Identifying space with a superconducting cosmic dust, the structure of the ‘world crystal’ is built. Particles moving on open geodesics are assimilated with the ‘electron gas’ in a lattice, and those moving on closed geodesics with nodes of the same lattice. By considering that this kind of structure may be achieved by pinning on a background gravitomagnetic field, the properties of such fields are studied. From this analysis and computing the background energy we are lead to a Cantorian-fractal structure of space–time, which allows one to interpret the gravitational interaction in terms of a mechanism similar to the composite fermions mechanism.