Dan Christensen, Sam Derbyshire and I are writing a paper on a fun topic I’ve discussed here before: the beauty of roots. We’re getting help from Greg Egan, but he’s too busy writing his next novel to commit to being a coauthor! Anyway, I’m giving a talk about this stuff today, and I think you’ll at least enjoy the pretty pictures:
• The Beauty of Roots: easy fun version.
• The Beauty of Roots: version for mathematicians.
The version for mathematicians has some proofs; the easy fun version states a few theorems, but it’s mainly pictures.
For mathematicians, I think the coolest part is the close relation between our main object of interest:
namely the set of all roots of all polynomials whose coefficients are ±1, and the Cantor set, which you get by taking a closed interval and repeatedly chopping out the middle third of each piece, forever:
They’re related because a point in the Cantor set can be seen as an infinite string of 0’s and 1’s, while a power series with coefficients ±1 can be seen as an infinite string of 1’s and -1’s. The sets called ‘dragons’, like the one at the top of this post, are also images of the Cantor set under continuous maps to the complex plane. But to understand how these facts understand our set of roots of polynomials with coefficients ±1, read the mathematician’s version of the talk.
Proof of Theorem 1 looks like it has a couple of typos: e.g. |z| < 1/2, and 'the annulus 1/2 < z < 2' (missing modulus bars).
Wow, just in time—the talk is in one hour! Thanks!
The talk went well—but amusingly, as I was walking out, I heard someone say “that was pretty different from the Cobordism Hypothesis”.
I just had a 20y flashback to a fascinating talk I once heard about Douady’s Banach analytic spaces and the Cantor set. But I don’t remember having seen anything of this sort in Douady’s work on Mandelbrot set stuff.
John Baez has a series of posts on the roots of polynomials having integer coefficients, titled “The Beauty of Roots”: part one, part two, part three, easy version of slides from a talk, similar slides with some proofs.