The Mandelbrot Set

The Mandelbrot set is a collection of numbers, a mathematical set, named after its discoverer; Polish born mathematician Benoit B. Mandelbrot. The set produces, with the aid of a computer, an infinitely scalable and complex fractal pattern like the one seen at the upper right of this page. Mandelbrot in fact, is widely considered to be the father of fractal geometry.

The numbers used to produce this fractal image are called "Complex numbers" because they include both real and imaginary components. The imaginary component is required because no real number can be squared and result in a negative number. (Confusing, I know, but stay with me.) So, in order to circumvent this limitation, the imaginary number i was invented and defined as the square root of -1 (negative one). i is used in combination with a real number (for example: 3i squared would equal -9), to form an equation that the computer then runs, producing the image.
In other words:
"The Mandelbrot Set is generated by the iterated function:

z_n+1 = z_n² + 
      c

where both z and c are complex numbers."

Got that?

In any case, higher mathematics gives me a screaming headache (hell, Long Division gives me a headache - I'm certainly no mathematician), so let's dispense with the technical stuff and come a bit closer to Earth...

I first learned about the Mandelbrot set years ago while watching a science program on PBS (NOVA, probably), and was enthralled enough to at least remember the name. Subsequent to my first computer acquisition, I discovered that the Mandelbrot set was included in a screen saver program called "After Dark". (Anyone remember "Flying Toasters"?) Unfortunately that program was designed to run only on the Mac and Windows 9x, and the company that published it is no longer on the scene. To my knowledge, no one has yet ported the program to NT based systems. And although I don't use a screen saver much these days, I do miss the Mandelbrot set in that context. Watching the vibrantly colored moving images could be quite relaxing after a long hard day in the salt mines...
OK, ...so I'm easily amused.

Still, there are lots of free programs currently available that will let you both explore the images, and experiment in great detail with the Mandelbrot set, and you don't even need to know how to add 2 + 2 in order to use them. Many are Java based and tied to a specific web page, others are downloadable executable files for installation on your computer. All are interesting, fun, and definitely different, producing strangely beautiful colors and images.

Choosing an area and 'drilling down' within the image (either by clicking and dragging to create a box, or just by clicking) produces a very real mathematical infinity that can be explored to whatever level one has patience (and computing power) enough to endure. The results can be really quite amazing, as the examples below illustrate;

Each of the numbered images above represents a four-fold magnification of a section of the previous image. No matter how deep one goes, the original form continues to repeat at successively smaller and smaller scales, with no end ever reachable. A true mathematical 'Infinity'.

A basic Google search for the Mandelbrot set will return about 203,000 hits, so those interested in the geometry and mathematics behind the pictures will easily find answers to all of their questions on the web. Obviously, there is a great deal of interest in fractals in the on-line community.

I had considered including one of the Java applets on this page, but there are already dozens of them available on the web (an example of one can be found here), so I opted to cheap out and save the bandwidth instead. :)

An excellent downloadable program; "Mandelbrot for Windows" - ('Careware'), can be found by clicking here. Mandelbrot for Windows was used to produce the trio of images seen below.