Mathematical Visualization of Fractal-Based Batik Designs

Abstract

In this manuscript, we use a rational-type mapping of complex polynomial to generate and visualize Julia and Mandelbrot fractals. We use the generalized viscosity approximation-type iterative scheme, extended with s-convexity improves convergence behavior and gives more control over the morphology of the resultant sets. Under this generalized scheme, a customized escape criterion is presented to precisely find orbit divergence. The structure, complexity, and symmetry of the produced fractals are investigated in relation to changing iterative parameters. We show a broad spectrum of fractal behaviors, from symmetric floral-like patterns to fragmented, fractal geometries, by means of extensive numerical simulations and graphical analysis performed in MATLAB R2024a. Moreover, some Julia and Mandelbrot sets are created to generate Batik-inspired designs, so highlighting the visual harmony and design possibilities of fractal geometry. This work not only broadens the theoretical underpinnings of fixed-point-based fractal generation but also links mathematical computation with useful applications in art and design.