Triakis tetrahedron

Triakis tetrahedron
(Click here for rotating model)
Type	Catalan solid
Face type	isosceles triangle
Faces	12
Edges	18
Vertices	8
Vertices by type	4{3}+4{6}
Face configuration	V3.6.6
Symmetry group	Td, [3,3], *332
Dihedral angle	129.521196° = arccos(-7/11)
Properties	convex, face-transitive
Truncated tetrahedron (dual polyhedron)	Net

In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.

It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron. This interpretation is expressed in the name.

If the triakis tetrahedron has shorter edge lengths 1, it has area $\tfrac{5}{3} \scriptstyle{\sqrt{11}}$ and volume $\tfrac{25}{36} \scriptstyle{\sqrt{2}}$.

Variations

A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell.

Stellations

This chiral figure is one of thirteen stellations allowed by Miller's rules.

See also

• Truncated triakis tetrahedron

References

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208  (The thirteen semiregular convex polyhedra and their duals, Page 14, Triakistetrahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron )

External links

• Eric W. Weisstein, Triakis tetrahedron (Catalan solid) at MathWorld.