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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 !

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    $\begingroup$ To start and get some non-rigorous background on fractals I would suggest watching this video youtube.com/watch?v=gB9n2gHsHN4 $\endgroup$
    – Varun
    Aug 18, 2017 at 1:03
  • $\begingroup$ Are you in college? Or university? Or where? (I'm surprised you are to write a paper on fractals, but yet haven't before encountered the concept of base 3 numbering systems as shown in comments on your other question. How did that happen?) $\endgroup$
    – Wildcard
    Aug 31, 2017 at 4:45
  • $\begingroup$ I know what base 3 is, and what logarithms and exponents are, I just didn't know (or I probably just forgot at some point) that subscript meant base. I mean, I'm just so used to saying log BASE something, that when someone writes a number under another number, I get/got confused. It happens to the best of us! I'm in highschool. Base 2 is pretty interesting, and doing math in other bases really screws with how you see numbers. I just made a mistake, as is oft to happen when you're tired and you work on your papers instead of homework. Thanks again! $\endgroup$
    – user463013
    Aug 31, 2017 at 4:52

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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.

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  • $\begingroup$ thanks! I've seen that fractals are related to chaos theory and topology, so it really helps to have your references ! $\endgroup$
    – user463013
    Aug 18, 2017 at 1:07
  • $\begingroup$ No problem, I used this book for my Dynamical Systems class in undergrad. It is fantastic. $\endgroup$ Aug 18, 2017 at 1:13
  • $\begingroup$ Do you have any nagging questions related to fractals? I don't completely understand the idea of fractional dimensions, but what really interests me is how parts of the mandelbrot set look like parts of the julia set. But that just leaves me with the question of how the mandelbrot set and it corresponding julia sets are related, and how julia sets and fatou sets are related. I have to go into depth with my topic, but it also has to be explainable. I'm a student tutor, so I'm used to this predicament, but unfortunately it's still a predicament. I hope I understand this enough to explain it! $\endgroup$
    – user463013
    Aug 18, 2017 at 1:47
  • $\begingroup$ For fractal dimension, consider reading a little on wikipedia regarding Hausdorff dimension. At the moment I'd have to review fractals to consider whether I have any further questions, but the mandelbrot set looks like parts of the julia set because the julia set is actually part of the mandelbrot set or vice-versa in a way. I don't know much about Fatou sets unfortunately, that relates more to the topological aspect of fractals. $\endgroup$ Aug 18, 2017 at 1:54
  • $\begingroup$ For direct application see en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension $\endgroup$ Aug 18, 2017 at 1:55
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  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.

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  • $\begingroup$ Thanks a lot! You're comment really helps! I'll have to look into theory behind complex dimensions, and all your recommendations! I might look into writing a definition for fractals - I'm actually writing another paper on Zeno's paradox - specifically the dichotomy paradox, and how it relates to the epsilon-delta definition of a limit. Papers never end, do they? Oh well, thanks for the help. P.S. Do you know how to change my username? It bugs me that I'm just a random user with a number. Thanks! $\endgroup$
    – user463013
    Aug 18, 2017 at 17:44
  • $\begingroup$ Glad I could help. No idea how to change your username---there is probably something in your profile that will let you do it. $\endgroup$
    – Xander Henderson
    Aug 18, 2017 at 17:46
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See the classic books: The Beauty of Fractals and The Science of Fractal Images.

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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. ..

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