Fractal Data Compression

## Question:

I am a college student and planning to graduate this May with a BS in Mathematics/Computer Science. As a graduation requirement, I am expected to produce a senior paper on a computer related subject of my choice and present it to a panel of Math/Comp Sci professors.

My tentative choice of topic is fractal compression in relation to graphics display. The study of chaos is the first of what I expect will be many related sub-topics. At present, I am lacking a conceptual grasp of complex numbers. I can handle the basic arithmetic operations, but still do not understand where complex numbers come from and why.

In the interest of successfully completing this project, I would appreciate any general insight you could offer (on complex numbers) in addition to other related sub-topics I should pursue. Thanks much!

Welcome to the group. What M. Casco Associates is all about is launching extraordinary success in the science careers of our clients. You are well along with your education at this point. I will be happy to help you fill in some blanks if you like.

Complex numbers first arose as a mathematical curiosity, some way to deal with the square root of negative numbers. Later it was found that many real physical systems are best described in terms of complex numbers, particularly oscillatory systems like radio waves and other alternating current electronics and electricity.

Recently with the work of Mandelbrot and others, complex numbers have re-entered the mathematical "gee whiz" world in the form of sets under iteration. The tie in to fractals is that the boundaries between complex numbers in a particular set and those not in it are fractal objects.

Fractal image compression is based on the proposition that very complex images can be expressed in a tiny amount of information, which when fed back onto itself repeatedly reproduces the desired image. It may be the same trick used by God to store the plan for an entire oak tree in an acorn.

The Mandelbrot set which is one of the most famous fractal images is contained entirely in the formula z=z^2+c. This is interpreted as "replace the complex number z with itself squared plus a complex constant c". When this process is repeated over and over, z either grows larger without limit or not. If it does, the original z was not in the Mandelbrot set. If it does not the original number was in the Mandelbrot set. By coloring the spot on the screen corresponding to the original z with a different color depending on how many repetitions of the formula it took to determine that a number was not in the set we get the image.

Another good reference work which is less theoretical and more practical than my material is the book FractalVision by Dick Oliver. It is no longer in print but the college library may have a copy.

This information is brought to you by M. Casco Associates, a company dedicated to helping humankind reach the stars through understanding how the universe works. My name is James D. Jones. If I can be of more help, please let me know.

JDJ