The Mandelbrot set
The Mandelbrot set, discovered in 1980 by Benoit Mandelbrot, is probably the most famous fractal. Like Julia sets, it is generated by a very simple formula, but it is incredibly complex.
The Mandelbrot set is loosely self-similar: parts of the original fractal appear again when zooming in, but often deformed and with different ornaments. This is what makes it so rewarding to zoom into this fractal: you never know what you will see next.
This is illustrated by the following short zoom, starting at the very left of the Mandelbrot set shown above. As you zoom in, you see copies of the original Mandelbrot set, but with different surroundings.
Another interesting aspect of the Mandelbrot set is that it is actually a map of all Julia sets. Each point corresponds to a Julia set. Points inside the Mandelbrot set (here shown as black) are connected Julia sets; points outside the Mandelbrot set tend to give more disorganized Julia sets.
With the switch feature in Ultra Fractal, you can easily pick a point of a Mandelbrot fractal to see the corresponding Julia set. This is the best way to discover interesting Julia sets.
Next: Fractals today