JMM, year-end reflections, and what I've been up to

JMM

I'll be at the Joint Mathematics Meeting in Boston next week. If any of you are attending and would like to meet up, please feel free to send me a DM or email me.

On Thursday, January 5th, I'm giving a lecture there in association with having been chosen for the 2023 JPBM Communications Award. It was quite the honor to receive, given that many of the past winners are math exposition heroes of mine. If any of you are there and able to make it do come to say hello!

Reflections on the year

Looking back on 2022 for 3blue1brown, I admit I'm disappointed. Often if I write something apologetic about not having output more videos, many patrons are very nice, reassuring me that one should aim for quality over quantity, and to not be too hard on myself. But I don't mean this just as an apology. It's natural to reflect on things at the end of a year, and seeing room for how things could have gone better, I'd like to share how I'm thinking about it.

The most disappointing thing is the sheer amount of time put into projects which were never finished. For instance, earlier this year, I made a small contribution to a Mark Rober video where he made a talking piano, which I guess is technically the most viewed thing I've ever contributed to (because, you know, Mark Rober). Originally, I planned to make a whole companion video talking about the algorithms behind getting a piano to synthesize speech like this, and how we could improve on a naive use of Fourier Transforms.

The hope was for something like the Wordle video. A programming saga of trying to find a improve an algorithm for a fun task, but wrapped around a central math lesson worth teaching, which in the case of Wordle was information theory. I spent a while messing around with writing different algorithms for the piano project and trying them out, but ultimately it felt like there wasn't any compelling narrative, or any central math lesson underlying it, so I set it aside. Ironically, the most compelling piece of math involved was the notion of a convolution, which ultimately I did make a video about for completely separate reasons.

I also spent way more time than I care to admit sketching out different possible explanations for the unsolvability of the quintic. I had a dream of marrying Abel's original proof (pre-group-theory) with a completely different approach from Vladimir Arnold in the 1960s, but ultimately anything I came up with had some caveat making the explanation more complicated than what I was aiming for. Unlike the piano project, this is one that almost certainly will result in a video. Enough animation infrastructure has been put into place that it'd be a waste not to. But more likely than not, it will just focus on describing Arnold's proof in as well-motivated a way as I can, rather than insisting on something sufficiently new.

That project led me to research more about the actual history of solving polynomials, including the life story of Galois, which brings us to the granddaddy of unfinished projects for the year. 

I've mentioned this in previous Patreon posts. In the spirit of shaking up my usual style, and armed with a new artist on hand to help illustrate topics outside what I might usually cover, I decided to make something like a documentary on the history of how attempts to find formulas for solving polynomials led to the invention of group theory. This idea is something very different from the usual fare of the channel, with visuals dominated by painted artwork and filmed interviews.

The story of Galois, one of the main figures in all of this, is quite famous in math circles. He died in a duel when 20 years old, how could it not be? But until reading more into it, I did not realize how great his life story is even before the dramatic end. It's all complicated by the fact that numerous biographies about him just kind of make up facts, or otherwise grossly exaggerated certain scenes and details, meaning much more time than I expected went into sussing out the truth from fiction in all this.

The scope of the project grew to be a bit daunting, and we spent a while doing different experiments to try to figure out what exactly the style of the piece should be. There's a natural trade off with quality of artwork and the time it'll take to do the project, and the advent of AI art this year threw an entirely new factor into the experiments.

I wish I had something more concrete to share with you for the partial progress on this, like a 10-minute snippet of the piece we have in mind, but sadly the process is not quite that linear. One thing I'm determined to do next year is work in a way that offers better previews for patrons along the way.  One good candidate here would be all the filmed interviews, which are largely done and edited together, but they don't make a ton of sense without the artwork and interspersed narration that each one will cut to.

It's still looking like a few months before the final version of this comes out, and at least right now the plan is to reduce the scope to focus more on the life story of Galois, and to only carry out the remainder of the plans about the history leading up to and following him if it seems like there's strong audience appetite for it.

What does this mean for next year? The most obvious implied goal is to make sure I get both the Galois project and the quintic out the door. But more broadly, the aim will be to better assess the scope of a given project ahead of time rather than letting it surprise me and to be more deliberate about what I choose to pour time into. Trying new formats, new processes, and going down the occasional rabbit hole is all well and good, but it can be taken too far.

To be clear, it's not all disappointment. I'm happy with the projects that did go out, with the Summer of Math Exposition #2, and certain other little experiments along the way, like making manim more interactive or introducing artwork into videos.

Also, I can't remember if I mentioned this on Patreon before, but one other unexpected new thing was a short contribution I made to a Netflix documentary "A Trip to Infinity". It's just a short clip involving using Fourier analysis to decode a message from a cardboard Godzilla which represents Steven Strogatz's conception of infinity before calculus. I promise that sentence makes sense in context!

What have I been up to this month?

I left off the most recent video on convolutions with a teaser about how the next video will continue the discussion in the continuous case, with a focus on the theory of probability and what makes the normal distribution so special. Instead of framing this as "convolutions part 2", I think the better approach here is to make a video (or perhaps two) dedicated to the central limit theorem. 

Convolutions play a key role, of course, but the driving question will be "why does the function e^{-x^2} of all functions crop up so much in statistics?" There's one visual intuition I'm excited to share. I doubt it's completely novel, but it's certainly not a typical one, and it offers a nice connection to why π shows up when normalizing this curve.

This will be the next video, and I'm excited about it, but a lot of my month has actually not been spent working on it directly. Over the year I've accrued a somewhat long list of to-dos for manim, the animation software behind the channel. The list contains a lot of little things which have never been urgent at the moment, but which stand to reduce friction in the video creation process in the future.

For whatever reason, as I started to animate the central limit theorem video, I felt compelled to chip away at a few of them. And then a few more. And, well, at some point I decided it would feel nice to wrap up the year clearing off as much of the list as I could to set a clean slate for next year's animating.

I'll admit to there being at least a little indulgence when I shift gears to work on manim. To me, it's a little bit like having a self-built car sitting in the garage as a hobbyist project. Does it make sense to build your own car? Maybe not. Is it the highest priority thing I could work on right this moment, especially considering the relative dearth of videos this year and the very thing I was criticizing myself for in the previous section? Hard to say. But it is undeniably satisfying, and it's something I know to be useful for all future videos.

One friend of mine suggested this kind of work on manim would be something perfect to delegate to someone else. I can understand why from the outside looking in they'd say that, but aside from the fact that I take pleasure in it, it's hard to overstate how nice it is when animating a video to be completely aware of how everything in the stack works and to feel a sense of confidence in readily making changes. This is especially true for videos that require introducing new capabilities to the animation tool, which the best videos often do.

In the past, I've felt well-rewarded for combing back through the library to refamiliarize myself with how everything works as I work on the following projects.

What's the upshot?

I'm actually incredibly excited for 2023. It feels like a lot of potential energy is built up, with seeds planted for future crops. Naturally, one goal for next year is to actually capitalize on that potential energy and output more videos.

Another goal is to re-tune my focus to the core mission of the channel That is, explain pieces of math that people are commonly confused by, where well-chosen visuals stand to add a lot of clarity. As enticing as it is to be drawn in by a good story (e.g. Galois), by a clever bit of problem-solving (e.g. the "olympiad level counting" video), or by an aesthetically enticing bit of math visualization, all of that should be secondary to whether the core topic is something people actually want to understand, and whether the visuals actually serve to clarify. For instance, maybe next year is the time to finally just sit down and make that series on probability without overthinking it.

And as I said, another goal is to find workflows that lend themselves to more previews of partial work on the way. You should all feel abundantly free to yell at me if 2023 doesn't see a higher frequency of posts to this feed. I'll do my best to share the partial progress of the Galois piece as it starts to coalesce and to get earlier drafts of lessons out to you for feedback.

Happy new year!
Grant