2.38 Magnetic Fractals
(type=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 Fractint 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 Fractint !
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 (p. 166). See Finite Attractors
(p. 185) for more information on this aspect of Fractint internals.
(Thanks to Kevin Allen for Magnetic type documentation above).