Old Ben

Mandelbrot Set:

The Mandelbrot set is an infinitely complex fractal, composed by using the following seemingly simple equation:




When various numbers are used in place of the variable C, after several iterations, they will either tend to lead toward infinity, or they will lead toward zero. When constructing the Mandelbrot set, not only are Real numbers used, but also Imaginary numbers (such as i, 2i, 3i etc... where 'i' is defined as the square root of negative 1). Any number that leads toward infinity is discarded, and all other numbers remaining are considered elements of the Mandelbrot set.


For example, c = 1 gives the sequence 0, 1, 2, 5, 26; which leads to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.


On the other hand, c = i gives the sequence 0, i, (-1 + i), -i, (-1 + i), -i, which is bounded, and so it belongs to the Mandelbrot set.




Fun Fact:
Only after infinite numbers have been placed into the equation Z = Z² + C, will the full set be truly created. Since it is impossible for humans (or even computers for that matter) to input INFINITE numbers into the equation, we will never know (and CAN never know) the FULL mandelbrot set!!


However...


When computed and graphed on the complex plane using several hundred, thousand, million etc... numbers, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal.

The infinite complexity of the Mandelbrot set is illustrated below:

The video above shows a magnification of the Mandelbrot set. The song playing in the video is Jonathan Coulton's "Mandelbrot Set". If you listen to the lyrics, he explains how to construct a Mandelbrot set. He also mentions other fractals such as the Koch curve, Seirpinski gasket, and Cantor Ternary Set. ENJOY!!