IFS FRACTAL GENERATION EXERCISE

Richard L. Bowman, PhD
937 College Ave., Harrisonburg, VA 22802, USA
richard.bowman@edtechbybowman.net

Introduction

Fractals were first named by Benoit Mandelbrot in the mid 1970's but these strange figures and been studied years before by other mathematicians. By definition a fractal has two primary characteristics: (a) a fractional dimension and (b) self-similarity. Generally students have studied one-dimensional (lines), two-dimensional (filled figures such as squares, trapizodes, and circles), and three-dimensional (filled objects such as cubes and spheres) objects. Fractals have dimensions that are in between these three integer dimensions. Also, as one looks more deeply or more closely at a fractal, its inner parts have a similar design to that of the whole object.

Iterative function system (IFS) fractals are often esthetically-pleasing geometric figures. The actual appearance of the fractal is determined by the number and positions (XF, YF) of fixed points chosen and the scaling factor (SC) specified. The algorithm for generating the fractal from this information is very straight-forward, but thousands of iterations of the algorithm are necessary before the shape of a particular fractal emerges.

The procedure requires the placing of a point (XP, YP) within the space defined by the set of fixed points and then randomly selecting one of the fixed points. A new point is plotted at a scaled distance from the chosen fixed point toward the original point. Thus the coordinates of the new point are given by:

This procedure is then repeated thousands of times. While the procedure involves randomness, the resulting geometry is always determined by the chosen set of fixed points and the specified scaling factor. Follow the directions below to create some simple IFS fractals and to study how the fixed points and scaling factor effect the resulting fractal.

Some Definitions

• Fixed Points - each IFS fractal uses a set of fixed points toward which to randomly move as the series of calculations are executed; the geometry defined by the positions of these points will determine the shape of the resulting fractal.
• Fractal - mathematical constructs characterized by self-similarity (that is, as one examines finer and finer details of the object, the magnified area is similar to the original but is not identical to it) and apparent chaos within reproducible bounds.
• IFS (Iterative Function System) - a method for generating fractals that involves performing a large number of calculations of a simple formula; the point used in any one calculation is randomly selected from a given set of fixed points.
• Scaling Factor - a fraction, with a value of less than 1.0, used to specify the distance from one plotted point to the next plotted point relative to the distance from the original plotted point to one of the fixed points; the scaling factor ultimately governs how diffuse or focused the resulting fractal pattern will be.

Explorations

1. Print out a copy of these directions for each member of the lab group, unless your instructor has already made copies for your use. Read the introductory material above before continuing with this exercise!!

2. At the fractal generator web page ( http://www.edtechbybowman.net/PhysAstroSims/fractals/FractalGenerator.html ), read all of the directions and hints, and execute the Sierpinski Triangle Fractal. Print out a copy of the resulting page for everyone in the lab group. (All students must hand in their own lab worksheet, but collaborate with each other in your lab group as you work through this activity.)

3. At the top of a blank piece of paper, write the title, "IFS Fractal Lab," and then your name and the names of the persons in your lab group. Place a check mark by your name for identification purposes.

Note: All of your responses in the steps below should be recorded on this blank sheet or directly on the printout of the associated fractal. Where appropriate, precede your responses with the number of the step to which you are responding.

4. While you wait for the printouts, experiment with different values for the scaling factor on the fractal generator web page. Try three or four different values within the suggested range of numbers. On your lab sheet describe what happens as you move from a scaling factor of 1/1.5 to 1/1.7 to 1/2 to 1/3. Print out at least one of these resulting fractals and enclose it with your report.

5. Systematically explore the specific fractals listed on the fractal generator page, moving from the top of the list to the bottom..(Leave designing your own fractal until later in this activity.) Select several scaling values in addition to the default values. Print out representative samples of these fractals as appropriate., or as directed by your instructor

6.. From the results of step #5, describe what happens in all of these fractals as one moves from a smaller scaling value to a larger one. (Remember that the scaling factor is actually the inverse of the scaling value.) Comment on any patterns you see in the size of the scaling value as number of fixed points is increased.

7. Also, from the results of step #5, describe how the positioning of the fixed points determines the shape of the resulting fractal.

Note: Except for a small margin around the fractal plot area, the outer corners of the plot area have the following coordinates, beginning at the upper left-corner and proceeding clockwise: (-1, 1), (1, 1), (1, -1) and (-1, -1).

8. After studying the pre-defined fractals and experimenting with their scaling factors, design a few fractals of your own to explore the conclusions you wrote in #6 and #7. Try 9 fixed points, a scaling value of 4, and fixed point coordinates of {( -1 , 0 ), ( -0.5 , 0 ), ( 0.5 , 0 ), ( 1 , 0 ), ( 0 , 1 ), ( 0 , 0.5 ), ( 0 , -0.5 ), ( 0 , -1 ), ( 0 , 0 )}. Then let your imagination run free and try some none "geometrically-shaped" designs. Points can even be repeated to emphasize a particular region of a fractal.

Conclusion

Collect all of your lab sheets, including all printouts that you used; staple them together; and hand them into your instructor. You have only begun to explore the depths of the IFS method of generating fractals. Check out the references below or others from your library to learn more. IFS fractals can be used to mimic natural objects such as leaves or manufactured objects such as submarines. Have fun exploring!

Bibliography

• Richard L. Bowman, "Fractal Metamorphosis: A Brief Student Tutorial," Comput. & Graphics, 19, 157-164 (1995).

• R. L. Devany, A First Course In Chaotic Dynamical Systems Addison-Wesley, 1992.

• H.-O. Peitgen, H. Jurgens, and D. Saupe, Fractals For The Classroom, corrected ed. Springer-Verlag, 1991.

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Created and maintained by: Richard L. Bowman (1997-2011; last updated: 14-Sep-11)