|f(x) = x2 for all real and complex x values|
|x-value||Real x2||Complex x2|
|a||b||f(a)||f(a + bi)|
|2||2||4||(2 + 2i)(2 + 2i) = 8i|
|3||-2||9||(3 - 2i)(3 - 2i) = 5 + 12i|
|6||3||36||(6 + 3i)(6 + 3i) = 27 + 36i|
|-2||1||4||(-2 + i)(-2 + i) = 3 - 4i|
The Mandelbrot Set
The Mandelbrot Set is similar to the Cantor Set, in that both sets consist of the points that did not escape when iterated through a function. However, the Mandelbrot Set is obtained using the complex function f(z) = z2 + c, where z is the complex independent variable and c is the complex parameter. The actual points in the Mandelbrot Set represent the values that did not escape to infinity (diverge). As you will see in the Java applet, the resulting pattern is quite beautiful.
The Julia Set
The Julia Set uses the function f(z) = z2 + c, while z is the complex independent variable and c is the complex parameter that is constant throughout the set. An interesting fact about the Julia Set is that if you pick a constant parameter that is inside the Mandelbrot Set, the Julia Set will be connected. Conversely, if you pick a constant parameter that is outside of the Mandelbrot Set, the Julia Set will be an unconnected 'dust' of points.