Structural properties of fractals with positive Lebesgue measures

TL;DR

This paper develops a theoretical model for fat fractals with positive Lebesgue measure, enabling the extraction of detailed structural information from small-angle scattering data for well-known fractals like Vicsek and Menger sponge.

Contribution

It introduces a new analytical model for fat fractals and demonstrates how to derive structural parameters from SAS data, advancing understanding of fractals with positive measure.

Findings

Analytical structure factor for fat fractals derived

Method to determine fractal dimension at each structural level

Extraction of particle count and inter-particle distances from SAS data

Abstract

Small-angle scattering (SAS) data which show a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). In this work we present a theoretical model for fat fractals and show how one can extract structural information about the underlying fractal using SAS method, for the well known fractals existing in the literature: Vicsek and Menger sponge. We calculate analytically the fractal structure factor and study its properties in momentum space. The models allow us to obtain the fractal dimension at each structural level inside the fractal, the number of particles inside the fractal and about the most common distances between the center of mass of the particles.