Central configurations of nested rotated regular tetrahedra

In this paper we prove that there are only two different classes of central configurations with convenient masses located at the vertices of two nested regular tetrahedra: either when one of the tetrahedra is a homothecy of the other one, or when one of the tetrahedra is a homothecy followed by a rotation of Euler angles α=γ=0α=γ=0 and β=πβ=π of the other one.We also analyze the central configurations with convenient masses located at the vertices of three nested regular tetrahedra when one them is a homothecy of the other one, and the third one is a homothecy followed by a rotation of Euler angles α=γ=0α=γ=0 and β=πβ=π of the other two.In all of these cases we have assumed that the masses on each tetrahedron are equal but masses on different tetrahedra could be different.