Fractal Explorer

Mathematical Principles & Theorems

Investigates holomorphic fractal geometry. The Mandelbrot Set \(M = \{c \in \mathbb{C} : z_{n+1} = z_n^2 + c \text{ remains bounded as } n \to \infty, z_0 = 0\}\). Julia Sets \(J_c\) fix parameter \(c\) and iterate over initial points \(z_0\). Escape-time algorithm tests whether \(|z_n| > 2\). Self-similarity exhibits non-integer Hausdorff dimension \(D = \lim_{\epsilon \to 0}\frac{\log N(\epsilon)}{\log(1/\epsilon)}\) (e.g., Sierpiński triangle \(D = \frac{\log 3}{\log 2} \approx 1.585\)).

Operating Instructions

• Select fractal mode: Mandelbrot Set, Julia Set, or Sierpiński Gasket.
• Set iteration depth \(N\) via the numeric value box to balance resolution and render speed.
• Click and drag on the canvas to pan; use scroll wheel or zoom buttons to explore infinite self-similarity.
• Observe boundary filament structures, cardioids, and hyperbolic components.