Fractals part 2

Fractals can be much more complex than the simple iterated shapes shown on the previous page. One of the most studied fractals is the Mandelbrot set, named after the very same Mandelbrot who defined the term 'fractal'. While some fractals can be understood without the use of computer's speed and precision, such as those described on the previous page, the Mandelbrot set could not have been explored without their aid.

 In The Computational Beauty of Nature, Flake writes "Benoit Mandelbrot was the first person to study the function x(t)^2 + c in great detail with a computer. To see just how pathological this iterative equation is, Mandelbrot wrote a simple program that preformed" an algorithm which gave complex numbers according to certain rules. Flake continues, "When one tries to visualize a set of natural, real, or complex numbers that has some mathematical property it is usually the case that one imagines the set to be somehow 'cold' because it represents a mathematical truth. yet, the Mandelbrot set is distinctly ... organic.

 Mandelbrot set

Mandelbrot used computers to show him sets of complex numbers much beyond the range of human computation, and arrived at the beautiful fractal pictured above. The set is a fractal -- as you get closer to it the colors and shapes become more and more exciting, but always retain a degree of self-similarity and a number of patterns which is astonishing. The number of hidden goodies in this set is remarkable. Each spiral and spike makes sense when you understand the math behind the object, and it becomes possible to know where you are at any depth of iteration simply by looking at the shapes on the screen. Yet the set is self-similar, although it does not have the ridged symmetricalness of the Koch snowflake.

In the zoom below, look for patterns. See if you can find the same basic shapes reoccurring at many different levels.

 Mandelbrot set zoom and song

All of this astonishing and mind-boggling beauty of math would not, indeed could not be known without the computer. The systems to be modeled are too complex, yet their very complexity is what makes them so amazing.

However, fractals are not just a diagram of mathematical systems; they are related to many phenomena in daily life. There are papers relating fractals to finance, road profiles, sound diffusion, image decompression, cognitive processes, fluctuations of sea levels, solar magnetic fields, biological aging, electropolished surfaces, arts across cultures, and epidermal ridges.