Fractal-inspired textile designs using the viscosity approximation method

In this work, we explore a novel computational method for the production and dynamic depiction of fractal structures, specifically Julia sets, Mandelbrot sets, and biomorphs, to generate innovative and artistically captivating textile designs customized for batik painting. This study uses an extended rational-type complex-valued function, which is clearly stated as Q(z) = sin zm + A zn + B. The constants A, B ∈ C and the parameters m, n ∈ N \ {1} give fractal patterns a lot of freedom. We use the viscosity approximation method to make these fractals work interesting. This is a complicated and quick iterative method that works well for finding fixed points in nonlinear mappings. The invention of new escape criteria, specifically designed to improve the clarity and complexity of the resulting fractal sets, is at the heart of the method. Using dynamic visualization techniques on the resulting graphs allows us to highlight how things change over time and how they respond to parameter variations. This helps us understand the complex geometric details that are always there in fractal shapes. Along with these visuals are detailed numerical analyses that show how changes in the structure of Julia sets, Mandelbrot sets, and biomorphs are related to changes in parameters. Our study also demonstrates that these fractal shapes can be applied in real-world contexts, such as batik textile patterns. Fractals generated by biomorphs create many interesting and unusual visual patterns and demonstrate that the proposed method not only makes batik designs more visually appealing but also offers textile artists new ways to express their creativity. Moreover, a comparative study with generative AI-based methods demonstrates that our deterministic, mathematically based method is more visually accurate, the parametric interpretation is far more interpretable, and the computational costs are much lower, thus rendering it a highly viable and scalable tool to the textile designing business.