abstract & topological properties
Flag count4320
Euler characteristic2
OrientableYes
Properties
SymmetryA3×A2, order 144
ConvexYes
NatureTame

The triangular-truncated tetrahedral duoprism or tratut is a convex uniform duoprism that consists of 3 truncated tetrahedral prisms, 4 triangular-hexagonal duoprisms, and 4 triangular duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular duoprism, and 2 triangular-hexagonal duoprisms. It is a duoprism based on a triangle and a truncated tetrahedron, which makes it a convex segmentoteron.

## Vertex coordinates

The vertices of a triangular-truncated tetrahedral duoprism of edge length 1 are given by all permutations and even sign changes of the last three coordinates of:

• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)}$.

## Representations

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

• x3o x3x3o () (full symmetry)
• ox oo3xx3xx&#x (tut atop tuttip)
• ooo3xxx3xxx&#x