# Triangular-tetrahedral duoprism

Triangular-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTratet
Coxeter diagramx3o x3o3o ()
Tapertopic notation1211
Elements
Tera3 tetrahedral prisms, 4 triangular duoprisms
Cells3 tetrahedra, 6+12 triangular prisms
Faces4+12 triangles, 18 squares
Edges12+18
Vertices12
Vertex figureTriangular scalene, edge lengths 1 (base triangle and top edge) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {102}}{12}}\approx 0.84163}$
Hypervolume${\displaystyle {\frac {\sqrt {6}}{48}}\approx 0.051031}$
Diteral anglesTepe–trip–triddip: 90°
triddip–trip–triddip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
Tepe–tet–tepe: 60°
HeightsTet atop tepe: ${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Trig atop triddip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Trip atop perp trip: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces7
Level of complexity10
Related polytopes
ArmyTratet
RegimentTratet
DualTriangular-tetrahedral duotegum
ConjugateNone
Abstract & topological properties
Flag count1440
Euler characteristic2
OrientableYes
Properties
SymmetryA3×A2, order 144
Flag orbits10
ConvexYes
NatureTame

The triangular-tetrahedral duoprism or tratet is a convex uniform duoprism that consists of 3 tetrahedral prisms and 4 triangular duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular duoprisms. It is a duoprism based on a triangle and a tetrahedron, and is thus also a convex segmentoteron, as a tetrahedron atop tetrahedral prism.

## Vertex coordinates

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

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

## Representations

A triangular-tetrahedral duoprism has the following Coxeter diagrams:

• x3o x3o3o (full symmetry)
• xx3oo ox3oo&#x (A2×A2 symmetry, triangle atop triangular duoprism)
• ox xx3oo3oo&#x (A3×A1 symmetry, tetrahedron atop tetrahedral prism)
• ox xo xx3oo&#x (A2×A1×A1 symmetry, triangular prism atop orthogonal triangular prism)
• ooo3ooo3xxx&#x (A3 symmetry, tetrahedra considered different)
• oox ooo3xxx&#x (A2×A1 symmetry, tetrahedra have mirror symmetry only)
• oooo3xxxx&#x (A2 symmetry, tetrahedra have no symmetry)
• xxoo xoox3oooo&#xr (A2×A1 symmetry)