Anyone familiar with fractals?

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In summary, the conversation discusses the topic of fractals and the challenges of writing a 12 page essay on the subject. The speaker has researched and identified basic properties of fractals and provided examples such as the Koch curve and Cantor set. They are also looking for other properties and related material, as well as applications of fractals. Suggestions for further research include exploring the mathematical foundations, the use of fractals in CGI and turbulence theory, and their presence in nature. The conversation also mentions IFS as a potential topic to cover in the essay.
  • #1
blahblah8724
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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 :)
 
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  • #2
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.
 
  • #3
blahblah8724 said:
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.
 
  • #4
blahblah8724 said:
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.
 
  • #5
Fractals in nature could also be an interesting route to go down.
 
  • #6
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.
 
  • #7
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)...
 

1. What are fractals?

Fractals are complex geometric shapes that exhibit self-similarity at different scales. This means that the same pattern repeats itself at a smaller level, creating an infinite level of detail within the shape.

2. How are fractals used in science?

Fractals have many applications in science, including modeling natural phenomena such as coastlines, clouds, and mountains. They are also used in computer graphics, data compression, and chaos theory.

3. What is the Mandelbrot Set?

The Mandelbrot Set is one of the most famous fractals, named after mathematician Benoit Mandelbrot. It is a set of complex numbers that, when iteratively applied to a function, create a complex pattern that resembles a snowflake or a fern.

4. Are fractals found in nature?

Yes, fractals can be found in many natural phenomena such as trees, rivers, and even in the human body. They are also found in the structure of galaxies and the distribution of stars in the universe.

5. How are fractals related to chaos theory?

Fractals are closely related to chaos theory, which studies the behavior of complex systems that are highly sensitive to initial conditions. Fractals demonstrate chaotic behavior, as small changes in input can lead to vastly different outputs.

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