Cantor sets and Connectivity

Well I have not been good at keeping this updated... but that does not mean that we haven't been working hard. We have finally been looking at examples of the sierpinski relatives. At first we looked at things like why are there straight lines. Lately I have been looking at how certain Cantor sets can help determine the connectivity. It does not work in every case, but it is helpful for some, and I think the Cantor sets are interesting. Instead of the usual middle thirds Cantor set that can be related to ternary numbers between 0 and 1, now we have other Cantor sets, often related to base 4 or binary. Since it is already July (eek!) I think the goal now should be to try and writeup what we have done. So the next task is to come up with an outline for a paper. The paper needs to be in LaTex, so the students will have to spend some time figuring that out. I know that this can be very slow! But it will be a good skill for them to develop. They were very fast at figuring out how to generate the Sierpinski relatives with Maple, so perhaps it won't be so bad. I am used to using WinEdt to edit LaTex, but now that I have a Mac I should try to use something else... so that is something I need to do as well. I am really tempted to stay with what is familiar though...

On a somewhat unrelated note, last week I went to a Master's thesis defense for which I was the external examiner. The thesis was on the topological properties (connectedness and disklikeness) of tiles and digit sets. It was very interesting, and reminded me how useful iterated function systems can be. The student used "Beamer" to generate slides with LaTex, so I learned something new! I am glad I asked... I feel like a bit of a loser for not knowing about it already, but I think it will be helpful in the future.

Tomorrow I meet with the students and we will try to come up with an outline for a paper. And I will talk about LaTex.

I really should update this more often, because I have left a lot of details out about what the students have worked on...

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).

Fractal Summer 2009 begins...

I've been going through the book "Fractals, Graphics and Mathematics Education" by Mandelbrot and Frame. There are many interesting and accessible articles in it, and some useful links are also provided. Unfortunately most of them are no longer available.



The only link that I could still find up is Robert Devaney's:

http://math.bu.edu/DYSYS/dysys.html

Actually Robert Devaney has many interesting links that you can get to from his homepage:

http://math.bu.edu/people/bob/



A wish-list of activities/topics for the summer students:

• MathSciNet- get them to use this to search for research papers


• Google Scholar- another good source for finding research papers


• Mathematica- a software program that is useful for writing programs for fractals. I already have a program for binary trees.


• Explore binary fractal trees: symmetric, asymmetric, general- this can be done using mathematica, but also by going through papers (Mandelbrot and Frame, Brown, etc)


• LaTex- learn to write basic articles


• Sierpinski relatives- explore, make observations


• Learn basic background about fractals: contractive mappings, IFS, fractal dimension, self-similarity


• Learn basic background about topology: notions like connectedness, simply vs multiply connected, how topology can distinguish between 2 fractals that have the same dimension


• Learn about the golden ratio and connections between fractals and the golden ratio- my paper for the MAA book is a good start


• Learn about applications of fractals- what is interesting?


• Look at interesting examples/classes of fractals


• Antoine's necklace


• Indra's Peals


• Chaos Game

Some goals:

• Encourage students to pursue what interests them, to find their own questions


• Prepare for some kind of presentation- either at APICS or StFX student research day (or both)