Unraveling the Mysteries of Magnets and Field Lines

[Arnan]

I know that anything casting a field obeys the inverse square law and that field lines are an abstract used to help us visualise the vectors on objects effected by the field.

Why is it then that if I drop iron filings around a magnet they sort themselves into neat lines?
I would expect that to orient themselves according to the field but why separate into a layered pattern emmanating out from the source when the actual change in force is smooth.

 

[tim_lou]

The magnet induces a magnetic moment in the iron fillings which attracts other iron fillings. The iron fillings tend to clump together so that you get a nice layered pattern.