The Mandelbrot fractal is the subset of the complex plane such that, for every complex number p in the set, the sequence { z₀, z₁, z₂, ... } does not diverge, where z₀ = 0+0i and z_(k+1) = (z_k)² + p.

A Julia set is a parameterized subset of the complex plane, such that for every complex number p in the set, the sequence { z₀, z₁, z₂, ... } does not diverge, where z₀ = p and z_(k+1) = z_k² + q for some complex parameter q. As we change q, we change the shape of the Julia set. It turns out that the most interesting looking Julia sets are the ones where q is close to the boundary of the Mandelbrot set !

Below are two Java applets generously donated by Victor Wu. The left one draws the Mandelbrot fractal. Clicking on on the left applet will update the right applet with the complex coordinates of the selected point. Then, hitting the "Draw" button on the right applet will draw the Julia set corresponding to the selected q value.

Observe that if you select a q slightly inside the Mandelbrot set (i.e. inside the green set), the Julia set will be connected. However, if you select a q slightly outside the Mandelbrot set, the resulting Julia set will be disconnected (like islands, or dust).

Question: which q values result in Julia sets with rotational symmetry vs mirror symmetry ?

Source code