# Hexagonal-truncated octahedral duoprism

Hexagonal-truncated octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHatoe
Coxeter diagramx6o o4x3x
Elements
Tera6 square-hexagonal duoprisms, 8 hexagonal duoprisms, 6 truncated octahedral prisms
Cells36 cubes, 12+24+48 hexagonal prisms, 6 truncated octahedra
Faces36+72+144 squares, 24+48 hexagons
Edges72+144+144
Vertices144
Vertex figureDigonal disphenoidal pyramid, edge lengths 2, 3, 3 (base triangle), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {14}}{2}}\approx 1.87083}$
Hypervolume${\displaystyle 12{\sqrt {6}}\approx 29.39388}$
Diteral anglesShiddip–hip–hiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Tope–toe–tope: 120°
Hiddip–hip–hiddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Shiddip–cube–tope: 90°
Hiddip–hip–tope: 90°
Central density1
Number of external pieces20
Level of complexity30
Related polytopes
ArmyHatoe
RegimentHatoe
DualHexagonal-tetrakis hexahedral duotegum
ConjugateHexagonal-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×G2, order 576
ConvexYes
NatureTame

The hexagonal-truncated octahedral duoprism or hatoe is a convex uniform duoprism that consists of 6 truncated octahedral prisms, 8 hexagonal duoprisms, and 6 square-hexagonal duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 square-hexagonal duoprism, and 2 hexagonal duoprisms.

This polyteron can be alternated into a triangular-pyritohedral icosahedral duoantiprism, although it cannot be made uniform.

## Vertex coordinates

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

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

## Representations

A hexagonal-truncated octahedral duoprism has the following Coxeter diagrams:

• x6o o4x3x (full symmetry)
• x6o x3x3x
• x3x o4x3x (hexagons as ditrigons)
• x3x x3x3x