Triakis tetrahedron

Triakis tetrahedron
Type	Catalan solid, Kleetope, Non-ideal
Faces	12
Edges	18
Vertices	8
Symmetry group	tetrahedral symmetry ${\displaystyle \mathrm {T} _{\mathrm {d} }}$
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 ${\displaystyle \mathrm {T} _{\mathrm {d} }}$. Each dihedral angle between triangular faces is ${\displaystyle \arccos(-7/11)\approx 129.52^{\circ }}$.^[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]

• Truncated triakis tetrahedron

• Weisstein, Eric W., "Triakis tetrahedron" ("Catalan solid") at MathWorld.