Explore the relationship between Mandelbrot and Julia sets. Click on the Mandelbrot set to select Julia parameters, or manually enter values to see different Julia sets.
Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.
The Mandelbrot set is a set of complex numbers that, when iterated through a specific mathematical function, do not diverge to infinity. It's defined by the function:
zn+1 = zn2 + c
Where z starts at 0, and c is the complex number being tested. If the sequence remains bounded after many iterations, c is in the Mandelbrot set.
Julia sets are closely related to the Mandelbrot set. For each complex number c, there is a corresponding Julia set. While the Mandelbrot set tells us which values of c produce connected Julia sets, each Julia set shows the behavior of the iteration for a fixed c with different starting z values.
The relationship between the Mandelbrot set and Julia sets is fascinating: each point in the Mandelbrot set corresponds to a connected Julia set, while points outside the Mandelbrot set correspond to disconnected Julia sets (dust-like fractals).