Types of Fractals
Julia SetsWHAT ARE THEY?
Julia Sets are one of the most famous types of fractals formed using formula iteration. They were discovered by Gaston Julia early in the century, but their true analysis only became possible after the development of computers (see algorithms). Just like all formula fractals, Julia Sets can be very complex, and yet use a very simple formula. To draw a Julia Set, do the following:
1. Choose any complex number and make c equal to it. This will be our Julia Set
2. Choose another complex number and make z equal to it.
3. Make z equal to z2 + c and repeat this change a lot of times.
4. If the number went rapidly to infinity, do not mark the corresponding point on the complex plane. Otherwise, it belongs to the set and you can mark it.
5. Repeat steps 2-5 for different numbers until all points on the plane are checked.
The constant c can be any complex number, and every complex number produces a different Julia Set. What is even more interesting is that the same formula with different constants can produce fractals that look nowhere close to each other:
Notice that some Julia Sets are connected, while the others are just collections of disconnected points. This factor is determined by whether or not the point c is on the corresponding Mandelbrot Set. The above pictures all use the formula z = z2 + c with different values of c. Although this is the most common formula, any other formula can be used as well. Very interesting patterns can be formed by using trigonometric or logarithmic functions in the formulas:
Notice that all of the above pictures are colored. The colored area is actually not the Julia Set itself, but the border, which is colored depending on how fast the point escapes to infinity during iteration (see algorithms).
There are a 3D versions of Julia Sets, which are called quaternions.
|Julia Set - allows free movement mode which allows to see how julia fractals change with changing input values|
Windows 95 program that creates pictures of the Mandlebrot Set and Julia Sets
|Mandelbrot and Julia Set Explorer
An explorer of most famous fractals that allows huge magnification using server/client interface (cgi)
|Julia Sets: Pixel Algorithm|
|Julia Sets: Inverse Iteration Method|
Cosmos : The Art of Mathematical Design
You will not find a lot of explanations in this book, but it is great if you want to see the great variety and beauty of fractals, especially formula generated ones. Every fractal in the book is followed by a formula and parameters, letting you experiment on your own.