Oddbean

If you draw all roots of all polynomials whose coefficients are ±1, you get an amazing picture that raises lots of challenging puzzles! I really hope someone reads our short article: https://www.ams.org/journals/notices/202309/rnoti-p1495.pdf and solves the main puzzle: why do the fractal regions of this set look so much like "dragon sets"? We have a good heuristic explanation, but no proof yet. Read on for a bit more about this: (1/2) https://media.mathstodon.xyz/media_attachments/files/111/153/875/818/969/329/original/b538457645986dd7.jpg

At left you see two closeups of the set of all roots of polynomials of some large degree with coefficients ±1. At right you see "dragon sets" - fractals described in a simple way depending on where we do the closeup. They look very similar in character... but not exactly the same. Can you make this precise and prove it? You can see a heuristic explanation here: https://math.ucr.edu/home/baez/roots/ (Search the page for "dragon".) This has got to be the key to solving the puzzle - but nobody has yet turned this idea into a precise statement and proved it. If you ever wanted to prove an interesting theorem about fractals, this could be your chance. (2/2) https://media.mathstodon.xyz/media_attachments/files/111/153/932/996/109/180/original/dda0c7f75479ad33.jpg

@b4c50e1b IIRC when I played around with drawing this fractal some years ago, I didn't calculate roots of polynomials at a fixed degree, but eliminated regions (pixels) that couldn't possibly contain a root of any series extended from a given finite prefix (with the number of distinct prefixes limited in practice by GPU memory). I have no idea if this approach generates the same mathematical object in the limit, but it looked similar to my eyes. https://code.mathr.co.uk/littlewood has probably bitrotted

@b4c50e1b Wow! I am curious how many points are in the set for (say) polynomials of order 21 - there are \(2^{20}\) distinct polynomials, each with 21 complex roots - giving over 21,000,000 points! I found this very surprising. It's amazing to see such beautiful complexity arise from something so simple!