5 ms·

The Beauty of Roots (2023)

87 points by ykonstant 3y ago

JadeNB 3y ago
In case you recognize the Baez address and are wondering, it's unsurprisingly not this kind of root https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Root_system . Instead it's this kind of root https://en.wikipedia.org/wiki/Root_of_a_polynomial 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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evanb 3y ago
I have a little python package for making images of Littlewood polynomials (and others). https://github.com/evanberkowitz/littlewood https://github.com/evanberkowitz/littlewood
cscheid 3y ago
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_of_roots.html https://cscheid.github.io/lux/demos/beauty_of_roots/beauty_o...
- pvg 3y ago
  ~ 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 https://news.ycombinator.com/item?id=980043
Jun8 3y ago
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-mathematics-experiments 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.
- gjm11 3y ago
  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.
- abetusk 3y ago
  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_finds_in_mathematic_46.html https://golem.ph.utexas.edu/category/2009/12/this_weeks_find... [1] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence [2] https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.html https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.h...
anthk 3y ago
You can do mandelbrot even with awk and sh. https://gist.githubusercontent.com/hessch/5383798/raw/e8d099c0defc2348d297fede4674e3fd63f8bde3/mandelbrot.awk 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 https://github.com/noyesno/awka awka -f mandelbrot.awk > mandelbrot.c cc -O2 -lawka -lm -o mandelbrot mandelbrot.c
guyomes 3y ago
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/ http://www.bohemianmatrices.com/gallery/
DanielGeisler 3y ago
See 'Entropy bagels' and other complex structures emerge from simple rules at https://news.ycombinator.com/item?id=39529374 https://news.ycombinator.com/item?id=39529374 for a related post.
enriquto 3y ago
Fascinating to see the Greg Egan participating in these experiments!

87 points by ykonstant 3y ago

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

deleted 3y ago
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evanb 3y ago

I have a little python package for making images of Littlewood polynomials (and others). https://github.com/evanberkowitz/littlewood https://github.com/evanberkowitz/littlewood

cscheid 3y ago

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_of_roots.html https://cscheid.github.io/lux/demos/beauty_of_roots/beauty_o...

pvg 3y ago
~ 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 https://news.ycombinator.com/item?id=980043

Jun8 3y ago

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-mathematics-experiments 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.

gjm11 3y ago
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.
abetusk 3y ago
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_finds_in_mathematic_46.html https://golem.ph.utexas.edu/category/2009/12/this_weeks_find... [1] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence [2] https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.html https://mechaelephant.com/dev/Littlewood-Polynomials-Notes.h...

anthk 3y ago

You can do mandelbrot even with awk and sh. https://gist.githubusercontent.com/hessch/5383798/raw/e8d099c0defc2348d297fede4674e3fd63f8bde3/mandelbrot.awk 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 https://github.com/noyesno/awka awka -f mandelbrot.awk > mandelbrot.c cc -O2 -lawka -lm -o mandelbrot mandelbrot.c

guyomes 3y ago

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/ http://www.bohemianmatrices.com/gallery/

DanielGeisler 3y ago

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

enriquto 3y ago

Fascinating to see the Greg Egan participating in these experiments!