Mathematical Modeling and Recursive Algorithms for Constructing Complex Fractal Patterns

In this paper, we present mathematical geometric models and recursive algorithms to generate and design complex patterns using fractal structures. By applying analytical, iterative methods, iterative function systems (IFS), and L-systems to create geometric models of complicated fractals, we developed fractal construction models, visualization tools, and fractal measurement approaches. We introduced a novel recursive fractal modeling (RFM) method designed to generate intricate fractal patterns with enhanced control over symmetry, scaling, and self-similarity. The RFM method builds upon traditional fractal generation techniques but introduces adaptive recursion and symmetry-preserving transformations to produce fractals with applications in domains such as medical imaging, textile design, and digital art. Our approach differs from existing methods like Barnsley's IFS and Jacquin's fractal coding by offering faster convergence, higher precision, and increased flexibility in pattern customization. We used the RFM method to create a mathematical model of fractal objects that allowed for the viewing of polygonal, Koch curves, Cayley trees, Serpin curves, Cantor set, star shapes, circulars, intersecting circles, and tree-shaped fractals. Using the proposed models, the fractal dimensions of these shapes were found, which made it possible to create complex fractal patterns using a wide variety of complicated geometric shapes. Moreover, we created a software tool that automates the visualization of fractal structures. This tool may be used for a variety of applications, including the ornamentation of building items, interior and exterior design, and pattern construction in the textile industry.