
Magnetic Fractals

formulas: magnet1m/.../magnet2j


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.

And that's all the technical (?) background you're getting here!  For more
details (are you SERIOUS ?!) read "The Beauty of Fractals". When you
understand it all, you might like to rewrite this section, before you
start your new job as a professor of theoretical physics...

In xmfract terms, the important bits of the above are "Fractals",
"Complex Numbers", "Formulae", and "The Beauty of Fractals". Lifting the
Formulae straight out of the Book and iterating them over the Complex
plane (just like the Mandelbrot set) produces Fractals.

The formulae are a bit more complicated than the Z^2+C used for the
Mandelbrot Set, that's all. They are :

                  [               ] 2
                  |  Z^2 + (C-1)  |
        MAGNET1 : | ------------- |
                  |  2*Z + (C-2)  |
                  [               ]

                  [                                         ] 2
                  |      Z^3 + 3*(C-1)*Z + (C-1)*(C-2)      |
        MAGNET2 : | --------------------------------------- |
                  |  3*(Z^2) + 3*(C-2)*Z + (C-1)*(C-2) + 1  |
                  [                                         ]

These aren't quite as horrific as they look (oh yeah ?!) as they only
involve two variables (Z and C), but cubing things, doing division, and
eventually squaring the result (all in Complex Numbers) don't exactly
spell S-p-e-e-d !  These are NOT the fastest fractals in xmfract!

As you might expect, for both formulae there is a single related
Mandelbrot-type set (magnet1m, magnet2m) and an infinite number of related
Julia-type sets (magnet1j, magnet2j), with the usual toggle between the
corresponding Ms and Js via the spacebar.

If you fancy delving into the Julia-types by hand, you will be prompted
for the Real and Imaginary parts of the parameter denoted by C. The
result is symmetrical about the Real axis (and therefore the initial image
gets drawn in half the usual time) if you specify a value of Zero for the
Imaginary part of C.

Fractint Historical Note:  Another complication (besides the formulae) in
implementing these fractal types was that they all have a finite attractor
(1.0 + 0.0i), as well as the usual one (Infinity). This fact spurred the
development of Finite Attractor logic in Fractint. Without this code you
can still generate these fractals, but you usually end up with a pretty
boring image that is mostly deep blue "lake", courtesy of Fractint's
standard Periodicity Logic. See "Finite Attractors" for more
information on this aspect of Fractint internals.

(Thanks to Kevin Allen for Magnetic type documentation above).

