Fractals?

Alright, so this being my first real post I'll try and cover what I think have been some of the main points in class so far. To be honest I haven't gotten a great grasp on what a fractal actually is during class yet. I believe I've at least understood the concepts we've been going through thus far, and am assuming we are building to a definition.

Week 1 was spent as what seemed to be an overview of what the class would cover. Although I had a hard time keeping up with some things, I feel I have a good idea of what was going on at this point. I had a particularly hard time with "itinerary" and "orbits". They seemed to be rather abstract definitions that we just kind of breezed through.

One of the main items we covered in week 1/2 was the Middle Thirds Cantor Set. This is the set of points that upon each iteration of the sequence the middle third of the unit interval is removed from the set. Apparently this was our "first fractal". I thought that it was very interesting that this set had an infinite number of points, but a length of zero (by our definition of length).

I tried to read ahead on Chapter II about "Spaces". Spaces are one part of abstract algebra that I've always had trouble with. I can understand the spaces I can visualize, but tend to have a harder time with more abstract spaces, i.e. "metric spaces" and "code spaces". I feel that I have a light grasp of a metric space after discussing it in class, although, I could use more work with it. I though the books definition left something to be desired.

I'm very interested in the fact that dimensions apparently do not have to be whole numbers (If I understood what was said in class). This goes against everything I had thought previously, but made some sort of sense when we were discussing it. I believe the argument was that the Middle Thirds Cantor Set was not zero dimensional because it was an infinite amount of points, but also was not 1 dimensional, because it did not constitute a continuous line with any length.

Shaky Concepts :

1. Transformation
2. Self Similar
3. code space
4. itinerary
5. orbit