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Author Topic: Why does the Mandelbrot set work?  (Read 3035 times)
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Warp
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« on: May 03, 2007, 12:04:05 PM »

This question has puzzled me forever.

The Mandelbrot set has always fascinated me for one thing:
An incredibly simple iterative formula like "z = z^2 + c"
produces an astonishingly wide variety of organic shapes.
I have collected some of my findings in this page:

http://warp.povusers.org/snaps/fract/

  It's just incredible that all those images have been created
by simply iterating that simple formula and then coloring the
pixels according to how many iterations it takes to bailout.
Using a clever color palette produces these stunning images.

  If someone (who is not a math guru and has never heard of the
Mandelbrot set) was given the problem of iterating a formula like
"z = z + c", where c is a complex number in the complex plane and
then coloring the equivalent pixel according to the number of
iterations it takes for z to go outside a circle of a given radius,
and then this person is asked what would the result be like, he
would probably think a moment and guess, correctly, that the result
would be a series of colored concentric circles. As the formula is
iterated z simply grows in the direction of c until it goes outside
the circle and that's it.

  Now, tell this person "how about z = z^2 + c instead?". His first
guess will still most probably be "concentric circles, the radii
probably not being arranged linearly but in a quadratic way" or
something similar.
  Even if you point out that "z^2" doesn't actually just go in
the same direction as the previous iteration due to how complex
number multiplication works, he would still most probably just think
that the result is some simple pattern. If he thinks about that for
a bit he might perhaps guess that it maybe forms a spiral or something
like that.

  When this person is then shown what it actually produces, the
result is most astonishing and unexpected. For a complete layman
like me, who doesn't understand anything about complex number
dynamics, the result is most astonishing and unexpected.

  I just can't understand *why* that formula produces those results.
I just can't even begin to comprehend how it is even possible.

  If I had no experience whatsoever about the Mandelbrot set and
I was given any of the images on that webpage of mine I mentioned
above and was asked to guess the mathematical formula it was
produced with, and after having no idea the correct answer of
"iterating z = z^2 + c and coloring the result according to the
number of iterations until bailout" (more detailedly explained,
of course), I would not believe it. There's no way I could believe
it without actually trying it for myself by making a program which
tests the claim.
  The formula is just so astonishingly small for what it actually
produces. It's possible to write an executable (DOS) binary which
draws the Mandelbrot set in less than 100 bytes (I have actually
done that).

  I have tried to find the answer to this question in the internet.
I have failed miserably. There are tons of websites which explain
*how* to calculate the Mandelbrot set. That's not what I'm looking
for. There are also some sites which present some geometrical and
mathematical properties of the set (eg. related to the bulbs and
numbers of "antenna" branches in each, etc), and while interesting,
that still doesn't explain *why* it happens, so it doesn't answer
my question.

  Does anyone know the answer to this question, or any website which
explains it?
  And mind you, I'm not a mathematician. Most complex dynamics theory
papers would probably go well over my head. I would like a simpler
explanation to this phenomenon.
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lkmitch
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« Reply #1 on: May 03, 2007, 05:38:13 PM »

The short answer lies in considering the iteration of z = z2 + c to be an investigation of the dynamics of a simple system.  Then, the disks, midgets, branches, etc., are artifacts of the dynamics:  periodic behavior, stability, instability, chaos, etc.

Here are a couple of examples:

It's easy to see that there are lots of points with periodic orbits.  For example, iterate c = -1 and you get the orbit, 0, -1, 0, -1, etc.  How can one find those points?  For c = -1, the values repeat every second iteration, so solve z = (z2 + c)2 + c for c.  Taking into account the initial condition of z0 = c, you get c = 0 and c = -1, which are the "centers" of the main cardioid and the large disk to its left.

Why is the main cardioid a cardioid?  The boundary is the set of points for which fixed-point behavior (period = 1) is neutrally stable.  That is, points where z = z2 + c (fixed points) and where f'(z) = 2z has a magnitude of 1 (neutrally stable).  Again, solve this for c and you get the cardioid shape.

Obviously, the full story is much, much more involved than that, but just starting with periodic orbits and their boundaries can get you pretty far.

Kerry
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Nahee_Enterprises
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nahee_enterprises Nahee.Enterprises NaheeEnterprise
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« Reply #2 on: May 04, 2007, 12:13:05 AM »

I have collected some of my findings in this page:
    http://warp.povusers.org/snaps/fract/

First of all, greetings and welcome to this particular Forum!!    smiley

Secondly, on the web page that you list above, I agree with this portion of your first paragraph:
Quote from: Juha Nieminen (Warp)
One common problem with webpages which present zooms of the Mandelbrot set is that usually the people who have made them pay little to no attention to the color palette used in the images. This is why you so often see really ugly Mandelbrot set images...

But in the second paragraph, where you state this:
Quote from: Juha Nieminen (Warp)
This is often because people use very limited Mandelbrot drawing software which have very low-quality color palettes (often limited to just 256 colors)...
I must definitely disagree with you, since you probably have not really seen what can be done with some software which may be limited to only a 256-color palette.  I can only contribute such a statement to somebody that has either not used properly, been experienced enough, or been around long enough to see what others have done with such software.

If you care to have a list of websites where people have created some astounding images using only 256-color palettes, then I would be most happy to supply that to you for your own education, and hopefully enjoyment.    smiley

I do agree with you where you further state:
Quote from: Juha Nieminen (Warp)
...people are too lazy to search for good ones and instead just use the default palette offered by the program.  Another common problem is that people don't explore the set enough nor try to find truely marvelous zooms of it.

But this is usually true of the beginner and novice, or the hobbyist.  Those whom take fractal imagery a bit more seriously, once they have learned the capabilities of the software, no longer are "too lazy".  They tend to then take the time for searching/creating color palettes, and for exploring/zooming in other areas.

« Last Edit: May 04, 2007, 12:14:54 AM by Nahee_Enterprises » Logged

lycium
Fractal Supremo
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Posts: 1158



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« Reply #3 on: May 04, 2007, 09:06:58 PM »

still I would give that top honour to the Riemann zeta function but that's another story

a bit out of place as you've noted, but my vote goes to the monster group: http://en.wikipedia.org/wiki/Monster_group

of course it could be because i'm just taking my first steps into real algebra wink

Seems as if somehow very complex behaviour does not necessarily mean very complex laws.

the general correlation is undeniable though wink
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FractalMonster
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Posts: 21



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« Reply #4 on: May 15, 2007, 10:39:17 PM »

Warp, you have really in a very nice way expressed very fundamental questions,
questions about simplicity - complexety that thousends of laymens have asked
themselves. I, myselves, have travelled in the Mandelbrot set since 1990.
The results of articles I've read, own computer experiments etc, have resulted
in an illustrated series of articles, "The chaotic series of fractal articles",

http://klippan.seths.se/fractals/articles/

Of the more than 30 articles, I have devooted articles 10 - 14 to the combinatorics
of the M set. Note that I am also a layman, NOT a mathematician.

BTW. See in your gallery that you have REALLY a taste for interesting motives in the M set  cheesy

------------
Regards,
Ingvar
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