Help with writing a paper on fractals

Question

I have to write a paper on fractals, and my general idea is to focus on the mandelbrot set. Do you have any idea what I need to know? I have to be able to explain the ideas to a layman, but I also want to explain fractals in-depth.

Some difficulties in research have been that fractals feel less rigorous than any other field of math I've studied. They also go into topics of topology I'm not used to. I'm still trying to understand why the mandelbrot set is bounded in a circle from $2$ to $2i$ in the complex plane, but I know it's related to $f_{c}(z)= z^2 + c$ Some interesting parts of research have been the idea of using mathematical visualization to explain ideas why certain areas look the way they do.

My general question is: Do you know where I should research? What do I need to read to write an awesome paper? Thanks !

Answer 1

Check out Chaos-An Introduction to Dynamical Systems by Kathleen T. Alligood, Tim D. Sauer, and James A.Yorke. In it you'll find resources regarding the quadratic map as a function of a complex variable which generate the Mandelbrot set, also related to the Julia set.

Answer 2

As far as i know, the mandelbrot set has a famous recurring figure in it, known as the apfelmensch. It looks sort of like a man. It is quite beautiful, and like fractals in general, possesses the self similarity property, that when you magnify it a certain number of times, it reappears. I just thought I'd make sure it was brought to your attention... At one time I had a good book called fractals everywhere, by Michael barnsley. I believe he introduces gromov-hausdorff distance, the fractal dimension (hausdorff dimension, or whatever it's called; I believe it's fractional ), and other related concepts. (Speaking of dimension, the cantor set has dimension $\frac{\log2}{log3}$, just as an example. ..) Also james gleick has a book on chaos, etc... There are apparently quite a few good books on the subject. ..

Answer 3

See the classic books: The Beauty of Fractals and The Science of Fractal Images.

Answer 4

1. You state that "...fractals feel less rigorous than any other field of math I've studied." This is actually a rather insightful comment, and opens up an interesting can of worms for a paper topic (should one decide to go down that route). It turns out that the term "fractal" is not really well-defined in mathematics.

   On the interwebs, you will see a lot of "definitions" that refer to the self-similar structure of a fractal, or some notion of complexity at all scales. This gives a good intuition, but is hard to make rigorous.

   The first attempt at a rigorous definition with which I am aware is Mandelbrot's. If I recall correctly, Mandelbrot originally argued that a fractal should be any set with Hausdorff dimension differing from its topological dimension. This definition had a short life, as it was quickly pointed out that there are many objects that should be considered "fractal" that don't satisfy this definition.

   Since then, others have argued that a fractal should be any set that has "interesting" dimensional properties. In particular, we have many notions of dimension (topological, Hausdorff, Minkowski, Assouad, box, packing, etc); a fractal should be a set for which at least two of these notions disagree (e.g. a set with Hausdorff dimension strictly smaller than the upper-Minkowski dimension). Again, this fails to find all things that we might consider fractal, and includes things that we might not want to consider fractal (such as the closure of the set $\{ n^{-1} : n\in\mathbb{N}\}$).

   My advisor would argue that a fractal is any set that possesses at least one non-real complex dimension. This seems reasonable to me, but then you have to get your head around the theory of complex dimensions, which is no mean feat.

   At any rate, I would argue that the perceived lack of rigor is not a lack of rigor in the analysis of the sets that we would like to call fractal, but in the squishiness of the definition.

2. In addition to the books that others have recommended, you might also want to check out:

   • Barnsley, Michael, Fractals everywhere, Boston, MA etc.: Academic Press, Inc. xii, 394 p. £ 28.00; {$} 39.95 (1988). ZBL0691.58001.
   • Mandelbrot, Benoit B., The fractal geometry of nature. Rev. ed. of “Fractals”, 1977, San Francisco: W. H. Freeman and Company. 461 p. (1982). ZBL0504.28001.
   • Falconer, Kenneth, Fractals. A very short introduction, Very Short Introductions. Oxford: Oxford University Press (ISBN 978-0-19-967598-2/pbk). xv, 132 p. (2013). ZBL1279.28001.
   • David, Guy; Semmes, Stephen, Fractured fractals and broken dreams. Self-similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications. 7. Oxford: Clarendon Press. ix, 212 p. (1997). ZBL0887.54001.
   • Strogatz, Steven H., Nonlinear dynamics and chaos. With applications to physics, biology, chemistry, and engineering, Boulder, CO: Westview Press (ISBN 978-0-8133-4910-7/pbk; 978-0-8133-4911-4/ebook). xiii, 513 p. (2015). ZBL1343.37001.
     
     

   These are more-or-less ordered by "friendliness"---Mandelbrot and Barnsley are both pretty readable for the uninitiated, but neither is terribly rigorous. Falconer has written a number of books on the topic (including what I consider to be the bible of fractal geometry, The Geometry of Fractal Sets, but that is a relatively hard read unless you already know some real analysis and measure theory)---the "Very Short Introduction" is a good way to get your feet wet. David and Semmes is more rigorous, but still fairly approachable. The book is (I think) more focused on self-similarity and geometry than on dynamics, so it may not be quite what you are interested in. Finally, Strogatz is a nice introduction to dynamical systems, though fractality is not really the point of the book, so, again, your milage may vary vis-a-vis relevance.