Assignment 9/16:  due Tuesday 9/17 at 3PM

Work in groups of from today's class.  Turn in one solution for your group.  

1. Turn in the picture your group determined for the IFS you were assigned in class.  Include the picture of the original shape plus the first three iterations.   Have a copy of the third iteration on a piece of paper by itself to exchange with another group. 
2. One application of the deterministic algorithm for a particular three-function IFS takes the square on the left to the three squares on the right in the picture. Sketch the result of applying the same deterministic algorithm to the shape on the right. (Note: there might be more than one answer, so be sure to describe the IFS rules you use.)
3. Alter the rules for the construction of the Cantor set so that the middle 1/5 is removed
   rather than the middle 1/3.  Draw a few stages of the resulting Cantor set, and explicitly write down the rules for the deterministic IFS that gives it.   

Assignment 9/17:  due Thursday 9/19 at 3PM

Work in the same groups as on yesterday's assignment.  Turn in one solution for your group.

Chaos Under Control page 391 #2.7

1. Turn in the IFS rules that your group determined were needed to produce the IFS picture you were given by the other group.  You should check this solution with the other group before turning it in.
2. For each of the following pictures, find a five function IFS to generate the picture. That is, for each picture, give five descriptions, each of  the form "scale by 1/3 in the x direction then translate by ¼ in the y direction then rotate by 90° clockwise" such that the combination of the five maps applied to the figure will cover it as in the collage theorem.  Pictures at http://www.math.union.edu//research/chaos/Ex2_32ab.gif
3. Figure 2.22  in your text depicts the six transformations which make up the IFS code of the "Cherry Tree." The pieces are pulled slightly apart to emphasize the action of the respective transformations.  Locate, roughly, places in the Tree where the last transformations were 11, 25, 43, 55, 341, 144, 214.  Describe how you might go about coloring the Tree as shown in Color Plate 13 (in your text)
4. Try making an IFS image of a picture of a natural object of your own choosing. Leaves (especially maple), flowers, vegetables such as broccoli and cauliflower, all make good subjects. So, sometimes, do trees. To find the IFS parameters you will have to make a covering of the original picture. You can do this approximately by hand (make rough sketches), a little more accurately using a reducing copying machine, or even more accurately using a computer "paint" program which allows variable scaling. Keep track of the scalings, translations, and rotations. (One student made a nine-function IFS of his face -- a reasonable likeness, but the result of a lot of trial and error. Moral: don't start with really complicated pictures.)