Teaching the geometry of iterated function systems is much easier when we replace static demonstrations with in-class animations and links to software that can then be run on the students' computers in their rooms. I thought I understood all the implications of a web implementation of these ideas, but we shall see, I had a surprise.
| The first step in learning to grow fractal pictures is to understand some of the geometry of plane transformations. |
| We begin with learning rules to grow a right isosceles Sierpinski gasket. |
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| Then we continue to more complicated fractals. |
| The relations between the parts of an IFS image can be effectively revealed through some animations. |
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Finally, here is the surprise mentioned earlier. Complicated images have always been a challenge in homework. Many students do not have a sufficeint sense of mental geometry to rotate or reflect intircate fractal shapes in their minds. In earlier years, I have suggested making rough sketches on small bits of paper, front and back to allow reflections. While making this suggestion in the fall 2000 class, I realized there is another possibility now that the homework is on the web. This is so remarkably simple that others must have thought of it before, but I have not seen this approach mentioned anywhere.
* Bring up the homework web page on your computer.
* Place the mouse arrow over the fractal image whose IFS rules you wish to find, hold down the mouse button until the menu pops up, and select "copy this image."
* Now open your graphics program and paste the image. This is the target image.
* Paste another copy and use the graphics functions - scale, rotate, reflect - to cover one of the pieces of the target. You must be careful about the placement of the pieces under rotation and reflection, but the animations mentioned above seem to be good preparation for this.
* Continue until all the pieces of the target are covered.
* Test the transformations by running the IFS program.
Every time I have described this simple trick, the audience has been amazed. Some have responded with Homer Simpsons' "Doahh!" How could such a simple, effective trick not have occurred to everyone? What other similar web applications are waiting to be found? This is an exciting time to be a teacher.
In examinations that cover IFS problems, I always include a few that are more difficult than the standard examples. In previous years, only about 70% of my students succeeed with these problems. In the fall 2000 class, the first to use these web pages, the success rate was about 90%. This is hardly a careful study of the effectiveness of web-based instruction, but the class size (over 170 students) makes the result worth investigating.
Having established the basic vocabulary of fractals and IFS, next we discuss variations on the IFS formalism that have accounted for about a third of the course projects over the last few years.
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