Emergence of Surface Fractals in Cellular Automata

Abstract

Self‐similar (fractal) structures are present at every scale ranging from galaxies down to aggregates of atoms to elementary particles. For surface fractals, the self‐similarity is inherited from the superposition of non‐overlapping mass fractals. Despite long‐standing theoretical investigations, no generic framework exists yet to describe the nature and generation of surface fractal systems. Here, cellular automata (CA) are identified as a generic mathematical system and, by exploring the associated small‐angle scattering (neutrons, X‐rays, light) intensity curves, the emergence of surface fractals is reported. The observed decay of scattering intensity manifesting as a non‐coherent sum of scattering amplitudes of objects composing the surface fractal is in agreement with theoretical predictions and is a manifestation of the power‐law distribution of object's sizes. The non‐coherent sum of intensities of a randomly distributed system of objects provides an approximation to the surface fractal scattering intensity, which is expected from the interplay of distances between objects and their size. The finding on the emergence of surface fractals in CA will enrich the understanding of their structural properties while the approximation of independent objects can provide a route toward testing randomness generated by CA.