TITLE: Magnet Mandelbrot Set
QUESTION [8 upvotes]: 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

REPLY [9 votes]: 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.