Augmented truncated tetrahedron

Augmented truncated tetrahedron
(3D model) (OFF file)
Rank	3
Type	CRF
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	${\frac {11{\sqrt {2}}}{4}}\approx 3.88909$
Dihedral angles	3–4 join: $\arccos \left(-{\frac {5{\sqrt {3}}}{9}}\right)\approx 164.20683^{\circ }$
	3–6 join: $\arccos \left(-{\frac {7}{9}}\right)\approx 141.05756^{\circ }$
	3–4 cupolaic: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	3–6 tut: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	6–6: $\arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }$
Central density	1
Number of external pieces	14
Level of complexity	18
Related polytopes
Army	Autut
Regiment	Autut
Dual	Rhombirhombistellated triakis tetrahedron
Conjugate	Augmented truncated tetrahedron
Abstract & topological properties
Flag count	108
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×I, order 6
Convex	Yes
Nature	Tame

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:

• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)$,

plus all permutations of:

• $\left(-{\frac {11{\sqrt {2}}}{12}},\,-{\frac {5{\sqrt {2}}}{12}},\,-{\frac {5{\sqrt {2}}}{12}}\right)$.

External links

• Klitzing, Richard. "autut".
• Quickfur. "The Augmented Truncated Tetrahedron".

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