## Julia Sets

This page has animated gif files illustrating the changing shapes of Julia sets as the parameter c changes. For some examples of 'static' Julia sets, see the Julia page.

Julia sets are parameterized by a complex number c = (a,b). For some values of c the Julia set is one piece ('connected'), while for other values of c the Julia set is dust ('totally disconnected'). The Mandelbrot set classifies the complex numbers into two groups; those c's where the Julia set is connected, and those c's where the Julia set is disconnected (every complex number is in exactly one of the two groups). As c varies the shape of the Julia set varies. The animated gif files here show you several scenarios of Julia sets as c changes. See if you can tell when the Julia set breaks apart inot dust (if it does).

These animated gif files were produced with MAPLE6. You may download the program here , or view the code (in plain text) here .

A Julia set breaking apart into dust. Here, c goes from (0,0) to (0,1.6) in 26 steps. This is a path up through the main body of the Mandelbrot set past the Misiurewicz point (0,1). For c=(0,0) the Julia set is a circle (of radius 1). For c=(0,1.6) the Julia set is dust. At c=(0,1) the Julia set is a 'line'. Each Julia set drawn here contains 32,768 points.

Here, c goes from (-1,-0.8) to (-1,0.8) in 26 steps. This is a path up through the 'neck' of the Mandelbrot set. Note that the point where he 'head' joins the 'body' of the Mandelbrot set is a parabolic point; the associated Julia set is just about to break apart. Each Julia set drawn here contains 32,768 points.

Here, c goes from (1,0.05) to (-2.4,0.05) in 29 steps. This is a path from right to left through the centre of the Mandelbrot set. Each Julia set drawn here contains 2,048 points.

More animated Julia sets.