Fractals

Introduction

You probably have heard about fractals before. They are beautiful pictures such as the one shown blow (fig.1). Any fractal can be described using iterative formulas. So you can generate a fractal by evaluating these formulas and finding the color of each pixel. That is a large computational task, and drawing a fractal needs a fast CPU and a carefully optimized program.

Fractals and their properties are studied in non-linear dynamics. Besides the scientists, there are people who like to explore fractals simply because the pictures are graceful and quaint.

I will skip the mathematical background in this article. You can find necessary explanations here:

• Julia and Mandelbrot Sets by David E. Joyce.
• What Is Mandelbrot Set and other articles on fractals by Fabio Cesari.
• Your fractal, sir. - Thank you, James by q123456789.
• Julia and Mandelbrot Sets - Theory by Martin Pfingstl.
• Fractal Theory by Cygnus Software

The last three articles also provide some C code for fractal generation. If you are new to fractals, I recommend you read the articles by Fabio Cesari. His explanations are simple and clear, with tables and images that are worth a thousand words.

I will call this PnP mode (Picture iN Picture) for short.

And a few words about non-goals, namely the things that I am not going to implement:

• fancy colors and palettes
  
• any types of fractals other than Julia and Mandelbrot;
• support for 3DNow (SIMD extensions found in old AMD processors; modern Athlon XP uses SSE);
• porting to other OSes and processors (x86 and Win9x/XP were just enough for me);
• multiprocessor support.

My first try was Chaos Pro. It's a complicated freeware application that can generate Julia and Mandelbrot sets and a lot of other fractals. You can even add your own fractal formulas using the built-in programming language. While the program is able to create cool 3D images and animations, use different palettes, etc., it still lacks some essential features. First, you cannot view the orbit and axes. Second, the Mandelbrot set could be generated in about 1.6 seconds on my Pentium M 1.5 GHz laptop, and the Julia set with C = -0.12+0.74i takes 1.4 seconds to draw. Chaos Pro supports PnP mode, but it's inconvenient to use because the rendering is so slow. Fractals cannot be fully rendered in real time (you can't get the picture changing instantly as you move the cursor). Third (and it was the main reason for me), I often got "Access violation" error or other bugs with Chaos Pro. Certainly, it's okay for the freeware application written by a hobbyist, but I wanted something more reliable.

Then I came across FracInt, an good old DOS program for drawing fractals. It has been developed several years ago, and a 20.0 version of this application is available now. The program is quite sophisticated and powerful, with a lot of intriguing features such as ability to view several dozens of fractal types, play fractal music, use the PnP mode, and create stereoscopic images. Its main disadvantage is the outdated user interface. Zooming takes a lot of time: you drag the mouse to set the desired rectangle size, then move it to the point and, finally, you click twice (left button to zoom in, right button to zoom out). This was unacceptable for me, because an average student will spend half of the lesson comprehending how to use this zooming interface. Video modes up to 1600x1200 are available (1024x768 VESA should work with most of the video cards), but when I ran FracInt in Windows XP and pressed Alt+Tab, my video driver crashed. The output to the sound card didn't work, and I had to use my PC speaker. Needless to say about missing MMX or SSE support in this old application.