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Triakis tetrahedron

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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 Td, A3, [3,3], (*332)
Rotation group T, [3,3]+, (332)
Dihedral angle 129°31′16″
Properties convex, face-transitive

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


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 22. The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals arccos(–7/18) ≈ 112.88538047616° and the acute ones equal arccos(5/6) ≈ 33.55730976192°.

Tetartoid symmetry


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

Orthogonal projections (graphs)
Centered by Short edge Face Vertex Long edge
[1] [3] [4]



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

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.

In modular origami, this is the result to connecting six Sonobe modules to form a triakis tetrahedron.



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

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]
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}
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)
{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.

See also



  1. ^ Conway, Symmetries of things, p.284
  2. ^ "Triakis Tetrahedron - Geometry Calculator".