Triakis tetrahedron

In geometry, a triakis tetrahedron (or tristetrahedron, or kistetrahedron) is a solid constructed by attaching four triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron. 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. This interpretation is also expressed in the name, triakis, which is used for Kleetopes of polyhedra with triangular faces.
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 . Each dihedral angle between triangular faces is . 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. The triakis tetrahedron has the Rupert property.
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.