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## Julia and Mandelbrot Set - A Tutorial on Pixel Images

> 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:
> PaFractalImage.java of applet
> PjFractalImage.java of project
> PjFractalImage_IP.java of info panel.