Triakis tetrahedron

As a Kleetope

The triakis tetrahedron is 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.

As a Catalan solid

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 \mathrm{T}_\mathrm{d} . Each dihedral angle between triangular faces is \arccos(-7/11) \approx 129.52^\circ. 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 can pass through a copy of itself of the same size, but it is an exceptionally tight squeeze: the largest known triakis tetrahedron that can pass through is only about 1.000004 times larger.

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.

Related polyhedron

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.

References

| editor1-last = Emmer | editor1-first = Michele | editor2-last = Abate | editor2-first = Marco | hdl-access = free

| editor-last = Bobenko | editor-first = Alexander I.

References

1. "Augmented Tetrahedron".