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Author Topic: Mandelbrot set in "cross-sections of magnetic field borders" ?  (Read 9517 times)
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Chillheimer
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chilli.chillheimer chillheimer
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« on: January 27, 2014, 12:31:19 PM »


An old but still great site for information about fractals is
http://www.miqel.com/fractals_math_patterns/visual-math-mandelbrot-magic.html

There I found the following statement:

Also Mandelbrot curves have been discovered
in cross-sections of magnetic field borders,
implying there is a 3-D mandelbrot equivalent that is closely tied to electromagnetism and therefore a deep structural and fundamental
aspect of life, and physical space/time.
(note: I read this in "Turbulent Mirror" can anyone cite a reference for this?


Now THAT would be very cool, as it would mean that fractals are indeed all around and within us, as the earths/suns magnetic field are all around us. Or on tapes, loudspeakers, harddrives... everywhere the m-set?
I knew it's everywhere in nature - but this would add a whole new dimension, in my opinion.

I mailed Miquel if anyone was able to cite a reference for this, but haven't received an answer. This short statement is all I could find on the web.

Does anyone here know more about this?
Or does anyone have the book "Turbulent Mirror" and could cite/scan the part where it says so?

I really need to know this!  shocked
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hobold
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« Reply #1 on: January 27, 2014, 04:32:30 PM »

I vaguely remember that minibrots are known to appear in simulations of specific magnetic models. In other words, there exist formulas in physics that describe behaviours of magnetic fields, which when implemented in computer simulation result in shapes that look like the classical Mandelbrot fractal (cardioid, bulbs, and antenna).

I am unaware of physics lab experiments which would create Mandelbrot shapes in the real world. The aforementioned simulations might be a case of very exotic boundary conditions that cannot be reproduced in any lab.
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stereoman
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« Reply #2 on: January 27, 2014, 06:49:10 PM »

This recalls me some Escher drawings, where  boundaries disintegrate in smaller iterations.
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kram1032
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« Reply #3 on: January 27, 2014, 11:50:10 PM »

I suppose there are some similarities between fractals and hyperbolic geometry (which is what those of Escher's drawings are that "disinitegrate" in "smaller iterations") though that doesn't really fit in here.

Btw, the link below says 403 Forbidden. Not sure what's going on there.
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Tglad
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« Reply #4 on: January 28, 2014, 01:14:44 AM »

I believe the 'Mandelbrot in magnetic field borders' actually refers to plots of one variable against another in a magnetic field equation.

The fact that the Mandelbrot set turns up in dynamic system plots is not altogether surprising since the Mandelbrot set is 'universal':
McMullen, Curtis T., The Mandelbrot set is universal. In The Mandelbrot set, theme and variations, 1–17, London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, 2000. MR1765082 (2002f:37081)
also: http://www.math.binghamton.edu/topics/mandel/mandel_why.html

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stereoman
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« Reply #5 on: January 28, 2014, 01:25:58 AM »

I suppose there are some similarities between fractals and hyperbolic geometry (which is what those of Escher's drawings are that "disinitegrate" in "smaller iterations") though that doesn't really fit in here.

Btw, the link below says 403 Forbidden. Not sure what's going on there.

The electromagnetic field of the Earth consists of gnomons, that is the natural way  living things grow through  time.
         But, as we know very well, which seems an obvious limit, the skin, for example, turns out to be quite permeable, and a certain level, absolutely nonexistent, since we can always penetrate to a next level.
          So to me, this is the same phenomenon, the layers of the electromagnetic field, when you get close enough to them, they are not having a limit as defined as it looks from afar.
          Saturn's rings appear solid until you get close enough.
          I think Escher understood this and did the function to make it visible to us in an artistic way.
           From another point of view, it is quite clear that the Mandelbrot set itself, is the section of something.
          But probably I´m wrong, and the Mandelbrot set is not a form whose limits disintegrate in smaller iterations
« Last Edit: January 28, 2014, 09:08:57 AM by stereoman » Logged
Chillheimer
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chilli.chillheimer chillheimer
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« Reply #6 on: January 28, 2014, 12:28:40 PM »

Strange, the site is down indeed. It was still online when I started this thread (and the months before).. probably just temporarily.

Tglad, hobold, as I'm pretty bad with math and formulas, do you personally think it is possible that if the formulas describe it in theory, then the m-set also is there for real? It should be, if I understand it correctly.. if not, the formulas would be wrong?

Do you think it would be measurable if you measure in the right way/places? Or is the force maybe too weak?


------offtopic---
Oh my, tglad.. with your link you started something..
what are hubbard trees? can't find anything (helpful)on google.. or is that too hard for a non-mathematician to understand? (I was very bad in math back in school..)

and about the "locally connected" points in there: are they talking about the connection between the upper und lower part of the m-set?
I'd tend to say that the only connection is in the center of the m-sets ...öhh.. butt.. embarrass
But then again, you'll never reach it by zooming in, so it's somehow located in infinity that approaches 'certain coordinates' with indefinite digits
so if you can never reach that point, is it a point at all? does it 'exist'?
or is it just a concept that we make up to understand something that can't really be grasped according to our reality?

damn. why do i find that stuff so interesting but have such a hard time with the mathematics needed to understand.. sad
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stereoman: I'm not sure if I understand gnomons correctly. would you say that minibrots also are gnomons?
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Tglad
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« Reply #7 on: January 28, 2014, 01:09:03 PM »

Chillheimer, I don't think the m-set is ever visible in magnetic field lines because I think the original reference is actually referring to graphs of one magnetic field variable vs another, not x,y in space, but for example flux vs charge or field strength vs polarity... something like that.

A Hubbard tree is a tree which defines how many branches there are on each level. I don't think it is too hard.
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kram1032
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« Reply #8 on: January 28, 2014, 09:18:25 PM »

it is quite clear that the Mandelbrot set itself, is the section of something.
Well, yes. The Mandelbrot set is that part of a certain polynomial of, in the limit, infinite degree, which remains bounded. Its border, by the normal rendering methods, is that part where its complex norm is exactly 2.
So the Mandelbrot set, geometrically, is defined by a plane z=4, cutting through the squared norm of the polynomial defined by the familiar recursion, and taking what ever is below that plane as inside the set and what ever is above it is outside.
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youhn
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« Reply #9 on: January 28, 2014, 09:35:19 PM »

Chillheimer, I don't think the m-set is ever visible in magnetic field lines ...

Magnetic lines are (normally) invisible aswell. But something being invisible says little about it's existence.

How about the claim that the boundary of the Mandelbrot set has infinite amount of different shapes? Other than extrapolation ... how can one ever argument or even prove that? After having explored the set for about hours and hours ... sometimes I fall out of this belief.
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kram1032
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« Reply #10 on: January 29, 2014, 05:22:58 PM »

That depends on what you mean by "shape". Mathematically, there are various concepts of "shape" that can differ quite a lot.
In one interpretation, the shape would correspond to the topology in which case the M-Set simply is a circle.
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youhn
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Shapes only exists in our heads.


« Reply #11 on: January 29, 2014, 07:39:30 PM »

Since I'm not strong in math, I ment the general definition.

<looking up @ wikipedia>

I mean congruent shapes. For example; could every skyline of every city be found on the boundary of the Mandelbrot set?
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kram1032
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« Reply #12 on: January 29, 2014, 09:20:43 PM »

every skyline of every city? Hmm... That seems rather hard to imagine, since the MSet's borders just keep being infinitely jagged.
Though what is plausible is that you could approximate such a shape as closely as you want if you dive down in the right spot. Not sure if that's a property of the MSet. Might not be.
The Zeta function does this though. It's truely universal, which means that, along its main strip (Zeta(z) with 1/2<Re(z)<1), it can approximate (pretty much) ANY function to arbitrary accuracy (it's just a question of how far you want go go up or down the strip with more and more complex functions being represented by points that are more and more distant from the origin.
This works on a radius of 1/4 around that point. Beyond that distance, it won't work.
All the requirements are that the function must be holomorphic (which should be perfectly valid for a skyline if you take distributions, if I'm not mistaken) and it cannot pass through 0 within that radius (which, unless you have a function that shoots off to infinity, can always be corrected by adding a constant).

Note, however, that this doesn't mean a part of the Zeta function will match a 1/4 circle of a given function *exactly* - it's just possible to come indefinitely close to it.

One of the biggest questions of today's mathematics can be stated as whether the Zeta function can even approximate itself in the same way as it approximates any other function (which fulfills those conditions) - it's equivalent to the Riemann hypothesis which asks whether all non-trivial zeros of the Zeta function are, in fact, on that same above-mentioned strip.

Hmm, come to think about it, this might mean that the Zeta function actually contains cut-outs of the limit polynomial that defines the M-Set... well, maybe not: it has way too many zeros and probably doesn't count as holomorphic. (Not the limit case anyway. - Any finite iteration, however, should work perfectly fine)
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youhn
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Shapes only exists in our heads.


« Reply #13 on: January 29, 2014, 10:19:30 PM »

Your last comment between brackets also crossed my mind, though in a slightly other formulation. I thought some shapes might be found at the boundary of the M set, at a given iteration. Iterate more towards the limit case and the resemblance might fade.

The Zeta function is a new subject for me. Sounds interesting!
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kram1032
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« Reply #14 on: January 29, 2014, 10:49:30 PM »

Oh, I was talking about the reverse situation: The Zeta function approximating a part of the MSet (or the underlying polynomial), rather than the MSet approximating arbitrary shapes. Of course, if you require smoothness, you can just cut off iterations early. - Though straight lines would never quite happen, would they?
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