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.
Best Answer
The Mandelbrot set can't have infinite area, since the entire set is contained in the disk of radius $2$ centered at the origin.