# Triangular-octahedral duoprism

Triangular-octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTroct
Coxeter diagramx3o o4o3x ()
Elements
Tera8 triangular duoprisms, 3 octahedral prisms
Cells12+24 triangular prisms, 3 octahedra
Faces6+24 triangles, 36 squares
Edges18+36
Vertices18
Vertex figureSquare scalene, edge lengths 1 (base square and top edge) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {30}}{6}}\approx 0.91287}$
Hypervolume${\displaystyle {\frac {\sqrt {6}}{12}}\approx 0.20412}$
Diteral anglesTriddip–trip–triddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Triddip–trip–ope: 90°
Ope–oct–ope: 60°
HeightsOct atop ope: ${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Triddip atop gyro triddip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces11
Level of complexity10
Related polytopes
ArmyTroct
RegimentTroct
DualTriangular-cubic duotegum
ConjugateNone
Abstract & topological properties
Flag count2880
Euler characteristic2
OrientableYes
Properties
SymmetryB3×A2, order 288
Flag orbits10
ConvexYes
NatureTame

The triangular-octahedral duoprism or troct is a convex uniform duoprism that consists of 3 octahedral prisms and 8 triangular duoprisms. Each vertex joins 2 octahedral prisms and 3 triangular duoprisms. It is a duoprism based on a triangle and an octahedron, which also makes it a convex segmentoteron.

## Vertex coordinates

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

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

## Representations

A triangular-octahedral duoprism has the following Coxeter diagrams:

• x3o o4o3x (full symmetry)
• x3o o3x3o (A3×A2 symmetry, octahedron as tetratetrahedron)
• ox oo4oo3xx&#x (BC3×A1 symmetry, octahedron atop octahedral prism)
• ox oo3ox3oo&#x (A3×A1 symmetry)
• xo3ox xx3oo&#x (A2×A2 axial, triangular duoprism atop gyro triangular duoprism, triangular-triangular antiprismatic duoprism)
• ooo4ooo3xxx&#x (BC3 symmery, octahedra seen differently)
• ooo3xxx3ooo&#x (A3 symmetry only)