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 ri, the ones of the medium polyhedron Ri and the ones of the outer polyhedron Ri satisfy the relation Ri = ρ ri and Ri = R ri 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 m1, the masses of the medium one are equal to m2, and the masses of the outer one are equal to m3. We prove that if the ratios of the masses m2 / m1 and m3 / m1 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 m2 / m1 and m3 / m1, 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. © 2008 Elsevier B.V. All rights reserved.
- 3 n-body problem
- Nested regular polyhedra
- Spatial central configurations