What is the Effect of Putting Different Variable Values in the Fractal "Mandel's" Equation?

Researched by Addison F.
2000-01

PURPOSE

The purpose of this experiment was to determine if changing the variable values in equations by one, two, three, four, and five would change the calculation time of certain computer generated fractals.

I became interested in this idea because I have always had an interest in mathematics and by reading about what mathematicians and scientists do with fractals. I also like the way computer artists use them to make art.

The information gained from this experiment might help scientists discover new fractals and new ways to use them. It will also benefit computer artists in making different designs with fractals.

HYPOTHESIS

My hypothesis is if you put different variable values in equations, it will change the outcome of a fractal compared to not having anything change.

I base my hypothesis on reading pages on the Internet and in certain books that say how even one little change in an equation can have dramatic effects on the way fractals look.

EXPERIMENT DESIGN

The constants in this study were:

 
The same fractal program.
The same fractal.
The amount of changed variables.
The same computer. 
The same printer.


The manipulated variable was the changed variable values in the equation.

The responding variable was the calculation time.

To measure the responding variable I obtained the calculation time from the "image info" in the program read-out. I measured it in seconds.

MATERIALS


QUANTITY ITEM DESCRIPTION
1 Fractal Program (Fractint)
1 Type of Fractal
1 Computer
1 Printer
1  Microsoft Excel program
1 Image Software (for printing)

PROCEDURES

  1.  Turn on computer.
  2.  Open Microsoft Excel.
  3.  Open Fractint. 
  4.  Open fractal titled Mandel. 
  5.  View the fractal in video mode 16 bit EGA: high graphic resolution. 
  6.  Open up the "image info" command.
  7.  Type the calculation time in Microsoft Excel.
  8.  Save the fractal to the same folder as Fractint.
  9.  Restart Fractint. 
  10.  Open fractal titled mandel again. 
  11.  Change the first variable by one in the equation. 
  12.  View the new fractal in video mode 16 bit EGA: high graphic resolution. 
  13.  Open up the "image info" command.
  14.  Type the calculation time in Microsoft Excel for the new fractal.
  15.  Save the new fractal to the same folder as Fractint. 
  16.  Exit out of Fractint.
  17.  Open the fractal save file titled fractal001.gif.
  18.  Print out the fractal.
  19.  Open the fractal save file titled fractal002.gif. 
  20.  Print out the fractal.
  21.  Repeat steps 3 to 20 and change the variable from one, to two through 5 each in turn.

RESULTS

The original purpose of this experiment was to see if changing a variable in an equation would effect the ouctcome of a fractal.

The results of the experiment were that a changed variable in an equation did affect the fractalËs calculation time.

See the table and graph

CONCLUSION

My hypothesis was if you put different variable values in equations, it will change the outcome of a fractal compared to not having anything change.

The results indicate that this hypothesis should be accepted because there were changes in the fractalËs calculation time.

Because of the results of this experiment, I wonder if changing more than one variable in an equation would effect a fractal differently.

If I were to conduct this project again I would change more than one variable at a time. I would also consider more trials, and using more fractal equations in addition to "mandel".


 
 
RESEARCH REPORT

Introduction

 Fractals are computer-generated images. Computer artists use them to make things like landscapes. Weather forecasters use fractals to help in predicting the weather. Fractals involve concepts like self-similarity, chaos theory, and algorithms.

History

 The word "Fractal" was formed from the word fractus, meaning broken by a mathematician named Benoit B. Mandelbrot in 1975. He used it to describe highly irregular shapes and mathematical functions. Fractals gained popularity after his research.

Self-Similarity

Self-similarity is a pattern that repeats itself again and again on a smaller and smaller scale. The first pattern that is repeated in the fractal is called a seed. Think about a tree--a twig shape resembles a branch. Then a branch resembles a bough. Then a bough resembles the trunk. Fractals have self-similarity, meaning that a piece of a fractal will often look like the whole. 

Algorithm

An algorithm is a recipe for the image. Fractals are extremely compressible in the sense that an algorithm, or recipe for the image, is far simpler to store than the image itself. But since fractals are so closely tied to algorithms, some fractals have the same paradoxical properties as systems that can compute. 

Chaos Theory

Weather is almost completely unpredictable, making it part of the chaos theory. A person known as Edward Lorenz discovered this by running a weather model over time, but because of the slowness of the computers at the time (1960Ës) he rounded off the numbers. The two results for forecasts he got were drastically different. Earlier on in the model they were similar but as they progressed they split faster and faster. To truly predict weather you have to account for every possible variable, this includes the flutter of a butterflyËs wings in Africa. As you can see this is near impossible. While fractals wonËt make the perfect predictions, they are now helping, along with chaos theory, to more accurately predict weather.

Recent Developments

 Lately there have been many uses of fractals in creating digital landscapes. In fact, using fractals, some of the landscapes in the Star-wars trilogy were created. How this is done is similar to how they are drawn. Instead of the points being drawn on a 2D plain, they are drawn on a 3D plain.

Fractals in Nature

 The human Body is a good example of fractals in nature. The body is full of fractal geometry. Take a look at the circulatory system for example. The system enables the heart to supply blood to every single cell in the body. There are multiple planes, which you can zoom in on, starting with the arteries and going all the way to the capillaries and further. The same is with the nervous system. The overall idea of fractals in the body is the feedback loops which many of the body systems use. They are functions as are fractals.

Summary

 Fractals are used in many different ways. Mathematicians use fractals to explain chaos theory. Fractals can have many different colors, shapes and sizes. There are many uses for fractals.
BIBLIOGRAPHY

Fereydoan Family, Emory University, Fractal Geometry, Encyclopedia America, 1999

Fractal, world Book, 1999

"Fractals" available at
http://www.parker.org/appliedmathhan/fractals.htm

"Fractint" available at
http://spanky.triumf.ca/www/fractint/fractint.html

"Fractals" available at
http://search.britannica.com/search?query=fractals

"Fractal" available at
http://www.britannica.com/bcom/eb/article/4/0,5716,35724+1+35083,00.html?query=fractals

"The Computational Beauty of Nature" available at
http://mitpress.mit.edu/books/FLAOH/cbnhtml/toc.html
 

ACKNOWLEDGEMENTS
Mr. Smith for, transporting all of the projects safely to and from the science fair. I am glad that my display was well taken care of; and My scienc teachers Mr. Newkirk and Mrs. Pasckvale.

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