The Quadratic Map

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How to use this applet

In this applet you see on the right the Mandelbrot set (the parameter plane for the complex function z2 + c where c is a complex parameter) and, on the left, the corresponding (filled) Julia set for the particular chosen c-value. The default parameter value is c = -1. The filled Julia set is displayed in black; the Julia set is the boundary of the black region. The colored points are the points whose orbits escape to infinity, with red points escaping fastest, followed by orange, yellow, green, blue, indigo, and violet. In the Mandelbrot set, the black points represent c-values in the Mandelbrot set, while colored points represent c-values for which the corresponding orbit of 0 escapes (with the same coloring scheme as above).

Don't know what I'm talking about??? Visit the Mandelbrot Set Explorer for a primer on the mathematics behind Mandelbrot and Julia sets.

You can use this applet to:

Here is how to do these things:

To magnify portions of the Mandelbrot or Julia set, simply select a rectangular region with your mouse in the appropriate window and click on compute and the selected region will be displayed. You might wish to increase the number in the Max. iterations box below each figure to get a better image. Alternatively, you can enter the lower left and upper right points of your desired window manually in the appropriate boxes. Then click compute to see the desired magnification.

To display a Julia set for a different c-value, simply click on the desired c-value in the Mandelbrot set window or enter the c-value manually in the window called Function parameters. Then click compute.

To see a particular orbit in or around the filled Julia set, click on the desired point. You can display the first ten points on the orbit by clicking the View orbit button, or you can see the orbit one-by-one by successively hitting the Iterate button.

To make a movie of how the Julia sets change, select the starting c-value in the Mandelbrot set and then click Add current E. Then select another c value and click again Add current E. You will see a straight line displayed in the Mandelbrot set. If you click the Animate button, you will then compute nine Julia sets drawn from along this line and then see the animation in the Juli set window. Click animate to replay this animation. To produce longer animations, simply choose more points in the Mandelbrot set and don't forget to click Add current E after each choice. Each additional choice adds eight more frames to your animation. And no, to answer your question: you cannot save this animation.

This applet was created by Yakov Shapiro for Professor Robert Devaney.