# Hexagonal-octahedral duoprism

Hexagonal-octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHoct
Coxeter diagramx6o o4o3x ()
Elements
Tera6 octahedral prisms, 8 triangular-hexagonal duoprisms
Cells48 triangular prisms, 6 octahedra, 12 hexagonal prisms
Faces48 triangles, 72 squares, 6 hexagons
Edges36+72
Vertices36
Vertex figureSquare scalene, edge lengths 1 (base square), 3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{2}}\approx 1.22474}$
Hypervolume${\displaystyle {\frac {\sqrt {6}}{2}}\approx 1.22474}$
Diteral anglesOpe–oct–ope: 120°
Thiddip–hip–thiddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Thiddip–trip–ope: 90°
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces14
Level of complexity10
Related polytopes
ArmyHoct
RegimentHoct
DualHexagonal-cubic duotegum
ConjugateNone
Abstract & topological properties
Flag count5760
Euler characteristic2
OrientableYes
Properties
SymmetryB3×G2, order 576
ConvexYes
NatureTame

The hexagonal-octahedral duoprism or hoct is a convex uniform duoprism that consists of 6 octahedral prisms and 8 triangular-hexagonal duoprisms. Each vertex joins 2 octahedral prisms and 4 triangular-hexagonal duoprisms.

## Vertex coordinates

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

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

## Representations

A hexagonal-octahedral duoprism has the following Coxeter diagrams:

• x6o o4o3x (full symmetry)
• x3x o4o3x (hexagons as ditrigons)
• x6o o3x3o (octahedra as tetratetrahedra)
• x3x o3x3o (hexagons as ditrigons and octahedra as tetratetrahedra)
• xo3ox xx6oo&#x (triangular-hexagonal duoprism atop triangle-gyrated triangular-hexagonal duoprism)
• xo3ox xx3xx&#x