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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 <math> \mathrm{T}_\mathrm{d} </math>. Each dihedral angle between triangular faces is <math> \arccos(-7/11) \approx 129.52^\circ</math>.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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>