The term "Fractal" was coined by Benoit Mandelbrot and is connected with their fractured nature and the fact that they actually have a fractional dimension
Many objects in nature are fractal-like: clouds, trees, ferns, lightning bolts, mountain ranges, etc. But mathematical fractals are also infinitely complex - i.e. one can magnify them continuously without coming to an end.
Which of the above images is a satellite photo and which is a part of the Mandelbrot Set?
Other types of fractals, called Iterated Function System (or IFS) fractals have the ability to create realistic images with very small sets of numbers, based on fractal rules.
The IFS system has been used to encode a scene of almost any level of complexity and detail as a small group of numbers, thereby achieving amazing compression ratios of images of 100 or more.
IFS is based on a process called affine transformations. An affine transformation takes a point, say within a rectangle on a computer screen, and uses some mathematical function transform it to a new location in the same rectangles. These transformation can scale, stretch and generally transform an image. An IFS contains a set of numbers that define the transformation. In the random iteration algorithm each transformation is assigned a probability p. With each round of iteration one of the transformations is chosen randomly. As the points are plotted the image emerges.
The fractal that appears at the top of this page (as well as those on the FRACTAL IMAGES sites below) are quite different. They actually lie in the complex plane (where the x-axis represent real numbers and the y-axis represent imaginary numbers - (i.e. the square root of -1) and are constructed pixel-by-pixel. They are sometimes called orbit-trap fractals or escape-time fractals because the color of each pixel is determined by the number of iterations it takes to determine whether or not the starting complex number will lead to an escape off to infinity or to getting trapped into an orbit. What does this mean?
Each pixel on the screen is assigned a complex number value from the complex plane
That value is iterated (fed back) into a given mathematical function to produce some list of numbers (called the orbit)
If the list of numbers (orbit) begins to take off to infinity, it's escape time is how many iterations it took before it was clear it was going to take off to infinity
If the list of orbit does not take off to infinity (after some predetermined number of iterations) then it is a prisoner point
Each pixel is either colored black if it's a prisoner, or given a color based on how many iterations it took "to escape"
The boundary between the black points and the colored points is infinitely complex and fractal
The Mandelbrot Set (seen below) is the most commonly known of this type of fractal, but there are an infinite number of variations on it. Here, the dark blue inner points represent the prisoners. The outer dark blue represents the points that take zero iterations to know it is going to go off to infinty.
When one "zooms in" on one of these fractals, the computer merely computes values and colors for points that are closer together on the complex plane.
It can be difficult to create a consice and clear explanation of how the "M-Set" is constructed.
In addition, to really understand the M-Set it is important to explore it on a computer. This explanation that I give to my students may help in understanding this most complex of mathematical objects. (A background in imaginary numbers, complex numbers, and function notation would be helpful). There are alternative ways of thinking about what the M-Set represents; for instance, the points of the M-Set also summarize all the inifinite Julia Sets. This connection takes a little more work to understand, but you can try with these two sites:
DLA is a process of randomly choosing points in 2- or 3-dimensions and treating them as particles. Each time a new particle is introduced, it moves around ranomdly until it runs into and "sticks" to other particles that are part of an existing structure. This leads to a fractal-like structure that mimics many structures in nature (a variation can be used, for example, to model the diffusion of an oil spill through soil).
The first of sites below shows 3D DLAs that have been constructed so that as each particle gets attached, the previous particles grow in width, to give a more 3-dimensional image, with quite beautiful results:
UltraFractal NOTE: Ultrafractal is apparently no longer freeware (as of 5/15/05). Sorry! (however, you can get a Trial Version which has some limitations on it - but it is worth trying!)
Chaos - The Making of a New Science , by James Gleick
This was a seminal work that came out in the late 1980's that got many people turned on to Fractal Geometry and Chaos Theory. Gleick is a great author on scientific and mathematical topics.
Several books by Peitgen
These books have incredibly beautiful fractal images and touch on Fractals in Science, Nature, and Art. However, th
e text is highly technical.
(image courtesy of Paul Bourke)
(image courtesy of 
