By simple geometric arguments, we can define, in a given area around a circle, six other circles, with the same radius. In other words, density gradients of matter will be generated among the hexagons in the boundary regions. Finally, given an expanding universe, several dodecahedrons, remnants of a primordial gravitational phase transition, could yield a sort of honeycomb structure. However, we can obtain relevant results just taking into account balls with the same radius, that is, by considering bubble nucleation at the same density and expanding at a constant speed v . The positions of the objects are fixed in a random way by using a random number generator.
Quasars are point-like objects.
Furthermore, it is supposed that every quasar emits light in an isotropic way. Between the refraction channels, we have to distinguish between filamentary and planar structures. We produce randomly distributed filamentary structures with a dimension of about 100 Mpc along the major axis and a shape like a cylinder with 5 points, where the longitudinal axis of the cylinder changes its direction. A typical simulated channel is shown in Fig. 4 . In particular, the channels of the model are built by five cylinders with a length of Mpc and radius Kpc.