👉 At its core, the mathematics of fractals, particularly the Mandelbrot set, reveals an astonishing level of complexity and self-similarity. By iterating a simple quadratic equation (z_n+1 = z_n^2 + c), where z and c are complex numbers, mathematicians uncover intricate patterns that repeat at every scale. This process generates shapes with infinite detail and boundary structures that defy traditional Euclidean geometry, showcasing how simple rules can produce complex, organic forms. Fractals also connect deeply with chaos theory and dynamical systems, illustrating how small changes in initial conditions can lead to vastly different outcomes—a concept vividly captured by the "butterfly effect." This interplay of simplicity and complexity, self-similarity across scales, and ties into broader areas like topology and complex analysis, making fractals a profound testament to math's ability to model the universe's inherent intricacy.
