Exploring the Mandelbrot Set Mathematical Foundations and Visualisation

This report investigates the Mandelbrot set, a famous fractal structure arising from iterative complex quadratic mappings. The work explains the mathematical principles behind the set, methods of visualisation, and its connection to chaos theory and complex dynamics.Key topics covered:Complex numbers and iteration – How repeated application of the function <i>zₙ₊₁ = zₙ² + c</i> generates fractal boundaries.Escape time algorithm – Method for determining set membership and rendering the fractal.Boundary complexity – Exploration of self-similarity and infinite detail within the set’s edge.Mathematical significance – Links to stability in dynamical systems and the classification of Julia sets.Visual representation – Use of computational tools to generate high-resolution fractal imagery.The report combines mathematical explanation with computational implementation, making the abstract beauty of the Mandelbrot set accessible to both mathematically trained and general audiences.