The Mandelbrot fractal is a very well known fractal with very beautiful pictures of its border. Beside these pictures, we can also visualize the actual methodology how the fractal is computed.
The Mandelbrot set is produced by the simple formula
z(n) = z(n-1)² + C, with
z(0) = 0 and
C set to the current point in the Cartesian coordinate system. If after
n iterations the value of
z(n) has not diverged too far from the origin of the coordinate system, i.e. the distance is larger than a predefined value
d, the point is part of the Mandelbrot set. If the point diverges, we can color the point depending on the iteration where the distance is larger than
d and therefore get the famous renderings of the border.
The following animation shows the movements of the points for the first 50 iterations. The black points remain part of the Mandelbrot fractal while the red points will diverge. The lines in the background show the trace of the points.
This image shows the same part of the Mandelbrot fractal with a higher resolution and 2000 iterations: