The Mathematics of Fractals – The Mandelbrot Set

The Mandelbrot set is defined by iterating z → z² + c (z and c complex). Points c for which the orbit stays bounded form a fractal with infinite detail. Discovered by Benoit Mandelbrot in 1980, its boundary has Hausdorff dimension 2 (it’s area‑filling). The set is connected, but the coastline is infinitely long. The Mandelbrot set is the most famous fractal, containing copies of itself at different scales. It appears in the distribution of prime numbers, in the analysis of fluid dynamics, and in the artwork of countless computer graphics enthusiasts.