Fractal Homeomorphism

[Sizqu27]

Hi! I'm very new to this site, this is my first post. But I'm going into my last year of IB soon, and my extended essay choice is a challenging one.

The topic is a part of the chaos theory, but more geometric and proven than chaos. So my overall question has yet to be formed, but is along the lines of fractal homeomorphism and about the specific relationship between say the area of a disk before and after it has been transformed and between it's perimeter before and after it has been transformed, using the same (elementary) transformation. *This original suggestion is not mine, but recieved from M. Barnsley*

After looking into it, I do think its very interesting, and I was wondering if there is anyonen available on this site that would be able to help me with the transformation and a better understanding of the transformation and topic itself.

Thank you!

Sara

[aldld]

That does sound like it would be an interesting topic to look into. Though you'd have to be specific. When you say "fractal homeomorphism" do you mean a homeomorphism between the n-disk (or n-ball? n-sphere?) to some fractal set (in whatever cases (if any) where that would even exist). I gather from your post that you're looking at relating topological properties to geometric properties and measure.

As a starting point, for math it can be helpful to look at the trivial cases first and see what you can build up from there.

[macrofire]

If only you would bother looking for isomorphisms...if possible, that would be cool.

From what I gather, you're trying to investigate the change created by some mapping.

I'm wondering where the "fractal" component comes in, but you might want to look at rigid motions that fulfill your requirements...then look at projective geometry to see the difference in the image of whatever object you're trying to transform.

Like aldld said, look at the trivial case to investigate what, if any, implications are present.