Central configurations of three nested regular polyhedra for the spatial 3n-body problem

Three regular polyhedra are called nested if they have the same number of vertices nn, the same center and the positions of the vertices of the inner polyhedron ri, the ones of the medium polyhedron Ri and the ones of the outer polyhedron RiRi satisfy the relation Ri=ρri and Ri=Rri for some scale factors R>ρ>1R>ρ>1 and for all i=1,…,ni=1,…,n. We consider 3n3n masses located at the vertices of three nested regular polyhedra. We assume that the masses of the inner polyhedron are equal to m1m1, the masses of the medium one are equal to m2m2, and the masses of the outer one are equal to m3m3. We prove that if the ratios of the masses m2/m1m2/m1 and m3/m1m3/m1 and the scale factors ρρ and RR satisfy two convenient relations, then this configuration is central for the 3n3n-body problem. Moreover there is some numerical evidence that, first, fixed two values of the ratios m2/m1m2/m1 and m3/m1m3/m1, the 3n3n-body problem has a unique central configuration of this type; and second that the number of nested regular polyhedra with the same number of vertices forming a central configuration for convenient masses and sizes is arbitrary.