HYPERCOMPLEX FRACTAL VISUALIZATION: AN EXPANSIVE EXPLORATION OF JULIA AND MANDELBROT SETS THROUGH ADVANCED ITERATION SCHEMES

Abstract

Introduction: The complex dynamics and geometry of multi-dimensional hypercomplex fractals, particularly quaternions (4D), pose computational challenges but promise new symmetry revelations. This paper integrates advanced non-standard iteration schemes extracted from fixed-point theory. Methodology: This study examines the application of the Garodia-Uddin method (quaternion fractal), Picard-Thakur iteration with s-convection, and Viscosity Approximation. Rendering was executed through an escape-time algorithm, prototyping high-resolution matrix-based analytics via R. Results: Algorithm modification significantly lowered the Mean Escape Time (AET). The distortion of non-linear control parameters (viscosity, convexity, rotation) directly controls the Fractal Dimension, allowing the synthesis of layered organic "Biomorphs" shapes and dense and stable 4D filament structures. Prototyping experiments with the R syntax also resulted in high-resolution aesthetic parametric curve projections. Discussion: The implications of this finding go beyond pure mathematics. State-of-the-art hyper-dimensional fractal representations are now relevant across disciplines, applied in holographic visualization of Artificial Neural Network (AI) architecture, macroeconomic turbulence simulations, and the development of high-precision generative art designs.