For background on Julia and Mandelbrot sets, see the introduction.

If you continue to click on the Mandelbrot set, the Julia set image will be replaced by other Julia sets. You can pretty easily get a pretty good idea how the Julia sets are related to the Mandelbrot set.

When 1/µ is displayed, the image becomes a teardrop shape. The original cardioid is transformed into the outside of the teardrop.

When 1/(µ+0.25) is displayed, the image appears parabolic. The original cardioid is transformed into the outside of the parabola.

When 1/lambda is displayed, the image appears as a circle, or, better stated, as a crescent, since one of the two primary circles in the lambda plane is converted to the outside of the circle, and the other is converted to the interior of a smaller circle inside that circle.

When 1/(lambda-1) is displayed, the image appears as an infinite strip. The two circles of the lambda plane are converted to the two sides outside the strip.

When 1/(µ-1.40115) is displayed, the cardioid is turned around and distorted a bit, but the real difference is that the sequence of circles attached to the right of the cardioid are expanded, each bigger then the previous. Their radii increase to infinity instead of decrease to 0.

You can magnify a piece of the Mandelbrot set, too. Just press the clickbox labelled "magnify the Mandelbrot set" before clicking on the Mandelbrot image. You'll also get a Julia set at the same time unless you press the clickbox labelled "get a Julia set".

You can also change the size of the image you get. To begin with the images are squares 300 pixels on a side. You can change that square to any size rectangle you like. The new shape will be sent on your next image request. That gives you another way to magnify an image. Please note that big images take noticeably longer to compute than small images.

You can add some detail to images by changing the "escape" conditions for the algorithm. The usual condition is that the iterate of the function eventually gets outside a sufficiently large circle. The mathematics then guarantees further iterates will go off to infinity. But if you prefer, you can change the circle into a square. The boundaries between levels will become scalloped instead of smooth. You can also "feather" the image, either with a square or a circular boundary. Feathering adds black so that the levels look something like they're colored with feathers.

September, 1994. © 1994.

David E. Joyce

Department of Mathematics and Computer Science

Clark University

Worcester, MA 01610

These pages are located at http://aleph0.clarku.edu/~djoyce/julia/julia.html