Fractals, Chaos and the Mandelbrot Set Workshop - Pt. 1

The Mandelbrot Set - Where Does It Come From?

The Mandelbrot Set is an image that results from repeatedly feeding the output of a simple equation back into the equation to get a new result. After this is done a certain number of times, perhaps 100 or as many as 1000 or more, if the number that results is still small (between 0 and 2), the coordinate values of the original point is coloured black. If the number becomes large before 100 repetitions is reached, the original point is coloured some other colour, according to a predetermined colour scheme. If a simple black and white image is desired, points that result in large numbers are coloured white. The amazing shape that is called the Mandelbrot Set, then, is the group of points that have coordinate values that remain small after many repetitions through the simple equation.

The cartesian number plane is defined by the familiar x and y axes that we know from typical graphs. Any point on the cartesian number plane can be described with an x and y value, such as (3,7) or (-5,-9). The Mandelbrot Set results from manipulating numbers in a different number plane, called the complex number plane, because points in this plane are made up of complex numbers. In the complex number plane, instead of having an x and a y axis, we have an imaginary number and a real number axis. The vertical axis is for the imaginary numbers, and the horizontal axis is for the real numbers. A real number is a regular number, the kind we are all familiar with. Imaginary numbers are similar, but have a unique twist. The basis of imaginary numbers is the square root of -1. The square root of 9 is 3. The square root of 4 is 2, and the square root of 1 is 1. But what is the square root of -9? It can't be -3, because -3 x -3 is +9. Because the square root of -1 cannot be any regular, or "real" number, we make up an imaginary number to be the square root of -1. This number, which only exists in our imagination, is denoted by the letter i, much as we use the letter x to denote many real numbers. So now we can have an axis of imaginary numbers that would have values of 7i, or 10i, or -3i, or -8i.

The simple equation that is used to generate the Mandelbrot Set is this:
zn+1 = zn^2 + c

What this means is that the new value for z will become equal to the current value of z squared, plus c. In .the case of the Mandelbrot Set, z starts out as 0, and we pick any value in the complex number plane for c. For example, point A in the complex number plane shown has coordinates of about 0.4i on the vertical, or imaginary axis, and 0.2 on the horizontal, or real axis.

{Image here...}

If we choose this value for c, and z starts at zero, the first new value for z becomes equal to c, because z^2 is zero. The next new value for z becomes different, because now z has a value. As the equation is applied time and again, different values result for z. If the value starts to increase, it may increase quickly, and "explode" to infinity. (See the program, "Orbit and Dwell of Mandelbrot Set Points" on the InfoPak disk.) If this happens we say that the original point, c, that we started with is outside the Mandelbrot Set and we colour it white. (Or some other colour.) If the value for z never gets big, even after many repetitions, we say that the original point, c, is inside the Mandelbrot Set and we colour it black. After going through this process with hundreds or thousands of points, the unique and fantastic shape appears!

The program "Numeric Values of Mandelbrot Set Points" on the InfoPak disk allows you to track the changing "size" of complex numbers as they are iterated through the formula.