In the weak field approximation to the gravitational field equations, we study gravitational paramagnetism and diamagnetism, the gravitational Meissner effect and gravitational superconductivity.The spontaneous symmetry breaking corresponds to crossing from closed geodesics to open ones, and to the existence of a critical temperature in the frame of a gauge model at finite temperature. In this later case one can obtain expressions giving the dependence of several superconducting parameters on temperature.

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.

Our visual cosmic spectacle is populated with matter formations of sophisticated geometrical shapes. Many of these are flat, or at least tend to be flat. The geometrical description of these started, historically, with circles, going through conic sections and ending with spirals. The flat formations, of matter both static and in motion, helped creating the modern theory of gravitation. They decided the modern image of the world we live in. Are they connected? Yes! The geometry of this connection is presented, starting from its main physical reasons: we cannot see them but in projections on the canopy.

In a fractal spacetime, the absence of a gravitational Meissner effect is thought of as ordering space as a crystal, at both a microscopic and a macroscopic scale. A gravitational Meissner effect keeps a wormhole open and penetrable and, in the same context, a gravitational superconductor levitates in an external gravitomagnetic field, an external gravitomagnetic field induces quantised vortices in a gravitational superconductor and gravitational rotons in a superfluid, and the planetary systems are self-organised as superconducting structures.