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.

Triakis tetrahedron

(Click here for rotating model)
TypeCatalan solid
Coxeter diagram
Conway notationkT
Face typeV3.6.6
isosceles triangle
Vertices by type4{3}+4{6}
Symmetry groupTd, A3, [3,3], (*332)
Rotation groupT, [3,3]+, (332)
Dihedral angle129°31′16″
Propertiesconvex, face-transitive

Truncated tetrahedron
(dual polyhedron)


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.

Tetartoid symmetry

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

Example tetartoid variations

Orthogonal projections

Orthogonal projection
Centered by Edge normal Face normal Face/vertex Edge
[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.

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.


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

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.