# 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"
$\arccos(-\frac{7}{11})$
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.

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

Orthogonal projection
Centered by Edge normal Face normal Face/vertex Edge
Truncated
tetrahedron
Triakis
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.

## 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.

Dimensional family of truncated spherical polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]...

*∞32
[∞,3]

[12i,3] [9i,3] [6i,3] [3i,3]
Figures
Schläfli 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} t{3i,3}
Coxeter
Dual
figures
Triakis
figures

V3.4.4

V3.6.6

V3.8.8

V3.10.10

V3.12.12

V3.14.14

V3.16.16

V3.∞.∞
Coxeter
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 V3.3.3.3.3