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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!  :o

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.

This recalls me some Escher drawings, where  boundaries disintegrate in smaller iterations.

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.

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

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

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.. :(
-----------

stereoman: I'm not sure if I understand gnomons correctly. would you say that minibrots also are gnomons?

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.

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.

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.

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.

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?

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)

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!

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?

 
I made this drawing to try to make the concept of gnomon visible.
     The origin is a point in the center of the thalamus , from there , in successive layers , the whole human organism is constructed.
     We can see how the layers have different densities according to function, and we can see how each layer has different forms , also according to their function.
     But this is only a design , in fact all of the layers are interpenetrated into one another , just as in our animal organism contain minerals and  vegetable parts .
      It is this interpenetration which blurs the boundaries in all fields since, from another point of view , being all the same body ,our way of setting limits is entirely artificial , and are never exact.
      Ie, there are marine mammals, and there are also fishes with lungs.
     In a rainbow , we can see different colors, but it is impossible to find the exact boundary between two of them because in reality do not exist.
     So , geometrically, the planet , ends where  the magnetosphere ends, but the planet's magnetosphere is part of the set of magnetospheres in the solar system , forming a unit , etc. .
     I think the Mandelbrot set is like  something ideal and perfect in conditions that can never be given in the real world, in the real world we have the romanescu , we can think of as an ideal fractal, but that may not manifest if not by the power of the gnomon , which always introduces other conditions.
      In other words, the gnomon carries some info that directs the recursive process to a definite shape, as I understand it, the M-set, can´t do this.
       At this point, I think the challenge is to create a digital gnomon wich can be preset and applied  to the M-set or other fractal generator programs.
     ¿Any takers?

Here is an article about the fractal geometry of the zeta function:

http://dhushara.com/DarkHeart/geozeta/zetageo.htm (http://dhushara.com/DarkHeart/geozeta/zetageo.htm)

Oh, that's nice :)

Though I wonder why the author thought that this (http://dhushara.com/DarkHeart/geozeta/zetageo3el_files/image057.png) was a good idea. You can't really read the exponent.

Edit: With a function like the Zeta function I'd imagine it's really hard to get proper bailout conditions, right?
Since you can't just say it diverges at some value, for Zeta(+infinity)=1

Or is the bailout condition something like "if Re(Zeta(z))>1, you are definitely inside the set"?

Edit: What happens if you bring the Zeta function's lines onto circles instead? For instance, you could do a Riemann sphere-like transform, mapping the circle |z|=1 to the line Re(z)=1, apply the Zeta function to that, and then do the back-transformation.

I applied the idea of a "MetaMandelbrotSet" to the Zeta function in Mathematica. This is my result (for now) :

Real Part	(http://i.imgur.com/VvkVChh.png)
Imaginary Part	(http://i.imgur.com/QV1VwEv.png)
Absolute	(http://i.imgur.com/hRbNmTk.png)

200 iterations, only rendered for the point (0,0)
I'm sure one can do significantly better, though if the full limit Meta Mandelbrotset is a challenge, the correspoinding Zeta Meta must be a nightmare.

Here is a nice reply on Math.SE concerning the(*) "magnetic Mandelbrot set"
http://math.stackexchange.com/a/140812/49989

Apparently, in those models, x corresponds to the temperature, while c corresponds to how many different spin states there are in a magnetic metal.
In real life, that means that c should be 2 (according to the Ising Model) and x should be somewhere in the range from 250-330K for "real life" materials under every-day-life temperatures.

(*) there isn't "the" magnetic MSet" though - there are many which are all models for different metals, most of which do not actually exist.

So those Minibrots do not actually occur as cross-sections of a magnetic field, but rather as seas of stability within a chaotic function which models the behavior of metals.
And within the real-life-relevant ranges, they do not occur at all.
- We at least never actually had anything that could classify as complex-valued temperature, to my knowledge. And we certainly never had anything with irrational, let alone complex spin. (Electrons, Protons and Neutrons all have +/- 1/2 spin)

Still, perhaps we could come up with some interesting interpretation in which those complex values actually do make sense within real life. As of right now, they seem to mostly be a mathematical curiosity highlighting the complexity surrounding the curie-temperature of magnetic materials.

thanks kram.. although I don't understand all, this answers my initial question, although I had hoped for a 'more wonderful' outcome.
but then again, the fractal nature around is is enough wonder for a lifetime.. :)

Don't give up hope yet. It wouldn't be the first time that some real value was extended into a complex value in a sensible, physically relevant way :)

Since temperature is just the average kinetic Energy in a system, and there is a correspondence between energy and pulse (with pulse being a vector quantity while energy is a scalar), perhaps there would be a description in terms of quaternions, where the scalar (real) part would be the average kinetic Energy while the vectorial part would be the average directed pulse. A description like that is possible (if not always ideal) though for a still metal, this average pulse would typically be 0, since if it wasn't, that would mean that the crystal would be moving.

In that case, perhaps (this is purely speculative) the complex model describes moving magnetic materials. If this is correct, maybe the Curie temperature would somehow be dependent on the velocity of the crystal? - Somehow doubtful and I've never actually heard of that but maybe?
It would mean that, for specific, well-defined speeds, you could have magnetic properties in metals that are way too hot to expect such properties.

However, I can't really see how this would apply to the c value: A complex number of spin configurations certainly makes little sense and even an non-integer or negative number is rather hard to grasp. If it just so happens that no islands of stability (regions "inside" the magnetic set) happen along the line c=2, this could still be a valid interpretation.

Or else, perhaps the models that are used are too simple to fully describe the phenomenon (or my naive use of quaternions here really doesn't apply)

  After reading all that has been said, it strikes me that we haven´t a definition of what is infinity.
   This may be the key issue in my view.
   Numerically, the infinite seems a clear concept, we can keep adding numbers indefinitely without reaching a limit.
   But this has little practical use for us, and also prevents to form a closed concept, the unit, which was the starting point of the ancient mathematicians.
   In ancient mathematics there was not  the zero concept, wich we relate to infinity.
   Geometrically, which is what interests us, because all the numbers have to finish defining forms, a line is the infinite for a point, a plane is the infinite for a line, and a solid is the infinite for a plane , in the same way an hypersolid is infinity for a three-dimensional solid, lets define the hypersolid as the solid extended in time.
    Closely related are logarithmic spirals wich allow for growing while still  inside some " infinite limits", if this can be said
    Seen this way, infinity is a handy and useful concept, my two cents.

For me this sequence shows all signs of "fractal behaviour"
I wonder if it itself goes on into infinity.. ;)


---the rest is too mathematical for me.. I feel like I don't need to be an expert in everything, so I'll leave that to you guys, instead of answering to stuff I don't understand--- ;)

yours is the right question.
    From one point of view, the sequence continues in an infinite hypothetical, but like with numbers, this does not help us much, things must be closed somewhat to make sense.
     But you will understand If I say that we have the octave, to bring order to  infinite levels of vibration.
     An octave is a closed process, take the color wheel, for us it is a closed circle, but we know that in the natural scale,is not, as there are vibrations above and below of visible colors.,This is also true in music, there are non audible sounds  and vibrations ,some harmonics are beyond our perception.
    Ancient philosophy stated that "man is the measure of everything", this is what we see here, the color wheel can´t be changed in any way, nor can the octave, because they are created by the human senses and mind from whatever is out there.
   In this case, the hypersolid contains all the possibilities for the solid, in the same way, the solid contains all the possibilities for the plane, and the plane contains all the possibilities for the line.
     Imagine how many drawings have been done  and can be done by man trough all times, these are the infinite possibilities the line has, but they all  always remains in the plane.
    

That's not true: We absolutely have concepts of infinity. In fact, infinitely many of them.
Infinity is highly context dependent and different kinds of infinity occur in different situations.
For instance, the infinity of 1/x as x goes to 0 is "weaker" than the infinity of 1/x²: You need a higher polynomial to "get rid of" the second infinity than for the first. (x²/x²=1 and x/x=1 but x/x²=1/x)
And then there are exponential infinities - ones that cannot be dealt with by any polynomial.

Other classes of infinity are the "infinitely small" or higher infinities which are like infinities of infinities. - Then there are the so-called Surreal Numbers which contain all numerically possible infinities.

It's not so much that we don't have concepts for infinity but rather that most infinities are not very important or too rare to be usually discussed in detail.

The way in which an (n+1)-plane is infinite compared to an n-plane is yet another very different concept of infinity.
We are mathematically also able to talk about infinity-planes - hyper planes of infinite dimensionality.

The most common forms of infinity nowadays have a very well established definition and underlying conception.

What is not true?
 Edit. Oh I see, you have definitions of infinite, but this is not the point, the point is how to deal with infinite in geometric terms, I suppose that having so many infinites in your box, there´s room for another one :dink:

Various cases of geometric infinity can also be dealt with.

No doubt, but I think the main problem with the gnomon concept, comes from two different concepts of infinity.
To me, infinity can be and is limited geometrically, but numbers can´t be limited this way, since I must talk with matemathics, and they have it´s own ideas , I try to make mine understandable, in search for the simplest approach.

When the idea of infinity is understood, some things in geometry and math become easier. For example an infinite line is easier to define and understand. Since it has no beginning or end, you don't have to think about it.

Everything in math and geometry is an approximated or extrapolated model from our percepted reality.

When talking about the infinity of numbers ... things get very confusing in my head. Take for example the number pi, which is said to be infinite. This does not refer to it's value, since pi is actually pretty defined. But to exactly express it in our decimal number system... that is a problem. I can agree to the statement that numbers are not infinite. But sets of numbers can be.

Infinity is something we though up to deal with the very big and ungrabbable. It is as real as the concept of "borders".

Which two concepts of infinity are you talking about?

I think we can speak of "open" and closed ".
Correct me if I'm wrong, as is the M-set, can be penetrated to  unthinkable depths , and presumably, with more powerful computers, they can continue to penetrate more and more, ad infinitum, in fact that is what some are doing.
   But in my view, this kind of infinity is not very useful beyond finding more or less attractive frames that differ very little from each other by beautiful they are, I would almost say it's infinitely boring.
   That address has no interest for me, what I want is an approximation to the infinite complexity of the hypersolid, at least, to his behaviour, where all happens trough closed cicles wich support each other.

Can you elaborate that? What, in your mind, is a hypersolid and how does it relate to any of this?
How does it involve an infinity of any kind, or especially one that can not be dealt with?

Returning to the design of the gnomon, three circles enclosed within one another, that touch at one point.
 
  They are continuously rotated at different speeds.
   The point where they meet is where the changes occur.
   Suppose that there are certain events scheduled in each cycle, since the speeds are different, the combination of the three series of events will give different results continuously.
    Now, imagine another inner level wich introduces changes into the events series, or  in the various speeds, the direction of time can even change, etc. .
 Of course, this can follow with inner levels that introduce further variations.

 This is the kind of infinity I talk about, and the way the nature works,  the hypersolid is the whole thing, limited by the numbers of cycles and layers, these limits are fixed by the concept of the octave, wich must fulfill all the possibilities , as we see in the musical scale, or the tonal scale, or the color wheel.
 Note that I don´t claim any discovery, the color wheel is one of my daily tools, like the musical scale is the tool for a musician, these tools can´t be changed nor discussed, but one can enter its structure to understand whats going really.

  

It seems thet recently some scientists have extended the concept of "magnetic fractals" to more general functions :



where m and n are integers (> 1) and a complex parameter.

See for instance :

Fei Yang, Jinsong Zeng, On the dynamics of a family of generated renormalization transformations, J. Math. Anal. Appl. 413 (2014) 361–377

Link to the paper:
http://arxiv.org/pdf/1312.1617.pdf

I have had a copy of that for years, of course, and it doesn't. I skimmed the sections to do with magnetism or the Mandelbrot set, finding them with the index; then resorted to a Google Books search of Google's copy. Nothing. Looks like Miqel got a bit confused about where he read that. I know The Beauty of Fractals references fractals arising in the complex-valued version of the Yang-Lee magnetic phase transition mapping, and Mandelbrot fractals appear in the parameter space for this system. They've been commonly implemented in fractal software for a while now, appearing in Fractint as the magnet1 and magnet2 types for instance.

The discussion in The Beauty of Fractals is on pages 129-149.

Just a footnote,
"formula" is a word wich means " small form", I didn´t knew it, and had real fun :evil1: