Metallic mean fractal systems and their tilings

Abstract

Fractals and quasiperiodic structures share self-similarity as a structural property. Motivated by the link between Fibonacci fractals and quasicrystals which are scaled by the golden mean ratio $\frac{1+\sqrt{5}}{2}$, we introduce and characterize a family of metallic-mean ratio fractals. We calculate the spatial properties of this generalized family, including their boundaries, which are also fractal. We then demonstrate how these fractals can be related to aperiodic tilings, and show how we can decorate them to produce new, fractal tilings.