Challenge Team Interim Report
Team Number: 040
School Name: Freedom High School
Area of Science:
Project Title:Order! What order?! Fractals and Chaos in Nature
When we first began our quest, it was very open and we had more questions than answers. Our subject, being so broad, forced us to first focus on something simpler in order to problem-solve for an answer. Well, we are not exactly at the point of pristine focus of our subject, but we are happy to say that we have narrowed our sights on the fractal aspect of our original subject matter, chaos. Let it be understood that while we have now decided to center in on fractals, this does not however mean that we have abandon all of the other aspect of chaos in our subject matter. Fractals are one specific area that models some of our worlds systems and in narrowing down a bit to that, we hope to achieve a greater understanding of chaos as a whole - as it would seem, chaos and fractals are indeed, inseparable. Fractals are one of the chaotic systems with a set of processes that can be taken "apart" and understanding of its components can be had. Still, as with so many other chaotic systems it is very difficult to predict the outcome of it in any situation. However, fractals have been shown to be the one system we have come closest to predicting. They seem to appear in many instances throughout nature and can be made into physical models.
The Definition of the Problem
Creating a program in order to help others use/understand fractals that will do multiple recursion quickly and efficiently.
If you take some mathematical expressions or constructions and repeat them over and over, using the answer back into the problem, you can create extremely complicated images. They then in turn, illustrate patterns of the complexity of which are believed to be models for the roughness or fuzziness of nature.
Our original questions were:
- does a fractal, like the Mandelbot Set display any pattern if we look at intersections of different lines with its edges?
- how sensitive to change are repeating functions or how fast do the changes to the system happen through time?
- can images be generated for any fractal system?
- can a SuperComputer be used as an interactive tool to help us better understand the fuzziness of natural processes?
Our current quest is:
- to gain an increased understanding of fractals, their patterns, how they model some aspects of nature, and their place in the modeling of chaotic system. With that knowledge, we then in turn, hope to create a way for super computers to help others better understand these questions.
By generating computer graphic images of fractals and their patterns, using their mathematical equations, we hope to achieve this broader understanding and make a new method for others to use so that they may share in this knowledge.
Description of Progress
Our progress has been marked by our creation of some small fractal pattern images using LOGO on Macintosh computers. We began our research on fractal systems named Julia Sets, of which the Mandelbot Set is apart of. To test a Julia Set you must pick every point in a plain defined by the real and imaginary number lines, apply a mathematical computation, such as raising it to a power, adding constants, etc., and then locate a new point defined by the output of your calculation. Over a large number of repeats the point will move away to infinity or settle down to a location in the plain called an attractor. You then mark the original point as an escaping point or a prisoner point.
Most of our information to date has come from the books Chaos and Fractals: New Frontiers of Science by Peitgen, Jurgens , and Saupe, Chaos, Making a New Science by Gleick, and a large number of Internet sites.
Our first simple LOGO program uses a small computer to manipulate the real and imaginary numbers on a coordinate plane in order to illustrate fractal images.
Current LOGO Program
TO MANDELBROT :S
MAKE "R 4 / :S
MAKE "X ( -2 ) MAKE "Y ( -2 )
SETXY ( :X * 100 ) ( :Y * 100 )
REPEAT :R [REPEAT :R [make "c 0 MAKE "XT :X MAKE "YT :Y FINDER MAKE "X :X + :S] MAKE "X ( -2 ) MAKE "Y :Y + :S]
SETXY ( :X * 100 ) ( :Y * 100 ) PC 20 DOT
SETXY ( :X * 100 ) ( :Y * 100 ) PC 1 DOT
MAKE "C :C + 1
MAKE "XT ( ( :XT * :XT ) - ( :YT * :YT ) )
MAKE "YT ( 2 * :XT * :YT )
IF ( DISTANCE 0 0 ) > 1450 [ESCDOT STOP]
IF ( DISTANCE :XT :YT ) < 1 [PRISDOT STOP]
IF ( :C > 5000 ) [PRISDOT STOP]
SETXY ( :XT * 100 ) ( :YT * 100 )
Using this program we hope to expand our current knowledge and take it a step further using super computers. It has quickly come to our attention that the only way to realistically experiment with Julia Set fractals was with a SuperComputer. Even our simplest images take over 30 minutes to create. So if the primary problem lies in the development of our existing knowledge/understanding of fractals we need the large machine speeds to advance. In order to alleviate this problem, a secondary one arises which is the creation of an interactive program on a super computer that will expand this existing knowledge and take it beyond it's borders for ourselves as well as others.
The next step for the team
Our next step is the refining of the LOGO program to allow modification of all variables. The team will then disassemble it into its fundamental components, refine these components and then set about translation into the FORTRAN language. This in turn will provide us with a SuperComputer program that can, much more quickly, test number plains. After building the foundation program we must them convert the output to bitmap images and develop a method to allow experimentation with all of the variables. In creating this interactive SuperComputer program, we hope to expand others knowledge of the workings and specific systems of fractals and clarify the manners in which fractals occur in mathematics as well as in nature.