The creative option: make your own self-similar fraction, and describe how you made it and how it is self-similar. Here are some ideas of how to make a new fractal:

1. Think about one of the fractals we've made in class.  They've all had some sort of iteration rule like:

• Take all of the straight lines in the shape and replace them with this shape:
• Repeat this process with every new straight line in the shape.

or maybe

• Connect the mid points of all of the sides of a triangle to get 4 new triangles, and erase the upside-down triangle
• Repeat this process with every new triangle you make.

You could make up your own new iteration rule by changing one of these rules (or maybe by coming up with something completely different).  You can get some ideas by looking at this web site: http://www.ph.biu.ac.il/~rapaport/java-apps/lsys.html. If you do this, you should:

• Draw out at least the first 3 iterations of your fractal.
• Write an explanation of the iteration rule you used to make your fractal
• On your third iteration picture, circle parts of the fractal that are similar to (same shape but smaller) the whole fractal.

2. Think about/read about the copy machine reduction fractal in the book and that we looked at in class.  This web site: http://static.pauldoo.com/fractals/IteratedFunctionSystem.html will let you make your own multiple copy machine reduction fractal.  To use this fractal making tool, you use the Add New Transform button to make several quadrilaterals, and position them on the screen.  The applet will make the fractal for you.  If you're going to do this, you should:

• First, play around and make a fractal that you like.
• Next, draw out by hand at least 2 iterations to show that you understand what the web site is making. Here's an example of me doing that for a fractal I made.
• Print out a copy of the fractal the computer made (including showing the rectangles) (you will either need to print directly from the web page, or you will need to use a screen capture utitility--you won't be able to copy and paste from the web site)
• On your print out, circle parts of the fractal that are similar to (same shape but smaller) the whole fractal. (Your self-similar parts will probably be a bit distorted compared to the whole thing unless you only use squares, but that's OK)

The non-creative option:

• Print out this worksheet.
• Go to a Mandelbrot-Julia web site like this one: http://gingerbooth.com/coursewareCBC/mandeljulia.html
• Click around on the Mandelbrot set, and observe the Julia sets that go with each point (if that isn't making any sense to you, maybe you should read about the Mandelbrot set in the book). 
• Fill out the worksheet, matching each marked point on the Mandelbrot set to its associated Julia set.
• Now, there should be one Julia set left that doesn't have a point associated with it. Click around on the Mandelbrot set until you find the missing Julia set.  Then draw in that point on the Mandelbrot set, and label it something like w, so I can see where you found it.