| Triakis tetrahedron | |
|---|---|
 | |
| Type | Catalan solid, Kleetope |
| 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 solid constructed by attaching four triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron. [3] This replaces the triangular faces with three, 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 the Kleetopes of polyhedra with triangular faces. [2]
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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 has no vertex-transitive, but rather face-transitive, meaning its solid appearance is unchanged by any transformation like reflecting and rotation between two triangular faces. [5] Whenever a triakis tetrahedron has a hole, it is possible for a polyhedron to exist with the same or larger size passing through it. [6]
