Triakis tetrahedron

The triakis tetrahedron is one of the 13 Catalan solids. It has 12 isosceles triangles as faces, with 4 order-6 and 4 order-3 vertices. It is the dual of the uniform truncated tetrahedron.

It can also be obtained as the convex hull of two dually-oriented tetrahedra, where one has edges exactly $$\frac53 ≈ 1.66667$$ times the length of those of the other. If the ratio of the edge lengths of the two tetrahedra is varied to be anything other than 1:1, a fully symmetric variant of the triakis tetrahedron is produced.

Each face of this polyhedron is an isosceles triangle with base side length $$\frac53 ≈ 1.66667$$ times those of the side edges. These triangles have apex angle $$\arccos\left(-\frac{7}{18}\right) ≈ 112.88538°$$ and base angles $$\arccos\left(\frac56\right) ≈ 33.55731°$$.

Vertex coordinates
A triakis tetrahedron with dual edge length 1 has vertex coordinates given by all even sign changes of: as well as all odd sign changes of:
 * $$\left(\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4}\right),$$
 * $$\left(\frac{9\sqrt2}{20},\,\frac{9\sqrt2}{20},\,\frac{9\sqrt2}{20}\right).$$