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Author Topic: The Mandelbrot set == infinite circles?  (Read 3701 times)
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Khaotik
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« on: January 25, 2011, 01:03:50 PM »

Is the boundary of the Mandelbrot set in complex plane really made of infinite circles? I mean, except for the centural cardioid, all other parts are separate circles.
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Kali
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« Reply #1 on: January 25, 2011, 03:00:58 PM »

As far as I know, yes. An endless chain of circles of infinite different sizes, describing all the possible 2D curves and forming infinite complex patterns. But keep in mind that the cardioid repeats itself in every minibrot copy.

(Please tell me anyone if you think I'm wrong)
« Last Edit: January 25, 2011, 03:03:21 PM by Kali » Logged

Khaotik
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« Reply #2 on: January 25, 2011, 03:21:22 PM »

If that's true, can we find a rule that generates all these circles and cardioids? Just like an
IFS fractal.
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Kali
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« Reply #3 on: January 25, 2011, 03:41:21 PM »

I don't think so, but who knows... I also have been thinking about a rule but to predict if a point will diverge or not, in order to obtain VERY deep zooms, with a limited amount of iterations. I wonder if it's mathematically possible...
« Last Edit: January 25, 2011, 03:46:49 PM by Kali » Logged

KRAFTWERK
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« Reply #4 on: January 25, 2011, 03:47:04 PM »

If that's true, can we find a rule that generates all these circles and cardioids? Just like an
IFS fractal.

...and exchange the circles for globes for the 3D transformation?  afro

Funny, I was thinking about (I think it was Daniel Whites idea) about building the "holy grail" 3D mandelbulb out of globes when I drove to work this morning.
(Dont know if the word globes are right, but you get the point wink )
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lkmitch
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« Reply #5 on: January 25, 2011, 04:20:43 PM »

Is the boundary of the Mandelbrot set in complex plane really made of infinite circles? I mean, except for the centural cardioid, all other parts are separate circles.

Except that they're not circles (except the largest one centered at -1).  And the boundary also includes the disks on disks, the disks on disks on disks, etc., and the dendrites, which lead to the midgets.  All those "excepts" rule out the possibility of a simple model (other than the actual equation) to define it all.  However, the main cardioid's boundary is well-defined and Mandelbrot himself came up with an approximate model for the sizes and locations of the disks around the cardioid.
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fractower
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« Reply #6 on: January 25, 2011, 06:43:06 PM »

I think someone has already answered this on the forum, but I could not find it. I hope I remembered this correctly.

Each circle can be associated with a zero of the iteration polynomial. I forget if the root is at the center of the circle.

The polynomials are defined as follows.

M_0 = z
M_1 = z^2 + z
M_2 = (z^2 + z)^2 + z
M_{n+1} = (M_n)^2 + z

Not all solutions are unique. Note z=0 is always a solution.

The order of the polonomial grows as 2^n so it blows up quite fast. I don't know of any way to calculate the size of the circle.
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Khaotik
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« Reply #7 on: January 26, 2011, 05:40:14 AM »

That's interesting. Thanks.
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jehovajah
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« Reply #8 on: January 27, 2011, 03:40:52 AM »

I hesitate to answer so definitely.
The circles are a description of what is called a cover of the mandelbrot set.

We have examples on this forum of different mandelbrots and mandelbulbs using differing ways of formulating the relationship. These often produce startling new images.

By a cover math types have done something you or i do everyday, we put something on we know the size of, and then look in the mirror and say whether we feel fat or not! grin

So we try to get to know something better by our covers, but our covers are not the thing we are covering.

I hope that makes sense.

If you change the bailout condition so that it is not a circle or sphere,but some other function, it impacts on the mandelbrot set.

For technical reasons which i won't go into, the type of curves found in the mandelbrot set are now called Roulettes. I still call them trochoids, but that's because i like the sound!

To generate the mandelbrot set you need a recursive definition. An explicit definition gives you a julia set.

Hope that helps and does not confuse.
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Adam Majewski
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« Reply #9 on: September 07, 2011, 03:37:45 PM »

Here is an image of boundaries of hyperbolic components with some description :
http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Components.jpg
HTH
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jehovajah
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« Reply #10 on: September 22, 2011, 01:40:54 AM »

Here is an image of boundaries of hyperbolic components with some description :
http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Components.jpg
HTH


Thanks for his link Adam. There is a lot to learn about the mandlebrot set still, and i have now got more detail to go on. Intuitively i feel that the set itself resists precise categorisation, even of its boundary because it is a fractal, and as we know with a fractal the possibilities are infinite! The fact that the boundary consists of closed curves,circle and cardioids, is intuitively what i would expect from the trochoidal relationships of the 2 main vectors z and c. As the power of the equation increases i would also expect to find further trochoid forms.

As your post shows it is quite a task to establish the intuition but here is a guy who makes it so intuitively obvious.
<a href="http://www.youtube.com/v/rz8A5l&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rz8A5l&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1</a>
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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