The Mandelbrot Set

An overview of the Mandelbrot set.

One of my favorite mathematical entities is a creature called the Mandelbrot set. Because of its complexity and the huge number of computations needed to compute or visualize the Mandelbrot set, it was not discovered until the late 20th century. The set was first visualized by Robert Brooks and Peter Matelski while they were working in a branch of abstract algebra. Starting in the late 1970s, Benoît Mandelbrot published some results regarding the set in relation to other research. Other mathematicians who worked with the set through the 1980s and ’90s named the set in his honor [1].

Since its discovery, images of the Mandelbrot set have made their way out of the mathematical ivory tower, and have appeared in popular culture—I have personally seen it on the covers of books (including, but not limited to, high school mathematics texts), on posters, in album art, printed on clothing, and in myriad other places. Clearly the images themselves speak to people.

In an effort to allow a lay audience to gain a deeper appreciation for images that they have almost certainly encountered, I have written several posts describing the Mandelbrot set. I would love for anyone with an interest in the Mandelbrot set to read through the series, so I have tried to make it easy to find by placing this page in a place where it is accessible from the front page.

The Mandelbrot Set Series

Source Code

For anyone who is interested, the source code for my fractal generating program is available. Please note that it is almost entirely undocumented, quirky as all hell, and likely riddled with bugs (off the top of my head, I can think of at least one fairly major bug in the way that bitmaps are read in as color palettes)—programming is not one of my strong suits, and the program has developed in a very piecemeal manner to provide me with the functionality that I require. I am happy to provide the parameter or color files used to generate any of the images on this website, assuming that I still have them. Feel free to send an email.

Have fun!

Notes:

[1]
Wikipedia contributors, “Mandelbrot Set,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Mandelbrot_set (19 November 2010).

11 Responses to The Mandelbrot Set

  1. Hi. Thx for nice images and src code.
    I see that you are graduate student in mathematics.
    Can you make images like that :
    http://www.math.nagoya-u.ac.jp/~kawahira/gallery/tiles/tiles.html
    ?
    I can’t so if you can please describe how to do it ?

    Best regards

    Adam

    • Xander says:

      To be honest, my Mandelbrot and Julia images are a long distance from my current area of research, which involves something called the Assouad dimension, and has been focusing more on self-similarities. I’m also only in my first year, so haven’t gotten as deep into things as I might. If you are interested in what Kawahira is doing, your best bet would probably be to read the thesis and paper linked at the top of the page you linked to.

      xander

  2. Graeme McRae says:

    Thank you for posting such a large and detailed fiery red and yellow view of the points not quite in the Mandelbrot Set. (Of course, the boring black bits are the set itself, including some cuddly litttle Mandelbrotlets being incinerated in the fire!) I “borrowed” a small part of your image to use as my Google+ banner, I hope you don’t mind. In the caption, I credited the page of your website where I found the image.

  3. Will says:

    Hi. I love your images – fractals have always been a source of wonder for me. Mathematics was a rather dry subject for myself at school I must admit, and fractals visual depict the wonder of mathematics in a form I can comprehend hahaha..

    I am a drumer in a band in New Zealand, and I am putting together a little music video project (just depicting the bands lyrics) and was wondering if I could use some of your images as the background. I may be mirroring them in a way to make them symmetrical and moving in the background etc… it’s completely non-profit of course – we don’t earn enough to even pay for drumsticks but we do what we can ;) Would that be Ok? I will place a credit and a link in the description of course. Your fractals are some fo the best I have seen.

    Thank you and best regards, Will.

  4. Michael Rothwell says:

    Hi. Xander
    What’s the difference between the functions J_K(z) and M_K(z) described on the Julia Sets post?
    Regards
    Michael

    • Xander says:

      The constant \(c\) in \(M_c(z)\) depends on the initial value. It perhaps might have been better if I had written \(M_{z_0}(z) = z^2 + z_0\). Essentially, a different function is used every for every initial value \(z_0\in\mathbb{C}\). On the the other hand, the value of \(K\) in \(J_k(z)\) is fixed—i.e. \(J_K(z) = z^2 + K\) for all initial values \(z_0\in\mathbb{C}\)—then the function is applied iteratively to every point in the complex plane.

  5. David Bock says:

    I just wanted to let you know, I found your Mandelbrot blog entries last week prepping for a presentation I was doing at a conference (railsconf2015) and I included a mention of your stuff for further reading… I’ll send you a link once the video is available.

  6. Cameron Palmer says:

    It would be very nice if the config files for any or all of the images in your images were added somewhere, even GitHub. I’m trying to figure out how you make the beautiful color transitions. I personally seem to consistently generate these hard color transitions.

    thanks.

    • Xander says:

      Unfortunately, I am not a great programmer and the code that I’ve written is pretty much for my own use. What this means is that whatever configuration files I still have are unlikely to work with the current version of the program. The color transitions come from spending a lot of time hand-tweaking the bitmap files that I use as input.

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