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

Abstract

Three regular polyhedra are called nested if they have the same number of vertices n , the same center and the positions of the vertices of the inner polyhedron r i , the ones of the medium polyhedron R i and the ones of the outer polyhedron R i satisfy the relation R i = ρ r i and R i = R r i for some scale factors R > ρ > 1 and for all i = 1 , … , n . We consider 3 n masses located at the vertices of three nested regular polyhedra. We assume that the masses of the inner polyhedron are equal to m 1 , the masses of the medium one are equal to m 2 , and the masses of the outer one are equal to m 3 . We prove that if the ratios of the masses m 2 / m 1 and m 3 / m 1 and the scale factors ρ and R satisfy two convenient relations, then this configuration is central for the 3 n -body problem. Moreover there is some numerical evidence that, first, fixed two values of the ratios m 2 / m 1 and m 3 / m 1 , the 3 n -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.