Comparation+and+Contrast

=** Comparation and Contrast **=

** Mandelbrot and Julia Sets **
Both Mandelbrot and Julia sets are types of fractals. However, these are more complicated fractals then the other fractals that have been mentioned (such as the[|Sierpinski's triangle]). Both these sets require the use of complex numbers. Thus before one can understand how the sets are created one needs to know some very simple properties of complex numbers. If you know the basics of complex numbers you may skip the section on them and continue reading. However, for those who are unfamiliar with complex numbers, there is a brief explanation that can be read by [|clicking here]. To compute the basic Mandelbrot (or Julia) set one uses the equation f(z) -- > z 2 + c, where both z and c are complex numbers. To describe what occurs it is easier to view the function f(z) to be a machine that squares a complex number and then adds c to it. Now to compute the sets one takes a starting value for z and places it in the "machine". The number is squared and c is added to it and a new number (most likely) comes out of our machine. Now, one places that new number in the machine and the process occurs again. This process is called iteration and it is how the Mandelbrot a nd Julia sets are computed. The purpose of the iteration is to determine the behaviour of the values that are put into the function, as will be shown in the following example. For the simplicity of the example we will use real numbers, but it should be noted that one could also use complex numbers to illustrate the point. f ( ½ ) = ¼ f ( ¼ ) = 1/16 f (1/16) = 1/256 It is obvious that if the iteration is continued the value will go towards 0. It is equally obvious that if one were to take a number greater than 1, after iterating it repeatedly, the value will go to infinity. For the Mandelbrot and Julia sets it can be proved (through a very complex proof) that if the distance, on the Cartesian plane (remember we are using complex numbers here), between the origin and a point resulting from the iteration of some initial value is greater than 2 then the behaviour of that initial value is that it will go to infinity. If, however, after numerous iterations (possibly hundreds, thousands or more) the distance between that origin and the point is never greater than two, it is said that this point is bounded. Then, knowing that, the definition of the Mandelbrot set is: the set of all the complex numbers, c, such that the iteration of f(z) -- > z 2 + c is bounded (starting with z =0 + 0i). More simply put, the Mandelbrot set is the graph of all the complex numbers c, that do not go to infinity when iterated in ** f(z) -- > z ** ** 2 ** ** + c **, with a starting value of z =0 + 0//i.// A Julia set is almost the same thing. It is defined to be: the set of all the complex numbers, z, such that the iteration of ** f(z) -- > z ** ** 2 ** ** + c ** is bounded for a particular value of c. Again, more simply put it is the graph of all the complex numbers z, that do not go to infinity when iterated in ** f(z) -- > z ** ** 2 ** ** + c **, where c is constant. It should be understood that these are simply the basic definitions of the two sets. The function that is iterated can be practically anything, as long as it uses complex numbers. Thus the basic difference between the Mandelbrot set and Julia set is that in any Mandelbrot set, you are plotting various values of c on a Cartesian plane, whereas for a Julia set, you are plotting various starting values of z, and c is kept constant. After looking at the fractals, you may be wondering why there are such a variety of beautiful colours. Well, the explanation is quite simple. As was mentioned previously, the functions used to create the fractals are iterated and the points that never result in a point further than 2 units from the origin are part of the set. However, the other points, which are not part of the set are the ones that result in the beautiful colours. The way the colours are computed is by seeing how many iterations it takes for the points that are not part of the set to reach infinity (this is determined by how many iterations it takes them to move a distance further than two units from the origin, in this case). For example, if a point were to move a distance further than two units from the origin after only 10 iterations, it could be coloured blue. Likewise, if it moved further than two units from the origin after 20 iterations is could be coloured red. The actual colours used are irrelevant, it is simply the fact that different colours are used to show the different behaviour of the points not in the set. It should be noted that the more colours used, the more details will be seen and the more spectacular the fractal will appear. If you wish to see more examples of Mandelbrot and Julia Sets, please check out the [|fractal gallery].
 * Example 1: ** f(z) -- > z 2 + c where the starting value for z is between 0 and 1 and c is 0.