The Mandelbrot Set

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800px-Mandelbrot_set_with_coloured_environment.png

The most well-known of the fractals is the infamous Mandelbrot set. It’s one of the first things that was really studied as a fractal. It was discovered by Benoit Mandelbrot during his early study of fractals in the context of the complex dynamics of quadratic polynomials the 1980s, and studied in greater detail by Douady and Hubbard in the early to mid-80s.

It’s a beautiful example of what makes fractals so attractive to us: it’s got an extremely simple definition; an incredibly complex structure; and it’s a rich source of amazing, beautiful images. It’s also been glommed onto by an amazing number of woo-meisters, who babble on about how it represents “fractal energies” – “fractal” has become a woo-term almost as prevalent as “quantum”, and every woo-site that babbles about fractals invariably uses an image of the Mandelbrot set. It’s also become a magnet for artists – the beauty of its structure, coming from a simple bit of math captures the interest of quite a lot of folks. Two musical examples are Jonathon Coulton and the post-rock band “Mandelbrot Set”. (If you like post-rock, I definitely recommend checking out MS; and a player for brilliant Mandelbrot set song is embedded below.)

So what is the Mandelbrot set?

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Take the set of functions
f_C(x) = x^2 + C where for each f_C, C is a complex constant. That gives an infinite set of simple functions over the complex numbers. For each possible complex number C, you look at the recurrence relation generated by repeatedly applying f, starting with x=0:

  1. m(0,C)=f_C(0)
  2. m(i+1,C)=f_C(m(i, C))

If m(i,C) doesn’t diverge (escape) towards infinity as i gets larger, then the complex number C is a member of the Mandelbrot set. That’s it – that simple definition – repeatedly apply f(x)=x^2 + C for complex numbers – produces the astonishing complexity of the Mandelbrot set.

If we use that definition of the Mandelbrot set, and draw the members of the set in black, we get an image like the one above. That’s nice, but it’s probably not what you expected. We’re all used to the beautiful colored bands and auras around that basic pointy black blob. Those colored regions are not really part of the set.

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The way we get the colored bands is by considering *how long* it takes for the points to start to diverge. Each color band is an escape interval – that is, some measure of how many iterations it takes for the repeated application of f(x) to diverge. Images like the ones to the right and below are generated using various variants of escape-interval colorings.

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My personal favorite rendering of the Mandelbrot set is an image called the Buddhabrot. In the Buddhabrot, what you do is look at values of C which *aren’t* in the mandebrot set. For each point m(i,C) before it escapes, plot a point. That gives you the escape path for the value C. If you take a large number of escape paths for randomly selected values of C, and you plot them so that the brightness of a pixel is determined by the number of escape paths that cross that pixel, you get the Budddhabrot. It’s fascinating because it reveals the structure in a particularly amazing way. If you look at a simple unzoomed image of the madelbrot set, what you see is a spiky black blob; the actually complexity of the structure isn’t obvious until you spend some time looking at it. The Buddhabrot is more obvious – you can see the astonishing complexity much more easily.

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0 thoughts on “The Mandelbrot Set”

  3. G Barnett

    So, is there a variant of the Mandelbrot that occupies a volume instead of the usual one which is on a plane?
    Sorta curious as to what something like that would look like.

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  4. Dan Vanderkam

    One of the problems with the Buddhabrot is that it’s completely non-local. In order to draw a single pixel, you have to calculate orbits for every other point. This makes zooming a pain. I rendered a huge “Nebulabrot” (10240×7680) a while back, see http://www.danvk.org/wp/?p=126 for details.
    Another cool thing about the Mandelbrot set: you can calculate an equation for the central lobe, secondary lobes, etc. I’ve never fully understood the math, but hey, that’s your job, right? =)

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  5. John Marley

    For some reason, I had a very hard time getting it into my head that the black areas are points in the Mandelbrot set, and the colored areas are not.

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  7. Lepht

    holy shit, the Buddhabrot is beautiful! and Dan Vanderkam, your Nebulabrot ain’t far behind, man. damn, that’s pretty math.
    Lepht
    (thanks MarkCC)

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  8. Torbjörn Larsson, OM

    How does one go about calculating the area of the Mandelbrot set?

    Mathematician: “Form the closure of the set, … um, wait.”
    Physicist: “Take the model systems lagrangian, … um, wait.”
    Engineer: “Weigh the inc cartridge before and after the plot.”
    [Note: The practical approach joke is just about ruined by computers. Now you would note the number of calculated set points to get a rough idea. Ah, well.]

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  9. Blake Stacey, OM

    Continuing the pattern established by Torbjörn Larsson, OM, one might say that the Web surfer will instruct you, “Google mandelbrot set area and click on the MathWorld link.” Yes, we true Web surfers can speak in hypertext.
    Intriguingly, it looks like the area computed by point-sampling (1.506 or thereabouts) is significantly smaller than the area expected on analytical grounds (between 1.66 and 1.71), although it is consistent with the upper bound established by theory (1.7274).
    The links at the bottom of the MathWorld article also look like interesting reading.

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  10. G Barnett

    Well, looks like a Nebulabrot is gonna be my new desktop pic for a while. That thing’s just insanely nice.

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  11. Sepia Mage

    @G Barnett
    The Mandelbrot set is defined on the complex plane, and there’s no 3D equivalent of the complex numbers. (In fact, there may be a theorem that there can’t be one. I can’t really remember.) There is a 4D equivalent though, called the quaternions.

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  12. genie arnold

    Do American mathematicians generally pronounce “Mandelbrot” with the “t”?
    Surely that’s not the French way.

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  13. clem

    Mark, if you are doing a fractal series, may I suggest a post or two on Iterated Function Systems? They may not have made the splash the early hype suggested but it is very simple to code up an IFS that creates Sierpinsky triangles and gaskets, Koch snowflakes, the famous fern and others and watching the thing develop from random points and rotations and contractions feels magical.

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  15. Torbjörn Larsson, OM

    Blake,
    Ironic for my unresearched joke that there is a closed form solution, but that it instead converge veeeeeery sloooooowly, so the practical solution is, ehrm, practical. (I think that was the appropriate math speak. Also known as “v(Ne)ry sl(No)wly, for N large”.)

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  16. Doug

    “Those colored regions are not really part of the set.”
    So, fractals really don’t have imprecise boundaries (fuzziness). Well, maybe fuzzy fractals do, but that’s a different story altogether.
    I wonder… why have so many mathematicians/artists represented fractals with those colored boundaries instead of including another picture of how long it takes things to diverge?

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  17. AdrianCockcroft

    There is also a band called Fractal, who play a sub-variant of prog rock known by some as math-rock. The music is characterized by extremely complex and varied time signatures, rather than overt usage of fractal mathematical forms. Other math-rock bands worth checking out include Bubblemath and Headshear. Fractal’s first CD “Continuum” came out in 2003 and they just released the first track from “Aftermath”. See http://www.fractal-continuum.com or (for the most free music tracks) http://fractal.bebo.com or http://www.myspace.com/fractalcontinuum

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  18. Mark Dow

    Here’s a plug for Cantor (e.g. Cantor set), and Peano and Hilbert (space-filling curves) and others who studied fractal properties long before the Mandelbrot set was studied. They recognized and specified the fractional dimension nature of these respective figures long before Mandelbrot used the word fractal. They might have framed the problem as studying the properties of numbers (number theory), but their techniques and terminology were isomorphic to many modern dynamical systems techniques.

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  19. Xanthir, FCD

    Doug: It’s because the colored boundaries are so much prettier!
    Seriously, that’s it. They look cool.

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  20. Eric Baird

    Re: post #3, “3D Mandelbrots”
    Well … There’s a solid fractal shape that’s based on a combination of the Mandelbrot set and the related “Tricorn” fractal, set at 90 degrees to one another: together these produce a complex surface that can be “cut” to produce either the standard Mandelbrot set or the Tricorn.
    Along with the Buddhabrot, that’s about the closest thing I can think of to a 3D Mandelbrot.
    Pics and Video
    Of course, you could try spinning a Mandelbrot on its axis, but that’s pretty boring.

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  21. Ruth Kimber

    Just read Death Qualified by Kate Wilhelm which was a novel about a mathematician who used the Mandelbrot sets to alter behaviour psychotically causing insane behaviour. It was a good novel and caused me to end up at this site searching about the Mandelbrot sets. Fantastic stuff. the Nebulabrot was beautiful. What more can you say.

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