Triakis tetrahedron

Triakis tetrahedron
(Click here for rotating model)
Type	Catalan solid
Coxeter diagram
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, A3, [3,3], *332
Rotation group	T, [3,3]+, 332
Dihedral angle	129° 31' 16" $\arccos(-\frac{7}{11})$
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.

Related polyhedra

The triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

Dimensional family of truncated polyhedra and tilings: 3.2n.2n

Symmetry *n32 [n,3]	Spherical				Euclidean	Hyperbolic...
	*232 [2,3] D3h	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Truncated figures	3.4.4	3.6.6	3.8.8	3.10.10	3.12.12	3.14.14	3.16.16	3.∞.∞
Coxeter Schläfli	t0,1{2,3}	t0,1{3,3}	t0,1{4,3}	t0,1{5,3}	t0,1{6,3}	t0,1{7,3}	t0,1{8,3}	t0,1{∞,3}
Uniform dual figures
Triakis figures	V3.4.4	V3.6.6	V3.8.8	V3.10.10	V3.12.12	V3.14.14	V3.16.16	V3.∞.∞
Coxeter

Family of uniform tetrahedral polyhedra

Symmetry: [3,3], (*332)							[3,3]+, (332)
{3,3}	t0,1{3,3}	t1{3,3}	t1,2{3,3}	t2{3,3}	t0,2{3,3}	t0,1,2{3,3}	s{3,3}
Duals to uniform polyhedra
V3.3.3	V3.6.6	V3.3.3.3	V3.6.6	V3.3.3	V3.4.3.4	V4.6.6	V3.3.3.3.3

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