Stochastic fractal by deterministic algorithm: Introducing the Möbius fractal

Abstract

The study of the structural properties of non-compact colloidal associates, as well as linear and branched polymers, is an important task of modern physical chemistry, since the structure at the nanoscale determines a number of important macroscopic features. Such systems often have fractal properties, i.e. they exhibit scale invariance in a number of characteristics. Numerous algorithms for constructing deterministic and stochastic fractal objects have been proposed recently. The former are based on an exact repetition of the shape at different scales, while when using the latter, the scaling ratios are observed only “on average”. In this paper, we propose a new algorithm for constructing a fractal object, called the Mobius fractal, which is essentially on the verge between regular and non-regular fractals. According to the correlation analysis, the fractal dimension of such a system is close to 1.75. The prospects for the further use in describing the results of experimental methods of structural diagnostics of nanomaterials, including small-angle scattering, are outlined.