Mandlebrot sets

Written by John Garnes on February 24, 2024

The Mandelbrot set is a visually captivating two-dimensional set defined in the complex plane. Its definition involves complex numbers ‘c’ and a function ‘f_c(z) = z^2 + c’. The set is formed by iterating this function starting at ‘z=0’ and checking if the sequence remains bounded in absolute value, without diverging to infinity.

Initially defined by Robert W. Brooks and Peter Matelski in 1978, the Mandelbrot set gained attention in 1980 when Benoit Mandelbrot created high-quality visualizations at IBM’s Thomas J. Watson Research Center. The images of the Mandelbrot set display a complex and infinitely detailed boundary, revealing finer recursive patterns upon zooming in. Mathematically, this boundary is a fractal curve.

To generate Mandelbrot set images, complex numbers are sampled, and for each point ‘c’, the sequence ‘f_c(0), f_c(f_c(0)), …’ is checked for divergence. The visual representation involves coloring pixels based on when the sequence crosses a chosen threshold. The Mandelbrot set’s popularity extends beyond mathematics due to its aesthetic appeal and as an example of complex structures emerging from simple rules. It stands as a renowned instance of mathematical visualization and beauty.