Introduction to Julia sets and the Mandelbrot set
In the applet below, the left image shows the Mandelbrot set M in the parameter plane. The current parameter c determines the Julia set K_{c} in the dynamic plane, which is shown in the right window. Suggestions for their exploration will be given after introducing the Java applet Juliette (click the gray column to give it keyboard focus if necessary):
- The current point z is mapped with the key f. Set c or z by clicking into the corresponding window, enter its coordinates after hitting i, or move it with the arrow keys. The program is guessing, which plane you are working in. To specify the plane, open the context menu on either side with a right-click. Or left-click once into the inactive plane (the second click would shift the point).
- Zoom in at the current point with the key z. When the “fractal” shape turns into a smooth band, increase the number of iterations with the key n. Zoom out with v and restart with h.
- To draw an external ray in the current plane, hit e and enter the angle as a fraction. (See below.)
- Hit the key r to renormalize in the current plane. Specify a preperiod to mark embedded Julia sets.
- Find special parameters c or points z with the key x. The (pre-)period is remembered for the following scaling commands.
- Hit p to rescale the dynamic plane according to local similarity or asymptotic similarity.
- Hit t to illustrate the asymptotic self-similarity by rescaling the parameter plane. Choose various levels from the context menu, or with the keys a, 0, 1, 2, 3, m.
Download Juliette 1.0 of July 2010.
The source code is included in the archive under the
GNU General Public License.
It was compiled with Java 1.6, compatibility level 1.5. If it does not run,
please enable Java in your browser, or
download
the Java plug-in.