The Mandelbrot Set Viewed through Precalculus

I’ve given this presentation in November 2014 at AMTNYS in Syracuse, and in April 2015 at NCTM in Boston. I’ll be giving the presentation in May 2015 at the NY Master Teacher Conference, and in July 2015 at Twitter Math Camp in Claremont CA. 

Drive

Why use the Mandelbrot fractal to get the students to learn some Precalculus topics?

  • How often do students study math that less than 40 years old? Think about it.
    What’s the most recently developed math topic that happens in your classroom? The Mandelbrot fractal was developed (discovered?) in the late 1970’s by Benoit Mandelbrot (amongst others, more information here) while working at IBM and through working with the Julia sets. His collegues at SUNY Stony Brook* used computers to create the first image of the Mandelbrot Set:
    Mandel
    The first picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978/1979*.
  • There are so many great Precalculus topics addressed by the Mandelbrot set, namely:
    • Complex numbers
    • Arithmetic with Complex numbers
    • Complex plane and Argand diagrams
    • Recursive sequences
    • Polar Form of Complex numbers
    • Graphing using the Polar plane
    • DeMoivre’s Theorem

    By themselves they are interesting topics with many diverse uses, but together they can do marvelous things.

  • The utter beauty and mysterious nature of the Mandelbrot set (and other related sets). It seems so inapproachable. Scary. It’s an infinite complex process and has all the crazy fractal properties of self-similarity, but not really. You can zoom in forever and get more and more detail as you go. You can create (and understand?) images like so (all of these images came from my amateur code).
    savedPicture
    savedPicture7-11-2014-21442
    savedPicture9-4-2015-22241
    savedPicture21-4-2015-121912
    But these images are only finitely interesting. They are just pixel representations of the actual fractal. We can zoom in and get as much detail as we can handle (or more correctly, as a 32bit processing.org decimal float can handle – duh).
  • Lastly, be selfish, wouldn’t you like to learn something new alongside your students? Isn’t trying new things inherently a sign of youth? This might not be your thing, but if you’re interested in bringing it back to the classroom, the students will be that much more interested to see the teacher learning alongside their students.

How

Here are several good videos that can help you get started with the Mandelbrot set:

Next, you can walk through the lesson outline that I’ve made (pdf) and through the presentation that I gave for NCTM Boston (pdf).

Here are links to all the interactives:

Everything is also found here: bit.ly/mandelbrotfractal, including all the presentation material, the walkthrough, the interactives (and all the source code).

I had a ton of fun with this project and presentation, and I hope the students did too. Please let me know if you have any questions or comments. Thanks!

*Errors fixed thanks to a comment by J. Peter Matelski (one of the original Mandelbrot developers!) 

5 thoughts on “The Mandelbrot Set Viewed through Precalculus

  1. Please do not suggest that I and my coauthor Brooks were employees at IBM.
    We were postdocs at a research university: SUNY Stony Brook. Our work was
    done in late 1978 and January 1979.
    I appreciate your attention to this detail. I am sure you are aware that
    the priority issue remains controversial.

    1. My apologies! I’ve fixed the errors in the post. Thank you for the help. Clearly your work has been an inspiration to me (and hopefully my students). Thank you!

  2. Hey glad you like the anim who needs drugs when you have teamfresh in aenswr to your questions the Mandelbrot shape at the end IS a direct result of the formula, no clever video editing was used!If you zoomed into the same area as before you would end up hitting even more complex patterns found within the set. The set itself is INFINITELY DEEP! So you could just go on zooming forever as long as you stay on the boundary of the set. If you know your way around the fractal you could then end up arriving at another one of the infinite amount of smaller Mandelbrot sets that are found within the original. So in short yes you could just jump there and then carry on zooming.want to know more about the mandelbrot set? click this link The only problem that is faced when the set is magnified to this extent (ie: e214) is the amount of time it takes to render each image/frame. Because the set is calculated using a mathematical formula- that is made from complex numbers the deeper you go the more precision you need regarding the maths.so e214 is a short way of saying there is 214 numbers in the string after the decimal- ie: 0.56685687967578 ..(214 digits) want to know more about the math? click this linkyou also have to take into account the amount of iterations that are used to create the image. (each iteration is the result of the formula being repeated) so the more iterations you have also affects how long it it takes to calculate.here is an interesting link to a post i made explaining iterations a little more In short this means that although I was able to render the very first frame of the animation in a fraction of a second, the final frame took more like 18 hours to render.As for the camera angle, it is possible to change it but in this instance I did not. The Mandelbrot set itself is totally 2D although some parts appear to have a three dimensional feel It is as flat as a pancake, so if you changed the camera angle it would look a bit like google earth when you tilt the camera- it would strech the image but still be flat. Any angles you saw were a direct result of the formulaAs for the set up of my rendering my exact methods are under wraps but I can tell you that I lovingly create the animations on my beat up laptop- then I have a separate dedicated machine (running a linux operating system) for the rendering of the animation.I render all my animations with anti aliasing. This means that an animation takes longer still to render but in my opinion it is worth it.Dont know what anti aliasing is? click this link to find out!hope this helps your curiosity oceans of loveteamfresh20 Feb 10 at 1:56am

Leave a Reply

Your email address will not be published. Required fields are marked *