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
3D model of a triakis tetrahedron

In geometry, a triakis tetrahedron (or kistetrahedron[1]) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron.

The triakis tetrahedron can be seen as a tetrahedron with a triangular pyramid 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.

The length of the shorter edges is 3/5 that of the longer edges.[2] If the triakis tetrahedron has shorter edge length 1, it has area 5/311 and volume 25/362.

Cartesian coordinates[edit]

Cartesian coordinates for the 8 vertices of a triakis tetrahedron centered at the origin, are the points (±5/3, ±5/3, ±5/3) with an even number of minus signs, along with the points (±1, ±1, ±1) with an odd number of minus signs:

  • (5/3, 5/3, 5/3), (5/3, -5/3, -5/3), (-5/3, 5/3, -5/3), (-5/3, -5/3, 5/3)
  • (-1, 1, 1), (1, -1, 1), (1, 1, -1), (-1, -1, -1)

The length of the shorter edges of this triakis tetrahedron equals . The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals and the acute ones equal .

Tetartoid symmetry[edit]

The triakis tetrahedron can be made as a degenerate limit of a tetartoid:

Example tetartoid variations
Tetartoid 0% (Regular Dodecahedron) Tetartoid 10% Tetartoid 20% Tetartoid 30%
Tetartoid 60% Tetartoid 80% Tetartoid 95% Tetartoid 100% (Triakis Tetrahedron)

Orthogonal projections[edit]

Orthogonal projections (graphs)
Centered by Short edge Face Vertex Long edge
Dual tetrahedron t01 ae.png Dual tetrahedron t01 af36.png Dual tetrahedron t01 A2.png Dual tetrahedron t01.png
Tetrahedron t01 ae.png Tetrahedron t01 af36.png 3-simplex t01 A2.svg 3-simplex t01.svg
[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

If the triangles are right-angled isosceles, the faces will be coplanar and form a cubic volume. This can be seen by adding the 6 edges of tetrahedron inside of a cube.

Rhombic disphenoid.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: t{n,3}
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.svg Truncated heptagonal tiling.svg H2-8-3-trunc-dual.svg H2 tiling 23i-3.png H2 tiling 23j12-3.png H2 tiling 23j9-3.png H2 tiling 23j6-3.png
Symbol 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}
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 Order-7 triakis triangular tiling.svg H2-8-3-kis-primal.svg 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.svg
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
  2. ^ "Triakis Tetrahedron - Geometry Calculator".

External links[edit]