The Beauty of Roots (2023) | Hacker News Reader

The Beauty of Roots (2023)

math.ucr.edu

87 pointsykonstant11 comments

WTF, this is amazing! I think this could be a great example of experimental mathematics, some examples for which are listed in SE (https://math.stackexchange.com/questions/264560/mind-blowing...) but this I believe is much better.

What I mean is that you can generate this without any a priori knowledge, then examine it like Galileo examined the moons of Jupiter, to seek interesting phenomena, which then you work to understand. For example: can one prove the empty space in the middle or around +1 and -1? Polynomials of degree <= 5 with integer coefficients in [-4, 4] do not have any roots with nonzero imaginary parts in |r|<r_0, where r_0 seems to be around 0.7.

The empty space in the middle is pretty easy. These are roots of polynomials of the form +- z^n +- z^(n-1) +- ... +- 1. If |z| is small then the earlier terms are smaller; the absolute value of their sum is at most |z|^n + ... + |z|^1 which is at most |z| / (1-|z|). If this is < 1 then the sum can't be large enough for adding it to +- 1 to give 0. So if |z| < 1-|z|, i.e., if |z| < 1/2, then z can't be a root.

That's for the "main" image. The first image in Baez's post is for polynomials whose coefficients are -4,-3,...,+4, and the analysis would be different for those, but it's still true that if |z| is small enough then the sort of calculation in the previous paragraph forces it not to be the root of any nonzero polynomial with small integer coefficients.

The holes near +- 1 are more complicated, I think.

The link references at the bottom have some more detail on the structure. For example, the N-Category Cafe talks about how the dragon curve shows up at the edges [0].

There are other gross level symmetries due to the symmetries of the Littlewood like polynomials involved. For example, if $p(x)$ is a Littlewood-like polynomial, then so is $p(1/x)$ etc.

I think the "holes" that show up on the unit disc because of the factors of the degrees of certain polynomials. If I'm not mistaken, the holes follow a Farey sequence [1].

I wrote a little blurb about it but it's still incomplete and haphazard. One thing I'm still curious about is how quickly the holes diminish in size.

[0] https://golem.ph.utexas.edu/category/2009/12/this_weeks_find...

[1] https://en.wikipedia.org/wiki/Farey_sequence

[2] https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.h...

I have a little python package for making images of Littlewood polynomials (and others).

https://github.com/evanberkowitz/littlewood

If you like these kind of pictures, you can also find very nice pictures showing the density of the eigenvalues values of integer matrices in the Bohemian matrices gallery [1].

[1]: http://www.bohemianmatrices.com/gallery/

Such a cool structure.

I was so amazed when I learned about it out ~10 years ago that I wrote a little interactive thing in javascript + webgl for it. I hope you'll forgive my self-indulging here: https://cscheid.github.io/lux/demos/beauty_of_roots/beauty_o...

~ 10 years ago

The post itself has had a long HN life although relatively light on commentary. The first submission is almost 15 years old!

https://news.ycombinator.com/item?id=980043

You can do mandelbrot even with awk and sh.

https://gist.githubusercontent.com/hessch/5383798/raw/e8d099...

For speed, you might be able to compile it with awka and any C compiler with the math library:

https://github.com/noyesno/awka

    awka -f mandelbrot.awk > mandelbrot.c

   cc -O2 -lawka -lm -o mandelbrot mandelbrot.c

See 'Entropy bagels' and other complex structures emerge from simple rules at https://news.ycombinator.com/item?id=39529374 for a related post.

Fascinating to see the Greg Egan participating in these experiments!

In case you recognize the Baez address and are wondering, it's unsurprisingly not this kind of root https://en.wikipedia.org/wiki/Root , but, somewhat surprisingly to me, also not this kind of root https://en.wikipedia.org/wiki/Root_system . Instead it's this kind of root https://en.wikipedia.org/wiki/Root_of_a_polynomial , which is really what I should have expected before thinking of root systems ….

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