Embedded Julia sets within the Mandelbrot set
Get acquainted with the phenomenon of embedded Julia sets in an interactive session with the Java applet Juliette below. Click the gray column to give it keyboard focus (do not click the images). Use the following commands or the buttons below:
- Note that the current parameter c is indicated by the yellow cross in the left window. It is within a “tiny” Mandelbrot set of period 75, which is close to a small Mandelbrot set of period 4. The latter is barely visible in the first image.
- Hit the key z repeatedly to zoom in smoothly. When the small Mandelbrot set of period 4 becomes prominent, hit r to mark it in the color magenta. Enter “4” or “0,4” in the dialog box.
- Zoom in further into the decorations of the small Mandelbrot set by hitting z repeatedly. You will see a thicker part on the decoration, which resembles the full Julia set in the right window. To mark this embedded Julia set in the parameter plane, hit r and enter “75,4” in the dialog box.
- Zoom in further with the key z, until you see the tiny Mandelbrot set of period 75. Mark it with r by entering “75” or “0,75”.
- You may see analogous structures in the dynamic plane by hitting p at any time, or by zooming in there. To zoom or to mark them with r, you must switch between the planes by left-clicking once into the inactive window, or by invoking the context menu with a right-click on the appropriate side. Restart by hitting h or by zooming out with v. (Since h is resetting the number of iterations to 100, you should change it to 1000 with n.)
- Now change the parameter c by left-clicking into the left window. Choose any small Mandelbrot set of primitive type. Determine its period p with the key x. Zoom in and look for embedded Julia sets in the decorations, just outside of the small Mandelbrot set, which is marked magenta with r. Zoom in further to the tiny Mandelbrot set and determine its period n with x. Zoom out again with v and mark the embedded Julia set with r by entering “n,p” in the dialog box.
Download Juliette. See the
context menu for an overview of commands and keyboard shortcuts.
Or read this introduction.