Math Exposition on YouTube

This summer, 3Blue1Brown launched the third installment of the Summer of Math Exposition contest. Every year, I’ve been impressed and inspired by the quality of some of the videos in this contest. Over the last couple of months, I worked with Boris Alexeev and John Jasper to throw our hat in the ring:

I was surprised by how much I learned from this experience. In what follows, I reflect on the process of making this video.

Part 1. The topic

We started brainstorming two months ago. We had a few cool ideas involving linear algebra and combinatorics, but we felt that the most interesting-to-normies video would discuss the mathematics of gerrymandering. Notably, since gerrymandering revolves around planar geometry, it’s well suited for the screen. We identified a few directions we could go:

1. Sometimes, fair districts are necessarily ugly.
2. You can probably gerrymander for the minority party using convex districts.
3. You can always gerrymander for the majority party using convex districts.
4. Typical maps can be sampled using MCMC.

These are listed in order of familiarity to us. First, 1 is based on our impossibility theorem for gerrymandering. Next, 2 is based on our paper that uses Brownian motion to determine the probability (under some random model of the voter distribution) of being able to gerrymander for the minority party with a straight line. The analysis in this paper is pretty slick, but Brownian motion is not terribly accessible, so it wasn’t a good fit. For 3, we would follow the proof ideas in this paper by Bespamyatnikh, Kirkpatrick, and Snoeyink. We decided that 4 was a little too mainstream to be novel.

At this point, it was a toss up between 1 and 3. Since we were already familiar with 1, I did a deep dive on 3. I decided the proof of the main result would be less technical if I modified the statement:

Theorem. Given sufficiently nice probability measures and over the plane and a natural number , there exists a partition of the plane into convex sets such that for each .

While the original paper focused on distributions of points, I would take and to have continuous distributions. This would allow me to mimic the proof idea in Figure 5 of the paper, but avoid annoying combinatorial arguments by instead passing through the intermediate value theorem. After writing out the argument, it was still too technical compared to the 3-paragraph proof of our impossibility theorem, so we decided to go with 1.

Part 2. The result

In the paper, we provide definitions of

(1) one person, one vote with parameter ,
(2) Polsby-Popper compactness with parameter , and
(3) bounded efficiency gap with parameters and

before stating the main result:

Theorem. Given , there exists a distribution of voters such that every partition into districts violates at least one of (1), (2), and (3).

This presentation of the result was particularly relevant since the efficiency gap metric was the subject of a SCOTUS case at the time. However, the efficiency gap is not terribly intuitive. Also, in our proof of the result, the second paragraph concludes that (1) and (2) together force every district to be won by the majority party, while the third paragraph just explains how this violates (3). This means we can simultaneously clarify the result and simplify its proof by replacing (3) with the more intuitive requirement

(3′) A minority party with at least of the vote gets at least one seat.

Next, the proof doesn’t use the full power of (1), but rather, that each district contains at least a fraction of the voters. Also, the argument focuses on an arbitrary district, which leads to yet another improvement in the theorem statement:

Theorem. Given , there exists a distribution of voters in which the minority party has at least of the vote, but any district with at least of the voters that has a Polsby-Popper score of at least is necessarily won by the majority party.

While this version of the result is much clearer than the original, it’s still not perfect. The statement would have more shock value if it took the form “fair districts must be ugly.” Also, the second paragraph of the proof uses multiple rounds of “it suffices to show”, which indicates that we’ll get a simpler argument after contraposition. This suggests yet another formulation:

Theorem. Given , there exists a distribution of voters in which the minority party has at least of the vote, but every majority-minority district with at least of the voters has a Polsby-Popper score less than .

Interestingly, the notion of “majority-minority district” can be applied not only to the political minority, but also to any racial minority. This application was not evident in the original formulation of the impossibility theorem since the efficiency gap is fundamentally a partisan metric. For exposition reasons, the bulk of the video takes , , and without affecting the idea of the proof.

Part 3. The proof

The above changes to the main result already give vast simplifications to the proof. We made two additional modifications to optimize for video.

First, our original proof doesn’t explicitly use the intuition captured in Figure 1 of the paper. In the figure, we take a grid of squares and distribute 5 majority voters and 4 minority voters in each square. For such an arrangement, the boundary of a majority-minority district needs to carefully split “cut squares” so as to aggregate minority population and make up for losses in the “inner squares”. This is implicitly used in the first display of the proof, but for the video, we made this explicit: At least a fifth of the squares that intersect the district need to be cut squares. This isolates how being majority-minority impacts the geometry of the district. Also, since the total number of squares corresponds to area and the cut squares correspond to perimeter, this naturally motivates the study of isoperimetry.

Next, we wanted to bound the number of cut squares in terms of the perimeter. Our original proof used an epsilon-net-type argument, but we wanted constants that weren’t so ugly. We investigated tight estimates from integral geometry, but we didn’t have any luck finding a slick proof of these. Instead, we found a bound with a clean constant along with a satisfying proof that uses the probabilistic method. We were heavily inspired by Buffon’s noodle, and I’m kinda surprised we pulled it off.

I didn’t expect that optimizing for exposition would have such a profound impact on my depth of understanding of my own research, but here we are. I suppose I should have expected this considering how much I learn when I teach.

Part 4. The content

Now that we fine tuned our result and its proof, it was time to make content.

For the video, I was pleasantly surprised by the capability of all the software that came with my MacBook Air. After storyboarding on Google Slides, I made animations in Keynote. Then I wrote a script and recorded using GarageBand. I’m not the best voice actor, so this took multiple takes and lots of editing. Then I used the Screenshot app to record my Keynote animations in presentation mode, and I edited this video to line up with the voiceover in iMovie. The lion’s share of this work was easily the animations, in part because I was simultaneously learning what all was possible. Before starting, we toyed with the idea of using Manim, but we didn’t know what the learning curve would be, so we went with a more familiar option. I assume Manim would give me more control of various elements a la LaTeX versus Microsoft Word.

While I cooked up the video, John put the web app together. He would send me iterations, and we would discuss possibilities. He seemed to enjoy solving the underlying programming puzzles while injecting a dose of artistry.

We shared a draft of our content to a few people for feedback and let it sit for a week before making the final edits. Some of the pacing was a little off in the initial version, but I was too close to notice, so it was nice to get outside opinions.

Part 5. The algorithm

The content was finalized by August 1, but we still had work to do. Not only do we want our video to be competitive in the SoME3 contest, we want it to perform well in the broader YouTube ecosystem. After a lot of research, it became clear that people won’t watch our video unless they click on it. Meanwhile, the only data that informs this decision is our title and thumbnail. So these need to be optimized.

First, the title should present a curiosity gap: give just enough information to entice the would-be viewer. Also, the title shouldn’t be much longer than 50 characters, since otherwise YouTube cuts it off and displays an ellipsis at the end. This led us to

Why You Want Voting Districts To Be Ugly

since this contradicts conventional wisdom. We thought the title would catch the eye better with numbers, so we decided to make a reference to the Polsby-Popper score:

Why You Want Voting Districts To Be 4% Pretty

Indeed, the main character of our story (IL-4) has a Polsby-Popper score of 0.04. Then we got a tip that this could be interpreted as “at least 4% pretty”, so we went with

Why You Want Voting Districts To Be Only 4% Pretty

notably, avoiding a split infinitive. We also added the hashtag #SoME3 to attract the 3Blue1Brown viewership.

The first draft of our thumbnail was eye-catching:

but we didn’t like all the text, and the background was too dark to be passable as a map when zoomed out. So then we went with

We liked that the green text contrasts with the light background, but when the picture is small, it’s hardly legible. So we flipped through a bunch of successful thumbnails and came up with this solution:

At this point, we’re pretty happy with the title and thumbnail, but they still might change if we learn more about what works. YouTube’s algorithm is pretty mysterious, and that’s a good thing (it keeps undesirable gamesmanship at bay). For example, I don’t know why, but our video is not showing up on the hashtag landing page.

Any pointers are welcome!