| Triakis tetrahedron | |
|---|---|
 | |
| Type | Catalan solid, Kleetope, Non-ideal |
| Faces | 12 |
| Edges | 18 |
| Vertices | 8 |
| Symmetry group | tetrahedral symmetry |
| Dihedral angle (degrees) | 129.52° |
| Dual polyhedron | truncated tetrahedron |
| Properties | convex, face-transitive, Rupert property |
| Net | |
 | |
In geometry, a triakis tetrahedron (or tristetrahedron [1] , or kistetrahedron [2] ) is a Catalan solid, constructed by attaching four triangular pyramids of a tetrahedron.
The triakis tetrahedron is constructed by attaching four triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron. [3] 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. [4] This interpretation is also expressed in the name, triakis, which is used for Kleetopes of polyhedra with triangular faces. [2]
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 . [4] 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. [5] The triakis tetrahedron can pass through a copy of itself of the same size. [6]
The triakis tetrahedron is the stacked polyhedron that is a non-ideal. Combinatorially, it has independent set of exactly half the vertices but is not bipartite, so neither can be realized as an ideal polyhedron. [7] [8]
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. [9]
