Triakis tetrahedron

Triakis tetrahedron

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″
arccos(−7/11)
Properties convex, face-transitive

Truncated tetrahedron
(dual polyhedron)

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.

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 tetaroid:

Orthogonal projections

Orthogonal projection
Centered by Edge normal Face normal Face/vertex Edge
Triakis
tetrahedron
(Dual)
Truncated
tetrahedron
Projective
symmetry
[1] [1] [3] [4]

Variations

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.

Stellations

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.