# Triakis tetrahedron

Triakis tetrahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Bowers style acronymTikit
Coxeter diagramm3m3o ()
Conway notationkT
Elements
Faces12 isosceles triangles
Edges6+12
Vertices4+4
Vertex figure4 hexagons, 4 triangles
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos\left(-\frac{7}{11}\right) ≈ 129.52120^\circ}$
Central density1
Number of external pieces12
Level of complexity3
Related polytopes
ArmyTikit
RegimentTikit
DualTruncated tetrahedron
ConjugateNone
Abstract & topological properties
Flag count72
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA3, order 24
ConvexYes
NatureTame

The triakis tetrahedron, or tikit, is one of the 13 Catalan solids. It has 12 isosceles triangles as faces, with 4 order-6 and 4 order-3 vertices. It is the dual of the uniform truncated tetrahedron.

It can also be obtained as the convex hull of two dually-oriented tetrahedra, where one has edges exactly ${\displaystyle \frac53 ≈ 1.66667}$ times the length of those of the other. If the ratio of the edge lengths of the two tetrahedra is varied to be anything between 1:1 (producing the cube) and 1:3 (in which case the vertices of the small tetrahedron are the face centers of the larger), a fully symmetric variant of the triakis tetrahedron is produced.

Each face of this polyhedron is an isosceles triangle with base side length ${\displaystyle \frac53 ≈ 1.66667}$ times those of the side edges. These triangles have apex angle ${\displaystyle \arccos\left(-\frac{7}{18}\right) ≈ 112.88538°}$ and base angles ${\displaystyle \arccos\left(\frac56\right) ≈ 33.55731°}$.

## Vertex coordinates

A triakis tetrahedron with dual edge length 1 has vertex coordinates given by all even sign changes of:

• ${\displaystyle \left(\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4}\right),}$

as well as all odd sign changes of:

• ${\displaystyle \left(\frac{9\sqrt2}{20},\,\frac{9\sqrt2}{20},\,\frac{9\sqrt2}{20}\right).}$