...From the figure 2 we see that the potential Y tends to a constant at the infinite. In the classical Eddington solution the potential owns a logarithmic growth. The figure 3 shows the association of a cluster of matter, located in the region s , sourrounded by a smooth hollow located in the region s*.

...In both regions matter attracts matter. But the negative sign, from the field equation and the Poisson equation, makes the matter and the "antipodal matter" to repel each other. This helps the confinement of the cluster. For a given thermal velocity the necessary quantity of matter to balance the pressure force is smaller. The smooth halo acts like a corset.

...A field equation provides a macroscopic description of the universe. It does not take account of the corpuscular nature of matter. The model implies that particles and antipodal-particles live in very distant, antipodal portions of space. In fact their natures are identical. The physical meaning of the field equation is the following : the particules and antipodal-particles interact by gravitational effect, but not by electromagnetic effect. We assume that the antipodal particles, clusters, rings, are not observable with a telescope, or a radiotelescope. The observation of antipodal structures should require some sort of gravitational telescope.

...From equation (22) clusters can be located in the antipodal region. Then, associated large halos, sourrounding wide rarefied regions, should exist in the observable universe too. In fact they do, for it corresponds, in our mind, to the observed large scale structure of the universe : the galaxies seem to be arranged around large rarefied bubbles. According to our model, large clouds of antipodal matter should exist in the corresponding associated antipodal regions.

...The universe was assumed to have a S3 x R1 topology. The reader has probabily some difficulties to understand this strange three-dimensional geometry. In fact the sphere S3 is simply shaped as the double cover of a projective space P3. In such arrangement each point s of the sphere is associated to its antipode A(s). The situation is similar for a sphere S2 covering a projective space P2, which can be represented in our space R3 as the well known Boy surface.

On the figure 10 we have figured the equator of a sphere and its location on the Boy Surface.

Figure 10 : The vicinity of the equator of a 2-sphere and its location on a Boy surface.

...The figure 11 shows how the equator of a S2 sphere can be glued on itself along a three half-turns Möbius belt. Locally the surface can be assimilated to a bundled manifold whose bundle owns two values + 1 and -1.

...In a 3-sphere S3, if one follows a geodesic, the antipodal point is at the half-way. If the 3-sphere is immersed in a four-dimensional space it is possible to make any point and its antipode to coincidate. These couples of points are associated through the antipodal diffential involutive maping A, but not identified.

...As shown on the figure 12 we can proceed continously from a "gruyère" structure to a cluster structure. This peculiar feature was illustrated before, through 2d numerical simulations. When a region of space is put "in front" of the antipodal region, as suggested in the figure 12, the clusters nest in the holes.

 
...This effect could act at the level of the galactic structure, as suggested in the figure 14, each galaxy nesting in a "hole" of the conjugated antipodal region.