Generalized CR-Iteration Scheme with Application in Textile Designing

Authors

Khaleel Ahmad Department of Mathematics, University of Management and Technology, Lahore, Pakistan
Umar Ishtiaq Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan https://orcid.org/0000-0002-5228-1073
Mohammad Akram Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia https://orcid.org/0000-0003-1416-5351
Ioan-Lucian Popa Department of Computing, Mathematics and Electronics, "1 Decembrie 1918" University of Alba Iulia, 510009 Alba Iulia, Romania

DOI:

https://doi.org/10.37256/cm.6620258340

Keywords:

complex function, escape criterion, Mandelbrot set fractal, iteration scheme, batik design

Abstract

In this manuscript, we develop a new iteration method that is generalized CR-iteration method. We generate the Julia and Mandelbrot set fractals for a complex function where c ∈ \{0}, p ∈ , p ≥ 2, and t ∈ , t ≥ 1 by using the generalized CR-iteration process. We discuss the escape criterion by using the generalized CR-iteration process for Julia and Mandelbrot set fractals generation. We generate the quadratic Julia set fractals, cubic Julia set fractals, quadratic Mandelbrot set fractals, and cubic Mandelbrot set fractals with MATLAB R2024a by using the generalized CR-iteration process with variation of the parameters α, γ, t, η, and c. Subtle variations in iteration parameters affect the structure, density, and symmetry of the fractals, as seen by a succession of graphical images. We generate the quadratic Julia and Mandelbrot set by using CR-iteration method and generalized CR-iteration method. We use Julia and Mandelbrot set patterns and repeat it to generate some beautiful batik designs. At the end, we compare this research with some existing literature.