

Beautiful plots of roots of polynomials - fogus
http://math.ucr.edu/home/baez/week285.html

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lmkg
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 =).

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

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defen
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> )

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RevRal
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._

\----

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

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

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camccann
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.

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carbocation
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.

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d0mine
Yet another way to generate pretty pictures using math.

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dmvaldman
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.

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lmkg
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.

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

