  Fractals, a fascinating mathematical concept made popular in the eighties,
remained for decades a beautiful scientific curiosity mainly. With the
tremendous advances in nanofabrication techniques, such as nanolithography, it
has become possible to design self-similar materials with fine structures down
to nanometer scale. Here, we investigate the effects of self similarity on
quantum electronic transport in graphene Sierpinski carpets. We find that a gap
opens up in the electron spectrum in the middle of which lies a flat band of
zeros energy modes. Although these states have a zero velocity, a supermetallic
phase is found at the neutrality point. For Fermi energy located in the
valence/conduction band and in the presence of a small inelastic scattering the
system stays metallic and the transport is found strongly anisotropic.
