The unique atomic positions in quasicrystals lead to peculiar self-similarity and fractal-like structural morphology. Accordingly, many of the material properties are supposed to manifest exceptional characteristics. In this Rapid Communication, we explain through numerical simulations the fundamental and peculiar aspects of quasicrystals wearless friction manifested in a 30° twisted bilayer graphene system. In particular, the sliding force exhibits a fractal structure with distinct area correlations due to the natural mixture between both periodic and aperiodic lateral modulations. In addition, zero power scaling of the sliding force with respect to the contact area is demonstrated for a geometric sequence of dodecagonal elements.