Julia and Mandelbrot Set

The Mandelbrot set shown to the right determines another set of fractals, so called Julia sets. Each point c in the Mandelbrot set determines a Julia set J(c). Picking and dragging with left mouse in the Mandelbrot image on the right will generate a new Julia set J(c) shown to the left. Mathematically we can say that the Mandelbrot set is the parameter space which describes the entirety of Julia sets.

For a given parameter c the Julia set J(c) is defined as the subset of points z of the complex plane for which the iteration

Iterate[z -> z^2+c]

does not diverge to infinity. Computationally, the algorithm checks for each pixel z if the sequence z,Iterate(z),Iterate(Iterate((z)),... stays within a certain region of the complex plane after a fixed number of iterations. The number of performed iterations can be adjusted using the slider "Number of iterations". The slider "Block Size" allows to work with a coarser pixel grid to speed up computations at the cost of resolution.

When the parameter c lies inside the black region of the Mandelbrot set then the corresponding Julia set has a non-empty black interior.

Some possible interactions:

• Pick inside the Mandelbrot image and see the Julia set being recomputed. The components of the picked complex parameter c is shown.
• Pick inside the Julia image and see the iterates of the picked point under the Julia map.
• Select a rectangular region in display to zoom into image: keep pressing the key 'm' while dragging a rectangle.
• Press the RESET button to initialize the applet, and to zoom out.
• The HUE slider changes the base value of the rainbow colors.

Note, this is applet is mainly a tutorial for studying image handling in JavaView rather than a full-fledged application to study fractals.

View source code:
PaJuliaSet.java of applet
PjJuliaSet.java of project
PjJuliaSet_IP.java of info panel.