Metallic mean fractal systems and their tilings

TL;DR

This paper introduces a family of fractals based on metallic-mean ratios, explores their properties, and demonstrates their connection to aperiodic tilings, expanding the understanding of quasiperiodic structures.

Contribution

It generalizes Fibonacci fractals to metallic-mean ratios and relates them to new fractal tilings, broadening the scope of quasiperiodic systems.

Findings

Calculated boundaries of metallic-mean fractals.

Established link between fractals and aperiodic tilings.

Produced new fractal tilings through decoration methods.

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.