MATLAB Code Implementation of Self-Similar Fractals

Fractal geometry is ubiquitous in nature and physics, appearing everywhere from coastlines to galactic distributions, with self-similarity being its core characteristic. MATLAB serves as an ideal numerical computation tool for implementing these recursive graphical structures due to its strong matrix manipulation capabilities. Fundamental Principles The essence of self-similar fractals lies in generating complex structures through infinite iterations of simple rules. Common implementation methods include: Iterated Function Systems (IFS): Generate fractals through repeated combinations of affine transformations (scaling/rotation/translation) using transformation matrices L-systems: Create tree-like fractals based on string rewriting rules through recursive string processing Escape-time algorithm: Used for complex-domain fractals like Mandelbrot sets with iterative divergence checks Implementation Key Points Physics students' implementations typically involve these critical steps: Define basic geometric units (e.g., line segments, triangles) as initial shapes Design recursive termination conditions (e.g., reaching preset iteration depth) Optimize coordinate transformations using MATLAB's matrix operations Enhance visual differentiation between iteration levels through colormap customization Optimization Recommendations Replace loops with vectorized operations to improve performance using arrayfun or vectorized approaches Add interactive controls (uicontrol) for real-time parameter adjustment Consider 3D fractal extensions (e.g., Menger sponge) using 3D plotting functions Output quantitative metrics like fractal dimension calculations Such implementations provide insights into understanding physical phenomena like phase transition criticality and diffusion-limited aggregation. Readers can experiment with transformation parameters to observe morphological evolution, experiencing the essence of nonlinear science where "simple rules generate complex behavior" through parameter modulation in the code.