Darren New wrote:
> Quantum physics, general relativity, *and* fractals!

Hmm. Well, I can't claim relativity, but...

Take one metallic lattice. Each atom is a tiny magnet. At low 
temperatures, the majority of the atoms line up, yielding a macroscopic 
magnetic field. At high temperatures, most atoms do not line up, 
yielding no macroscopic field. So, at low temperatures, the lattice is 
ferromagnetic, while at high temperatures it is paramagnetic. But at 
what temperature does it change from one to the other?

Ah, well, that depends. On a great many things. But one trick is to use 
"renormalisation". That is, take your lattice, step back from it, and 
"average out" the alignment of the individual atoms. At a low 
temperature, you have an essentially stable lattice with small 
variations. Averaging blurs out the small variations, leaving only 
overall uniformity - in other words, the lattice appears to be at a 
lower temperature after renormalisation. Similarly, a warm lattice 
appears hotter after renormalisation.

In short, repeatedly renormalising a given lattice will make its 
apparent temperature tend upwards or downwards. If you can write a 
formula for this, then you can take any starting temperature, repeatedly 
apply this formula to it, and eventually end up with either a very low 
or a very high number, indicating that the lattice is ferromagnetic or 
paramagnetic at the temperature you started with.

The formula for doing a renormalisation depends on the nature of the 
lattice and the number of possible atom alignments. Some scientists were 
looking at this, and found that the phase transition (the temperature at 
which ferromagnetic becomes paramagnetic) was surprisingly difficult to 
predict. So they decided to use complex-valued temperatures (which, 
obviously, don't really exist) to try to make the maths clearer.

Oh, they made the problem clearer alright. The phase transition is a 
fractal. (!) But hey, what do you expect, iterating a nonlinear function 
like that?

Even weirder, sometimes the model predicts additional magnetic phases 
which only occur at complex-valued temperatures.

Personally, I just use this stuff to make pretty pictures...

http://www.icd.com/tsd/fractals/beginner4.htm

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