Triakis tetrahedron

File:Triakis tetrahedron.stl

3D model of a triakis tetrahedron

In geometry, a triakis tetrahedron (or tristetrahedronTemplate:R, or kistetrahedronTemplate:R) is a solid constructed by attaching four triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron.Template:R This replaces the equilateral triangular faces of the regular tetrahedron with three isosceles triangles at each face, so there are twelve in total; eight vertices and eighteen edges form them.Template:R This interpretation is also expressed in the name, triakis, which is used for Kleetopes of polyhedra with triangular faces.Template:R

The triakis tetrahedron is a Catalan solid, the dual polyhedron of a truncated tetrahedron, an Archimedean solid with four hexagonal and four triangular faces, constructed by cutting off the vertices of a regular tetrahedron; it shares the same symmetry of full tetrahedral ${\mathrm{T}}_{\mathrm{d}}$. Each dihedral angle between triangular faces is $\arccos (-7/11)\approx 129.52^{\circ }$.Template:R Unlike its dual, the truncated tetrahedron is not vertex-transitive, but rather face-transitive, meaning its solid appearance is unchanged by any transformation like reflecting and rotation between two triangular faces.Template:R The triakis tetrahedron has the Rupert property.Template:R

A triakis tetrahedron is different from an augmented tetrahedron as latter is obtained by augmenting the four faces of a tetrahedron with four regular tetrahedra (instead of nonuniform triangular pyramids) resulting in an equilateral polyhedron which is a concave deltahedron (whose all faces are congruent equilateral triangles). The convex hull of an augmented tetrahedron is a triakis tetrahedron.

See also

• Truncated triakis tetrahedron

External links

• Template:Mathworld2