Around the roots of unity are large empty expanses, with a single root in the middle. This sounds like a classic unstable equilibrium to me. But an equilibrium of what?
Craziness like this is why I became a mathematician =).
i'd like to know the error involved, and if it has to do with the fractal patterns seen. finding roots of polynomials is difficult to do exactly. typically iterative methods are used that can only converge to the root.
Nonetheless, the iterative approximation algorithms are incredibly efficient and numerically stable. There's one really clever trick that actually uses floating-point imprecision to do something you couldn't do in exact arithmetic (inverting a degenerate matrix), that converges to within machine-epsilon in about three iterations.
[1] In the general case; special cases may be tractable, e.g. x^n-1=0.
Impossibility depends on your set of techniques. The Abel-Ruffini theorem only applies if you are trying to solve them in terms of radicals. However exact solutions to any degree polynomial are known in terms of multivariate hypergeometric functions, and also there is another solution in terms of Siegel theta functions.
I don't get why the Abel-Ruffini theorem is relevant here. Plotting sqrt(2) on the picture is impossible too. An n-th root is just shorthand for the root of the polynomial X^n - a. Introducing a shorthand doesn't change your ability to plot it on a picture.
As I read it, the WP article very nicely points out that exact roots are possible"
"The theorem says that not all higher-degree equations have solutions which can be expressed by performing a finite number of operations... Some polynomials of arbitrary degree are indeed solvable with a finite number of such operations."
Presumably he instructed mathematica to approximate the roots to an error proportional to the resolution of the image; in that (likely) case the error doesn't have anything to do with the fractals.
Craziness like this is why I became a mathematician =).