Fractals Everywhere!

Welcome to Comet Way's Fractal Page

still under construction

References and Background

Fractal mathematics is a relatively new branch of mathematics. It involves algebraic concepts and analysis. Some of the applications of fractals can be found in compression algorithms or other heuristics that approximate solutions for NP complete type problems. However, the potential of the power of fractals has yet to be uncovered.

For a basic introduction to fractals, I would suggest reading James Gleick's "Chaos". It glosses over the history and some of the major figures involved in fractals. This introduction assumes the reader does not have a solid mathematical background. Another good book about chaos theory by an author whose name escapes me at the moment is called "Complexity".
For those of you who are familiar with abstract algebra, topology, and some analysis, I highly suggest the following:

Peitgen, Jurgens, and Saupe, "Chaos and Fractals"
Michael F. Barnsley, "Fractals Everywhere"
G. A. Edgar, "Measure, Topology, and Fractal Geometry"

Computer Programs

Computer programs are written to study fractals and to apply the knowledge attained from fractals. Utilizing chaos theory in programs is a unique approach of a problem. For example, fractal compression algorithms for images achieve incredible compression ratios with little loss of image quality. Modeling physical properties on computers is aided by fractals. Matlab (short for Matrix Lab) is a program much like mathematical combined with a unix shell. You can write scripts and functions. The built in libraries allows you to solve difficult analytical equations. Also built in are functions to plot. Formulas derived from a given problem can be iterated and plotted in matlab.
Often times, the study of fractals involve studying the space which the fractal resides in. These are metric spaces and have certain properties. The actual points in these spaces can be pretty much anything, real numbers, rational numbers, complex numbers, strings of characters, etc... A very unique, and the most widely associated property of a fractal's space is the plot of its points by the number of iterations it takes for that point to reach infinity. If the space can be completely represented 2 dimensionally (space is R² or C), this plot can be view with colors to reveal many beautiful pictures. Fractint is the best, most fully featured fractal program I have ever used. It can be found for DOS, WIN, and can be easily compiled for UNIX.

I wrote a mandelbrot generator in Java which allows you to generate successive frames of fractals, while automatically modifying the fractal parameters along the way. There were a few movies which were converted from these frames. Please see the fractal movie page for more info.

Pictures

Here are some pictures generated in XFractint. The large gif can be viewed in fractint and you can see all the information as well as continue zooming. These images were generated on 2 SGI's, a P166, and Comet Way's Micro-Super Computer.

Most of these images are very large. There may be problems with loading them in xfractint if your display is smaller than the window size. Please keep submitting xfractint bugs so they eventually will get fixed.

Click on the image to get the fractint generated gif.

jonlin@andrew.cmu.edu