One of the amazing facts about fractals is that the pictures you see around are generated by very simple formulas.
Specifically, the Mandelbrot set is generated with the
following formula:
Z0 = 0
Zn+1 = Zn2 + C
This kind of function is called an iterative function. The input of the current equation is
the value of the previous evaluated equation, where Z and C are complex numbers of the form z = a + bi.
The point of the formula is that some values of C approach infinity
and some don't. If the values don't, then the set of co-ordinates (a,
b) are part of the Mandelbrot set. The picture is then generated depending
on the values for (a, b) together with a color, which is assigned
depending on the divergence to infinity. To summarise: if the value
of (a, b) goes to infinity, then the color is solid; otherwise, it receives
a different color and the most interesting part of the picture is generated.
For more details on the mathematics of Mandelbrot set go here
or here.
Vector representation and iterative functions connect the Mandelbrot
to complex numbers. Of course, this brief introduction has been greatly
simplified, but I hope that the magic and mystery is not gone. Fractal
theory is part of new Mathematics and is being used in very important
problems in computer science such as encryption, file compression, and image processing.
You can go directly to the interactive part of this section and play
with the Mandelbrot Applet I've coded, which rendered this
image. You can also find the source code in the next
page.
|
