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Triakis tetrahedron

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Triakis tetrahedron

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

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 1
  • Variations 2
  • Stellations 3
  • Related polyhedra 4
  • See also 5
  • References 6
  • External links 7

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.


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.

*n32 symmetry mutation of truncated tilings: 3.2n.2n
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
[12i,3] [9i,3] [6i,3]
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
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
  • (Section 3-9)  
  • (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 [2] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron )

External links

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