Mann-generated Julia fractals for textile design: AET, color entropy, and chromatic variation

Fractals exhibit intricate self-similarity patterns that arise naturally in both mathematical structures and physical phenomena. In this study, we explore a rational mapping F(z)=zn+c−1zk+r, where n≥2, c,r∈ℂ,z∈ℂ∖{0}, and k>0. The generation and analysis of Julia sets are investigated using the Mann iterative techniques. Special emphasis is placed on how the polynomial degree, control parameters, and coefficients influence the geometric and dynamical behavior of the resulting fractal structures. Analytical and numerical approaches to average escape time and color entropy are employed to examine their symmetries and boundary characteristics. Numerical simulations reveal how parameter variations produce complex transformations in fractal morphology, highlighting the interplay between iteration dynamics and visual complexity. Color entropy is computed for a set of fractal texture images to quantify chromatic and structural complexity. The entropy results distinguish highly complex, high-variation patterns from smoother, uniform textures, demonstrating the effectiveness of entropy as a complementary metric in fractal-based fabric design. The findings enrich the theoretical understanding of complex dynamics and extend fixed-point-based fractal synthesis toward real-world applications in computational modeling, visualization, and fabric design.