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We know that the Mandelbrot set is derived from the iterations of z^2 + c.

Do anyone know something about magnet Mandelbrot? I found it in the software UltraFractal, and it is much more beautiful than the original Mandelbrot, in my opinion.

Do you know anything about it? What is it iterating?

EDIT: Here's a picture of a magnet Julia set

Magnet Julia

alt text

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Links to pictures and other source material, please. –  Sam Nead Oct 12 '10 at 13:09
    
This seems to be the relevant part of the Ultra Fractal documentation, but it's not very illuminating: ultrafractal.com/help/index.html?/help/formulas/standard/… –  Rahul Oct 12 '10 at 15:48
    
Here's another link I found, that has the formulas and two pictures: icd.com/tsd/fractals/beginner4.htm I would love for someone to explain the gasket-like features of this fractal in terms of complex dynamics. –  Rahul Oct 12 '10 at 15:54
    
Sorry, the link is icd.com/tsd/fractals/beginner4.htm –  Rahul Oct 12 '10 at 17:08

2 Answers 2

up vote 5 down vote accepted

I searched Google for "magnetic fractal", and found the answer on the first hit. It quotes the Fractint documentation (I can't resist mentioning that Fractint is the grand-daddy of freeware fractal-generating software for personal computers -- it had its first release in 1988, and is still being maintained!):

These fractals use formulae derived from the study of hierarchical lattices, in the context of magnetic renormalisation transformations. This kinda stuff is useful in an area of theoretical physics that deals with magnetic phase-transitions (predicting at which temperatures a given substance will be magnetic, or non-magnetic). In an attempt to clarify the results obtained for Real temperatures (the kind that you and I can feel), the study moved into the realm of Complex Numbers, aiming to spot Real phase-transitions by finding the intersections of lines representing Complex phase-transitions with the Real Axis. The first people to try this were two physicists called Yang and Lee, who found the situation a bit more complex than first expected, as the phase boundaries for Complex temperatures are (surprise!) fractals.

The formulas for the two fractals are also given there. They are $$z \mapsto \left(\frac{z^2 + (c-1)}{2z + (c-2)}\right)^2$$ for magnet 1, and $$z \mapsto \left(\frac{z^3 + 3(c-1)z + (c-1)(c-2)}{3z^2 + 3(c-2)z + (c-1)(c-2) + 1}\right)^2$$ for magnet 2.

I'm by no means knowledgeable on this subject, but I've been looking at some of Robert Devaney's papers, which I came across via tetration.org. Looking at Devaney's images, I'd guess that the reason why these fractals have the beautiful Sierpinski-gasket-like structures, while the standard quadratic Julia and Mandelbrot sets don't, is that each of the formulas defining these fractals is a rational function, that is, a ratio of two polynomials, rather than a single polynomial. I believe the field that studies these things is called complex dynamics. My knowledge doesn't extend to how rational functions with poles give rise to Julia sets with gaskets, but you could try looking for the answer to that in some of Devaney's papers, or in the book Iteration of Rational Functions by Alan F. Beardon which is cited in the Wikipedia article on Julia sets.

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Rational functions and gaskets go hand in hand: The Apollonian gasket is very much related to iterations of rational maps, giyf. –  Per Alexandersson Aug 27 '12 at 21:30

Apparently, the idea is roughly this:

  • At low temperatures, the metallic crystal lattice is completely ordered, and the metal is magnetic.

  • At high temperatures, the metallic crystal lattice is completely disordered, and the metal is non-magnetic.

  • As you warm up a low temperature lattice, small, isolated pockets of disorder appear in the otherwise ordered lattice.

  • As you cool down a hot lattice, small pockets of order appear in the otherwise chaotic arrangement.

  • You can write a "magnetic phase renormalisation transform" which represents "zooming out" of the lattice.

  • At lowish temperatures, when you zoom out, you can't "see" the small pockets of disorder any more, and the lattice looks totally ordered. This tells you that at lowish temperatures, the metal is still magnetic.

  • In short, to figure out whether the metal is magnetic or non-magnetic at temperature $x$, you just feed $x$ through this renormalisation transform, again and again, until $x$ settles at a very high or very low value.

  • The actual renormalisation transform varies depending on the properties of the lattice. FractInt (God rest its soul) implements two such lattices, as per Rahul's answer. Notice that each lattice has a parameter, $c$, which represents the number of possible "quantum spins" the metal ions may have.

  • The guys were struggling to figure out how the properties of the lattice as a function of $c$. So they tried replacing the temperature $x$ with a complex number $z$. Now, obviously, the real world does not have complex-number temperatures. But they thought it might be illuminating to see what the maths does...

  • ...It was illuminating. They couldn't figure it out because it's a damned fractal!

  • If you look at a magnetic Julia set, the region across the line $\Re(z)=0$ corresponds to real-world temperatures. (I think there's some linear transformation between actual temperatures and function coordinates; I don't recall what it is off-hand.)

P.S. Apparently in the real world, $c=2$ (the "Ising spin model"), but they tried letting that be a complex number too. The results are the beautiful magnetic Mandelbrot fractals, to go with the magnetic Julias.

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