# The Mandelbrot/Julia Set Applet

This applet allows you to

• Compute and view various regions in the Mandelbrot set.
• Select a c-value from the Mandelbrot set and view the corresponding filled Julia set.
• View the orbit of any point, not just 0, in and around the filled Julia set.

When you first open the applet, you will see the control panel in the upper left window. You will also be computing the Mandelbrot set in the upper right window and a particular Julia set (known as "Douady's rabbit") in the lower right. The c-value (-0.125 + 0.75i) corresponding to this Julia set is displayed as a white dot in the Mandelbrot set as well as recorded numerically in the control panel. These computations will take some time, as will each successive magnification/computation. A timer tells you how much longer you have to wait. For people with old computers, either have patience or buy a new computer! On my little Macintosh G3, it takes less than a few seconds to compute the initial images.

The Mandelbrot set and filled Julia sets are displayed in black in these images. Since much of these sets consist of filaments that are too narrow to be seen at typical resolutions, we use a graduated color scheme so that your eye "zeroes in" on the black regions. Colors proceed from red to orange, yellow, green, blue, and then violet in order of increasing number of iterations necessary for escape.

Magnifying regions in the Mandelbrot set. To magnify portions of the Mandelbrot set, click and drag from the center of the region you wish to enlarge, just as in previous applets. Click the Zoom to Box button to compute this enlargement. Click Zoom Out to see a larger portion of the Mandelbrot set at any stage. Zoom out increases the linear dimensions of the image by a factor of 2 (so the area increases by 4). Click Reset to return to the original Mandelbrot set image. Change the Iteration count at any time to increase the resolution of your image.

Computing filled Julia sets. To compute a filled Julia set, click on a point in the Mandelbrot set. You see the new c-value displayed as a white dot in the Mandelbrot set. Its coordinates are displayed in the control panel. The new filled Julia set is computed and then displayed in the lower right window. With it you see displayed the "fate" of the orbit of 0 displayed as a blinking series of points (unless the orbit of 0 tends to infinity).

As you move the mouse over the filled Julia set image, you see displayed in the lower left window the corresponding orbit of the seed located where the mouse is pointing. The orbit is displayed as a sequence of points in the plane. Colors indicate higher and lower iterations. For example, the first few points on the orbit are colored red; the next few yellow, followed by green, blue, and violet. So, for example, if you only see red points in this window, you can be pretty sure that your selected orbit has escaped to infinity.