Introduction to Julia sets and the Mandelbrot set
These pages provide an interactive exploration of complex dynamics. The left image shows the Mandelbrot set M in the parameter plane. The current parameter c determines the Julia set Kc in the dynamic plane, which is shown in the right image; see the explanations below. The web application is implemented in JavaScript. For offline use you may save the html-files together with juliette.js and mandelwork.js. The application is controlled with the buttons beneath or by pressing the corresponding keys, and by clicking into the windows. Most browsers do not allow to save an image. Usage:- The current point z is mapped to z2+c with the key f. Set c or z by clicking into the corresponding window, enter its coordinates after hitting c or z, or move it with the arrow keys. To specify the plane you are working in, use z or c as well, or left-click once into the inactive plane.
- Zoom in at the current point with the key i. When the “fractal” shape turns into a smooth band, increase the number of iterations with the key n. Zoom out with o and restart with h.
- To draw an external ray in the current plane, hit e and enter the angle as a fraction.
- 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 asymptotic self-similarity by rescaling the parameter plane. Choose various levels with the keys 0, 1, 2, 3, m.
To do: implement x e p t 0 1 2 3 m.
Mathematical explanations under construction ...