Complex Numbers Imaginary or Infinite

To understand a fractal one must understand complex numbers. A complex number is a number that cannot be graphed on a number line. There are many complex numbers that exist in nature for whatever reason, and allow us to make sense of this chaotic universe. One of the most famous complex numbers, the imaginary number i, is crucial to the understanding of the Mandelbrot Set. The actual function of the Mandelbrot Set is [z -> z^2 + c]. All this function does is input a given constant c and filter the new number back through the same function infinitely many times. The Mandelbrot Set requires that z be grounded at zero to start, but many other fractals can be generated by experimenting with different starting values. The reason complex numbers are essential to fractal geometry is due to their mathematical properties that create infinite sets that are also bounded by parameters. Some of these concepts require a basic understanding of abstract math and being able to visualize the imaginary coordinate system. Regardless, these complex numbers do in fact exist and make possible many of the technological advances being seen in all fields of science. Harmonic motion requires the use of complex numbers in order to model certain frequencies that otherwise could not be explained with trigonometric functions. Consequently, the idea of infinity that has yet to be really comprehended is the central theme to appreciating the beauty of fractal geometry. The set generated by this function continues infinitely, plotting more and more precise points, and when graphed produces the Mandelbrot Set. The mind-blowing aspect of this visual image is the fact that one can zoom in infinitely and continue to see self similarity. As one zooms in on the intricate spirals they see previous patterns unfolding in a dynamically changing number set. These mysterious patterns are now being found in everything ranging from the arteries in your arm to the cyclical patterns of the economy. They are replacing many primitive systems as the blueprint for trying to understand complex dynamic systems. Fractal antennas are imbedded in tens of millions of communication devices that allow for any two people to simultaneously communicate from any two points on a globe. The evolution of computer animation uses fractals to create the movies and video games that are constantly pushing the limit to virtual reality. More importantly than how humans are manipulating fractals to their advantage, is the pre-existent nature to which they are being discovered all around us. Next post I will go into detail about how fractals are being used to model natural phenomenon.