# Hexagonal-truncated tetrahedral duoprism

Hexagonal-truncated tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHatut
Coxeter diagramx6o x3x3o ()
Elements
Tera4 triangular-hexagonal duoprisms, 4 hexagonal duoprisms, 6 truncated tetrahedral prisms
Cells24 triangular prisms, 6+12+24 hexagonal prisms, 6 truncated tetrahedra
Faces24 triangles, 36+72 squares, 12+24 hexagons
Edges36+72+72
Vertices72
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 3, 3 (base triangle), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {38}}{4}}\approx 1.54110}$
Hypervolume${\displaystyle {\frac {23{\sqrt {6}}}{8}}\approx 7.04228}$
Diteral anglesTuttip–tut–tuttip: 120°
Thiddip-hip-hiddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Thiddip–trip–tuttip: 90°
Hiddip-hip-tuttip: 90°
Hiddip–hip–hiddip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density1
Number of external pieces14
Level of complexity30
Related polytopes
ArmyHatut
RegimentHatut
DualHexagonal-triakis tetrahedral duotegum
ConjugateHexagonal-truncated tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×G2, order 288
ConvexYes
NatureTame

The hexagonal-truncated tetrahedral duoprism or hatut is a convex uniform duoprism that consists of 6 truncated tetrahedral prisms, 4 hexagonal duoprisms, and 4 triangular-hexagonal duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-hexagonal duoprism, and 2 hexagonal duoprisms.

## Vertex coordinates

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

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

## Representations

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

• x6o x3x3o () (full symmetry)
• x3x x3x3o () (hexagons as ditrigons)