# Triangular duoprism

Triangular duoprism
Rank4
TypeUniform
Notation
Bowers style acronymTriddip
Coxeter diagramx3o x3o ()
Tapertopic notation1111
Elements
Cells6 triangular prisms
Faces6 triangles, 9 squares
Edges18
Vertices9
Vertex figureTetragonal disphenoid, edge lengths 1 (base) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Inradius${\displaystyle {\frac {\sqrt {3}}{6}}\approx 0.28868}$
Hypervolume${\displaystyle {\frac {3}{16}}=0.1875}$
Dichoral anglesTrip–4–trip: 90°
Trip–3–trip: 60°
Height${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Central density1
Number of external pieces6
Level of complexity3
Related polytopes
ArmyTriddip
RegimentTriddip
DualTriangular duotegum
ConjugateNone
Abstract & topological properties
Flag count216
Euler characteristic0
OrientableYes
Properties
SymmetryA2≀S2, order 72
Flag orbits3
ConvexYes
NatureTame

The triangular duoprism or triddip, also known as the triangular-triangular duoprism, the 3 duoprism or the 3-3 duoprism, is a noble uniform duoprism that consists of 6 triangular prisms, with 4 meeting at each vertex. It is the simplest possible duoprism (excluding the degenerate dichora) and is also the 6-2 gyrochoron and digonal trigyrotegum. It is the first in an infinite family of isogonal triangular dihedral swirlchora and also the first in an infinite family of isochoric triangular hosohedral swirlchora.

It is also a convex segmentochoron (designated K-4.10 on Richard Klitzing's list), as it is a triangle atop a triangular prism.

## Vertex coordinates

Coordinates for the vertices of a triangular duoprism of edge length 1, centered at the origin, are given by:

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

Simpler coordinates can be given in 6-dimensions, as all permutations of

• ${\displaystyle \left({\dfrac {\sqrt {2}}{2}},0,0,{\dfrac {\sqrt {2}}{2}},0,0\right)}$,

where the sum of the first 3 coordinates is equal to the sum of the last 3.

## Representations

A triangular duoprism has the following Coxeter diagrams:

• x3o x3o () (full symmetry)
• ox xx3oo&#x (axial, triangle atop triangular prism)
• xxoo xoox&#xr (axial, vertex first)
• xxx3ooo&#x (A2 axial)

## Related polychora

### Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: