Triakis Tetrahedron
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Triakis Tetrahedron
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 that of the longer edges[2]. If the triakis tetrahedron has shorter edge length 1, it has area and volume .

Cartesian coordinates

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

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

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 projection
Centered by Edge normal Face normal Face/vertex 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] [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

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.

See also


  1. ^ Conway, Symmetries of things, p.284
  2. ^
  • 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, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (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

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.



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