Beautiful plots of roots of polynomials | Hacker News Reader

Beautiful plots of roots of polynomials

math.ucr.edu

102 pointsfogus16 comments

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 =).

As a mathematician, haven't you abstracted from the what? Isn't that the point of math?

Did anyone else get an intense feeling of dread when looking at the picture of the roots of all degree-24 polynomials with coefficients in {-1, 1} ? (this one: http://math.ucr.edu/home/baez/roots/polynomialrootssmall.png )

Re-posting my comment from the dupe submission:

Absolutely gorgeous. I especially like that they're beautiful in a much different way from fractals.

There is a sereneness here that is a fresh of breath air, much different than the imposing fractal figures.

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It could be that my reaction is from taking existentialism in stride, these days. Though, fractals are still scary/beautiful.

Wow, gorgeous! The structures in this image look like Hilbert curves: http://math.ucr.edu/home/baez/roots/polynomialroots05expi02....

Actually, parts of it look frighteningly similar to the Heighway dragon curve: http://en.wikipedia.org/wiki/Dragon_curve

I can't even begin to fathom what kind of connection there might be, but the degree of resemblance seems to go beyond just coincidence.

Yes! And the dragon curve can be written as an L-system, of which Hilbert curves are a special (space-filling) subset. Great tie-in, thank you.

Yet another way to generate pretty pictures using math.

It's scary how symmetric everything get's as N approaches one extreme or another.

puuuurty

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.

It says he used Mathematica, which uses arbitrary-precision arithmetic.

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.

Finding exact roots of high-order (>4) polynomials isn't just difficult; it's provably impossible[1].

http://en.wikipedia.org/wiki/Abel-Ruffini_theorem

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.

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."

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.

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