THE MANDELBROT SET

AN EXAMPLE OF COMPLEX NUMBERS IN ACTION

The Mandelbrot Set is a picture in the complex plane based on the iteration of the function  with  and c is a constant.

In the complex plane, the horizontal axis is the real part of the number

and the vertical axis the imaginary part.

By iteration we mean each new value of x is used to calculate

the next value of x.

The starting value of  is called the critical orbit.

If you test different values of c, there are three possible results :

1.       the iteration settles down to a fixed value

2.  the iteration cycles between a number of values ( called the period )

3.  the iteration continually increases and increases, until it becomes infinite

Points on the complex plane which do either 1 or 2 are the points ( in black on the two diagrams here ) which form the Mandelbrot Set.

Do the iteration for the following values of c, indicating into which category it fits. If periodic, state the period.

 c result 0 -1 i -0.5i -2 -0.5+0.25i