# Hexagonal-tetrahedral duoprism

Hexagonal-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHatet
Coxeter diagramx6o x3o3o ()
Elements
Tera6 tetrahedral prisms, 4 triangular-hexagonal duoprisms
Cells6 tetrahedra, 24 triangular prisms, 6 hexagonal prisms
Faces24 triangles, 36 squares, 4 hexagons
Edges24+36
Vertices24
Vertex figureTriangular scalene, edge lengths 1 (base triangle), 3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {22}}{4}}\approx 1.17260}$
Hypervolume${\displaystyle {\frac {\sqrt {6}}{8}}\approx 0.30619}$
Diteral anglesTepe–tet–tepe: 120°
Tepe–trip–thiddip: 90°
Thiddip–hip–thiddip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
HeightsHig atop thiddip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Hip atop perp hip: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces10
Level of complexity10
Related polytopes
ArmyHatet
RegimentHatet
DualHexagonal-tetrahedral duotegum
ConjugateNone
Abstract & topological properties
Flag count2880
Euler characteristic2
OrientableYes
Properties
SymmetryA3×G2, order 288
ConvexYes
NatureTame

The hexagonal-tetrahedral duoprism or hatet is a convex uniform duoprism that consists of 6 tetrahedral prisms and 4 triangular-hexagonal duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular-hexagonal duoprisms.

## Vertex coordinates

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

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

## Representations

A hexagonal-tetrahedral duoprism has the following Coxeter diagrams:

• x6o x3o3o (full symmetry)
• x3x x3o3o (hexagons as ditrigons)
• ox3oo xx6oo&#x (hexagon atop triangular-hexagonal duoprism)
• ox3oo xx3xx&#x
• ox xo xx6oo&#x (hexagonal prism atop orthogonal hexagonal prism)
• ox xo xx3xx&#x
• oox xxx3xxx&#x
• xxxx3xxxx&#x