  Scale symmetry is a fundamental symmetry of physics that seems however not to
be fully realized in the universe. Here, we focus on the astronomical scales
ruled by gravity, where scale symmetry holds and gives rise to a truly scale
invariant distribution of matter, namely it gives rise to a fractal geometry. A
suitable explanation of the features of the fractal cosmic mass distribution is
provided by the nonlinear Poisson--Boltzmann--Emden equation. An alternative
interpretation of this equation is connected with theories of quantum gravity.
We study the fractal solutions of the equation and connect them with the
statistical theory of random multiplicative cascades, which originated in the
theory of fluid turbulence. The type of multifractal mass distributions so
obtained agrees with results from the analysis of cosmological simulations and
of observations of the galaxy distribution.
