The Mandelbrot fractal is a fascinating mathematical set that exhibits intricate and infinitely complex patterns. It was discovered and popularized by the mathematician Benoît Mandelbrot in the 1970s
The Mandelbrot set is defined in the complex plane, which consists of numbers with real and imaginary parts. Each point in the complex plane represents a different complex number. For a given complex number \(c\), the behavior of the sequence \(z_{n+1}=z_n^{2}+c\) is studied.
The interesting and visually stunning aspect of the Mandelbrot set lies in the points that are outside the set. The iterations can lead to chaotic behavior, generating intricate patterns with intricate detail, no matter how closely you zoom in. These patterns exhibit self-similarity, meaning that smaller and smaller portions of the set resemble the overall shape of the set.
When graphically visualizing the Mandelbrot set, points inside the set are usually colored black, while the colors of the points outside the set are determined based on the number of iterations required to reach an escape condition or the magnitude of divergence.