Looking at the structure of our universe, you can find it to be very self-similar. It is composed of gigantic superclusters, which are in turn composed of clusters. Every cluster is composed of galaxies, which are in turn composed of star systems such as the solar system, which are further composed of planets with moons revolving around them. Truly, every detail of the universe shows the same clustering patterns. The cluster fractals, such as the Cantor Square below are indeed useful in modeling the universe:

Cluster fractals are formed by repeatedly cutting out pieces of a polygon. The fractal above is obviously not a good model, and making it more random helps a lot. The fractal dimension of such fractals can be found very easily using the similarity method. In the Cantor Square, for example there are 4 smaller squares, the sides of each of which are 1/3 of the entire picture. The fractal dimension will thus be log 4 / log 3 = 1.26. This is remarkably close to the fractal dimension of the universe according to one of the experiments, where it was found to be about 1.23. The fact that this is a fraction is yet another proof of the universe’s fractal geometry.