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.
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].
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[ 2.7 ms ] story [ 42.1 ms ] threadhttps://github.com/evanberkowitz/littlewood
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...
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
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.
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.
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...
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
[1]: http://www.bohemianmatrices.com/gallery/