Determination of Fractal Dimension of Partial Geometric Fractals

Abstract

Purpose. Research aims on determining the dependence of the fractal dimension of partial geometric fractals on the degree of distortion of the fractal structure. Such analysis is important for understanding behavior of complex objects in cases of partial violation of self-similarity, which frequently occurs in real-world tasks of computer graphics, modeling of natural processes, biomedical imaging, and materials analysis. Methodology. Research employed fractal construction technique based on L-systems with intermediate multi-symbol representation and defect seeding by randomly removing a certain percentage of symbols from generated sequence. Analysis was performed using box-counting method: fractal structure was repeatedly overlaid with square grids of variable side lengths, after which number of squares required to cover figure was determined. Fractal dimension was calculated from logarithmic relationship between number of squares and side length of squares (slope method on logarithmic plot). Graphical representations of fractals were saved in high-resolution BMP format to minimize rasterization errors. To improve reliability, each structure was analyzed through tenfold repeated grid coverage with varying parameters. Findings. Proposed methodology was tested on well-known geometric fractals: Koch snowflake, Koch square island, Sierpinski curve, dragon curve, and Lévy curve without distortions. Obtained fractal dimension values corresponded to theoretical predictions. For these fractals, relationship between fractal dimension and degree of damage (percentage of removed elements) was determined. In particular, increasing number of defects generally led to decrease in fractal dimension, indicating loss of self-similarity and simplification of geometric structure. These results allow quantitative description of fractal structure degradation and identification of critical limits of self-similarity stability. Originality. Research introduces new objects of study, namely partial geometric fractals, and provides new data on self-similarity levels of partial fractals with varying degrees of damage. Practical value. Results can be applied in computer graphics, image processing, 3D modeling, as well as in materials science and other fields where complexity and structure of materials are important. Study expands current approaches to fractal structure analysis and proposes new methods for their investigation and application.