How Can I Expand My Essay on Fractals?

[blahblah8724]

Part of our core mathematics course is a completing a 12 page essay on a chosen topic and I decided to write about Fractals. After a bit of research I started to question whether there was enough material at undergraduate level to be able to complete a comprehensive essay on the subject.


What I have so far is this:

1. Define the basic properties of Fractals such as the Hausdorff dimension and the Topological dimension.

2. Introducing a few examples of fractals such as the Koch curve and the Cantor set.

3. Applying the basic properties in 1. to the examples given i.e. computing the Hausdorff dimension of the fractals etc..


This doesn't seem enough to be able to fill 12 pages of maths on the subject and I was wondering if anyone else knew any other properties / related material or perhaps books on fractals which are an undergraduate level?

Also, does anyone possibly know any applications of fractals?

Help would be really appreciated!

Thanks :)

 

[turbo]

What are the mathematical foundations behind fractals? Have you studied complex numbers well enough (real and imaginary components) so that you can explain them in a paper? What relation does self-similarity have to the math? Also, if you start with the basics (Mandelbrot set) why does self-similarity break down as you look at smaller and smaller areas of the set? Self-similarity is still there, but every smaller and smaller area turns out to be unique. You could write a whole book about this.

 

[AlephZero]

Also, does anyone possibly know any applications of fractals?

The use of fractals in generating CGI images for movies and video games should be good for at least 12 pages.

 

[bigfooted]

Also, does anyone possibly know any applications of fractals?

Fractal theory can be used to explain some fundamental concepts in turbulence theory.

Some basic fractals are the result of applying numerical solution methods under different starting conditions, like in Newton's method for example to generate the Julia fractal. So they can be interpreted as visual pictures of the convergence of Newtons method.

 

[MrB8rPhysics]

Fractals in nature could also be an interesting route to go down.

 

[I like Serena]

Mandelbrot started his research with land measurements.
It turned out that the length of a coast line could not be measured accurately.

I imagine it is used for measurements of coast lines and such.

 

[micromass]

Hausdorff dimension and topological dimension is not enough to fill 12 pages? I wonder how small you wrote...

Maybe you can also do something on IFS (= Iterated Function Systems)...