Triakis tetrahedron

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Triakis tetrahedron
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation kT
Face type V3.6.6
DU02 facets.png

isosceles triangle
Faces 12
Edges 18
Vertices 8
Vertices by type 4{3}+4{6}
Symmetry group Td, A3, [3,3], (*332)
Rotation group T, [3,3]+, (332)
Dihedral angle 129° 31' 16"
Properties convex, face-transitive
Truncated tetrahedron.png
Truncated tetrahedron
(dual polyhedron)
Triakis tetrahedron 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[edit]

Orthogonal projection
Centered by Edge normal Face normal Face/vertex Edge
Tetrahedron t01 ae.png Tetrahedron t01 af36.png 3-simplex t01 A2.svg 3-simplex t01.svg
Dual tetrahedron t01 ae.png Dual tetrahedron t01 af36.png Dual tetrahedron t01 A2.png Dual tetrahedron t01.png
[1] [1] [3] [4]


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

5-cell net.png


Stellation of triakis tetrahedron.png

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

Related polyhedra[edit]

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.

*n32 symmetry mutation of truncated tilings: 3.2n.2n
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
[12i,3] [9i,3] [6i,3]
Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.png H2 tiling 237-3.png H2 tiling 238-3.png H2 tiling 23i-3.png H2 tiling 23j12-3.png H2 tiling 23j9-3.png H2 tiling 23j6-3.png
Config. 3.4.4 3.6.6 3.8.8 3.10.10 3.12.12 3.14.14 3.16.16 3.∞.∞ 3.24i.24i 3.18i.18i 3.12i.12i
Spherical trigonal bipyramid.png Spherical triakis tetrahedron.png Spherical triakis octahedron.png Spherical triakis icosahedron.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Ord7 triakis triang til.png Ord8 triakis triang til.png Ord-infin triakis triang til.png
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t12.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png Uniform polyhedron-33-s012.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
{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
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Dodecahedron.svg
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.

See also[edit]


  1. ^ Conway, Symmetries of things, p.284

External links[edit]