Augmented truncated tetrahedron

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..

Augmented truncated tetrahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Autut
Coxeter diagram	xxoo3oxux&#xt
Elements
Faces	1+1+3+3 triangles, 3 squares, 3 hexagons
Edges	3+3+3+3+3+6+6
Vertices	3+3+3+6
Vertex figures	3 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	$\frac{11\sqrt2}{4} ≈ 3.88909$
Dihedral angles	3–4 join: $\arccos\left(-\frac{5\sqrt3}{9}\right) ≈ 164.20683°$
	3–6 join: $\arccos\left(-\frac79\right) ≈ 141.05756°$
	3–4 cupolaic: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$
	3–6 tut: $\arccos\left(-\frac13\right) ≈ 109.47122°$
	6–6: $\arccos\left(\frac13\right) ≈ 70.52878°$
Central density	1
Related polytopes
Army	Autut
Regiment	Autut
Dual	Rhombirhombistellated triakis tetrahedron
Conjugate	Augmented truncated tetrahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Nature	Tame

Vertex coordinates

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

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

Plus all permutations of:

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

External links

• Klitzing, Richard. "autut".

• Quickfur. "The Augmented Truncated Tetrahedron".

• Wikipedia Contributors. "Augmented truncated tetrahedron".
• McCooey, David. "Augmented Truncated Tetrahedron"