What is a Fractal?
Here's a fun (well...sorta fun) video which explains fractals. (33 megabytes)
A fractal is a geometric shape that is complex and detailed in structure at any level of magnification. Many fractals are self-similar, meaning that each small portion of the fractal can be viewed as a reduced scale replica of the whole. (I found the previous statement on an abandoned web page...I don't know who wrote it, but it simply defines the concept perfectly.)
For example, a natural example of fractal geometry can be seen in snowflakes in the sense that all snowflakes are almost identical in form yet the structure of a snowflake will never be duplicated exactly. Each flake differs slightly from all other flakes. Moreover, when a snowflake is magnified, each isolated portion of the flake is nearly identical to the whole flake! Other natural examples of fractals include ashes, tree bark, tree branches, broccoli stems, bubbles, carpet.
Fractal mathematics is related in ways to chaos theory. Chaos theory is the study of forever-changing complex systems. Chaos theory is commonly misunderstood to be a belief that the universe is absolutely chaotic. However in fact, chaos theory suggests that objects, actions, environments, etc. that we perceive as chaotic may actually contain perfect order according to a complex system which we don't have the capacity to fully understand yet; therefore these systems, due to their complexity, seem like chaos.
I don't fully understand fractal geometry, as my focus is not calculus, but I feel that I have a good layman's understanding of the possibilities and implications of fractal mathematics.
The first implication, and perhaps most intriguing, is the notion that our entire universe - it's existence, as well it's past, present, on both macro and micro levels - may be explained via a complex system -- mathematical or otherwise.
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The second implication is far more simple...
The very nature of fractal formulae is to generate numbers. The numbers generated by a fractal equation can be used to define such things as coordinates, frequency, duration, amplitude, etc. for use in audio or visual art. For instance, the background image used on this page is in fact a fractal-based graphic.
Fractals as Coordinates
The most commonly known fractal is called the "Mandelbrot Set". The Mandelbrot set is based on the following quadratic equation:
f(z) = z2 + c
Note that both z and c can be complex numbers. (Complex numbers are pairs of real numbers.)
The actual equation is:
Z => Z ^ 2 + C
The first iteration of this equation will present a pair of coordinates (the exact coordinates all depend on the initial values you define for z and c - the initial value you define is often referred to as a "seed"). Then, on the second, third, fourth, (and so on) iterations, the equation presents related coordinates. If, let's say, you process 1,000,000 iterations of the equation, then you will be presented with 1,000,000 pairs of coordinates which are all very closely related and belong to a wonderfully complex pattern. These coordinates, when mapped to a simple X Y graph, create an image.
Here's the interesting part: No matter what seed is used in the above equation, the coordinates produces by multiple iterations will always be either exactly like the following picture or exactly like a tiny portion of the following picture:
Below is a section of the above coordinate map magnified approximately 3000 times (the magnification is caused simply by using a different seed in the original equation). You can see how each smaller portion of this image is closely related to the whole image above -- hence the definition: "self-similar". Like a snowflake, or carpet, or tree bark.
Fun: The Mandelbrot and Julia Set Explorer is a cool web site where you can create your own images online.
Fractals in Music
Would you like to listen to the pictures above?
Fractal equations and other complex systems are used in computer programming to generate random numbers (and more). However, it is absolutely impossible for a computer to generate a totally random number!! What actually happens is that computer is instructed to generate a number using a mathematical algorithm based on a particular seed (sometimes the value of the computer's internal clock). In other words, the computer is told "grab the number from the clock - perhaps 12:30 - and run that number through a complex equation to generate a number which seems random".
In this sense, fractals are wonderful tools to generate numbers to use as pitches, durations, amplitudes, and other elements in musical composition. The numbers seem random due to the complexity of the pattern yet they are related in such interesting ways that the musical result is often very pleasing...always fascinating.