Magnifying the fern, we see each frond is a shrunken copy of the whole fern.
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So the fern is made up of how many copies of the fern? Looks like a lot, doesn't it? An IFS with dozens of rules? Can this be right?
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As practice for the fern, consider the fractal on the left. The most obvious decomposition (on the right) involves a lerge number of copies of the fractal. In the limit, infiitely many transformations are required with this decomposition.
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However, there is another approach: the fractal can be divided into four small copies (the four bottom pieces, each shrunk by 1/4), and what remains is another copy, shrunk by 3/4. The five transformations are straightforward.
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Now the fern decomposition is reasonably clear. The only issue to mention is the stem. Years ago, presenting this example to a group of elementary and middle school children (without discussing rotations or reflections - just the decomposition), I convinced most of the students that only three rules were needed: the lower right and lower left fronds, and all the other fronds lumped into one piece. Then a little fourth grader asked "What about the bottom of the stem?" "It's just a single line. I'll draw it in by hand," I replied. "But what about the little stem connecting each frond to the middle? Oh, and what about the even littler stems connecting the frons of the fronds?" she asked. I don't know where you are now, and I never learned your name. But I remember you. These are the moments that make teaching such a centering experience.
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