New hire, and upcoming projects

Hey everyone, I thought I'd pass along a few updates for what's new and coming up in 3blue1brown land.

First off, I hired an artist this year, Kurt Bruns, both to help outsource some of the production and to extend the visual style in a new direction. He was the one who did the drawings for the Alice/Bob/Cube shadow video. One of the first main projects we've been working on will be a pretty big one, and the way it's shaping up now it'll more likely than not end up being a multi-part series covering some highlights from the history of algebra in the last ~600 years.

For a long time now, as you might know from past posts, one of the main topics on my radar has been the unsolvability of the quintic. That is, the theorem that you cannot write a general formula for solving polynomial equations with degree 5 or higher using just the operations of +, -, *, /, and n'th roots.  Most proofs of this theorem are quite complicated, and popular expositions end up either incomplete, in that they lean on statements like "mathematicians have shown that...", or inaccessible, in that they require a semester of buildup.

The most accessible and visually appealing one I've come across is due to Vladimir Arnold, from 1963, which has already had some good expository coverage online, including one of the SoME1 submissions. My initial aim was to modify this argument in the hopes of making it a bit simpler still and to give a slightly richer understanding of the relationship between permuting roots and the structure of formulas solving polynomials, in particular, the cubic and quartic formulas. The idea basically mixes together the essential ideas of Abel's original proof of this theorem from 1824 with those of Arnold a century and a half later.

One of the reasons I put this to the side last year is that the initial versions of the argument I was chewing on were either incomplete or shaped up to be more cumbersome to prove than I was hoping. I think it's now down to a point that can comfortably fit in an accessible video on YouTube, so I'll be excited to share it soon.

As part of this, I got more into the history underlying this problem, and boy oh boy are there some interesting human stories in the mix. The tale of Galois is the most famous, with his dramatic death at the age of 20 in a duel, although the retellings often exaggerate or invent certain facts that we simply don't know for sure about this event. What I didn't realize was how his life leading up to this point was no less dramatic and interesting, full of rebellion and unexpected academic failure for such a young genius. It also has been striking parallels with the life of Abel, who also died quite young, and struggled to get his work appreciated in his lifetime.

More than that, tracing the history of ideas that led to the initial proofs of the unsolvability of the quintic, most notably the work of Lagrange which set the stage for Abel and Galois, actually gave me a new appreciation for group theory, and how it really came about. What I would love to do, though I know it will take a lot of time, is to go through the history of the seemingly simple task of solving polynomial equations, and some of the fun human stories we find along the way and explain how it gave rise to modern algebra.

---

All that said, the next video will actually not be any of these, but a completely separate collaborative project with another YouTuber. I won't say too much here, other than to foreshadow that it'll be a fun mixture of Fourier analysis with mechanical engineering.

-Grant