# Augmented truncated tetrahedron

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Augmented truncated tetrahedron
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymAutut
Coxeter diagramxxoo3oxux&#xt
Elements
Faces1+1+3+3 triangles, 3 squares, 3 hexagons
Edges3+3+3+3+3+6+6
Vertices3+3+3+6
Vertex figures3 rectangle, edge length 1 and 2
6 irregular tetragons, edge length 1, 2, 1, 3
3+3 isosceles triangles, edge lengths 1, 3, 3
Measures (edge length 1)
Volume${\displaystyle \frac{11\sqrt2}{4} ≈ 3.88909}$
Dihedral angles3–4 join: ${\displaystyle \arccos\left(-\frac{5\sqrt3}{9}\right) ≈ 164.20683°}$
3–6 join: ${\displaystyle \arccos\left(-\frac79\right) ≈ 141.05756°}$
3–4 cupolaic: ${\displaystyle \arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°}$
3–6 tut: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122°}$
6–6: ${\displaystyle \arccos\left(\frac13\right) ≈ 70.52878°}$
Central density1
Related polytopes
ArmyAutut
RegimentAutut
DualRhombirhombistellated triakis tetrahedron
ConjugateAugmented truncated tetrahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexYes
NatureTame

The augmented truncated tetrahedron is one of the 92 Johnson solids (J65). It consists of 1+1+3+3 triangles, 3 squares, and 3 hexagons. It can be constructed by attaching a triangular cupola to one of the hexagonal faces of the truncated tetrahedron..

## Vertex coordinates

An augmented truncated tetrahedron of edge length 1 has vertices given by all even sign changes of:

• ${\displaystyle \left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right),}$

Plus all permutations of:

• ${\displaystyle \left(-\frac{11\sqrt2}{12},\,-\frac{5\sqrt2}{12},\,-\frac{5\sqrt2}{12}\right).}$