Triakis tetrahedron

Triakis tetrahedron
(Click here for rotating model)
Type	Catalan solid
Coxeter diagram
Conway notation	kT
Face type	V3.6.6 isosceles triangle
Faces	12
Edges	18
Vertices	8
Vertices by type	4{3}+4{6}
Symmetry group	T_d, 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 (or kistetrahedron^[1]) 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. It is very similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. 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 $\tfrac{25}{36} \scriptstyle{\sqrt{2}}$ .

Orthogonal projections

Orthogonal projection
Centered by	Edge normal	Face normal	Face/vertex	Edge
Truncated tetrahedron
Triakis tetrahedron
Projective symmetry	[1]	[1]	[3]	[4]

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

Spherical triakis tetrahedron

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 spherical polyhedra and tilings: *3.2n.2n*
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
Symmetry *n32 [n,3]	*232 [2,3] D_3h	*332 [3,3] T_d	*432 [4,3] O_h	*532 [5,3] I_h	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]	[3i,3]
Figures
Schläfli	t{2,3}	t{3,3}	t{4,3}	t{5,3}	t{6,3}	t{7,3}	t{8,3}	t{∞,3}	t{12i,3}	t{9i,3}	t{6i,3}	t{3i,3}
Coxeter
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}	t{3,3}	r{3,3}	t{3,3}	{3,3}	rr{3,3}	tr{3,3}	sr{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

References

↑ Conway, Symmetries of things, p.284

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, MR 730208 (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 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron )