Sierpinski Relatives

I have been looking at the Sierpinski Relatives for a few weeks now, trying to work out the details in some of the more interesting examples. Questions that keep coming up are:

• How can we determine that the fractal is connected?
• Is it possible that there is more than one component and there are holes?
• How can we determine that there are holes?
• Why do straight line segments show up- is it easy to determine different classes based on straight lines?
• What other criteria can we use to determine different classes?
• How can we use the self-similarity of the fractals to answer these questions?

And working out some examples by hand is rather tedious, I really need to figure out how to generate images using Maple and/or Mathematica. The more I look at examples the more complicated things seem- examples that I assumed were clearly connected don't seem as clear any more. This is frustrating but perhaps a good thing because it means that the structure is a lot more interesting. All these fractals with the exact same fractal dimension but very different geometrical and topological properties. So the broader goal is to use this class to come up with new ways to classify and characterize fractals that are different from dimension. Of course I could try the epsilon-neighbourhood approach that I used for the fractal trees, but in that case I could work out the specific details for where the points on a tree were. So far I am not able to do the same thing with the relatives, and I want to try to get away from that kind of number-crunching. I think using the self-similarity is the key.

I have finally introduced the students to the relatives so they can start looking at examples. We spent the first few weeks of May going over background, mostly the first part from Barnsley's "Fractals Everywhere". It is a great book, but I started to feel that maybe the students should just dive right in and play with the fractals before knowing all the mathematical details. I want them to have an intuitive idea about certain concepts like "compact" or "metric" or "affine map" rather than just go through the strict definitions. I am hoping that now that they have specific examples to play with it will be more interesting for them. There are so many little quirks about the relatives, I think we will all make many little observations (hopefully).