Math exposition on YouTube – Brief, Fats Matrices

2023-08-04 18:34:08

This summer season, 3Blue1Brown launched the third installment of the Summer of Math Exposition contest. Yearly, I’ve been impressed and impressed by the standard of a number of the movies on this contest. Over the past couple of months, I labored with Boris Alexeev and John Jasper to throw our hat within the ring:

I used to be shocked by how a lot I realized from this expertise. In what follows, I replicate on the method of constructing this video.

Half 1. The subject

We began brainstorming two months in the past. We had just a few cool concepts involving linear algebra and combinatorics, however we felt that probably the most interesting-to-normies video would talk about the arithmetic of gerrymandering. Notably, since gerrymandering revolves round planar geometry, it’s effectively suited to the display screen. We recognized just a few instructions we might go:

Typically, honest districts are essentially ugly.
You may most likely gerrymander for the minority get together utilizing convex districts.
You may at all times gerrymander for almost all get together utilizing convex districts.
Typical maps might be sampled utilizing MCMC.

These are listed so as of familiarity to us. First, 1 relies on our impossibility theorem for gerrymandering. Subsequent, 2 relies on our paper that makes use of Brownian movement to find out the chance (underneath some random mannequin of the voter distribution) of with the ability to gerrymander for the minority get together with a straight line. The evaluation on this paper is fairly slick, however Brownian movement is just not terribly accessible, so it wasn’t match. For 3, we’d observe the proof concepts in this paper by Bespamyatnikh, Kirkpatrick, and Snoeyink. We determined that 4 was a bit too mainstream to be novel.

At this level, it was a toss up between 1 and three. Since we have been already conversant in 1, I did a deep dive on 3. I made a decision the proof of the primary end result could be much less technical if I modified the assertion:

Theorem. Given sufficiently good chance measures $mu$ and $nu$ over the airplane and a pure quantity $k$ , there exists a partition of the airplane into convex units $C_1,ldots,C_k$ such that $mu(C_i)=nu(C_i)=frac{1}{k}$ for every $iin[k]$ .

Whereas the unique paper centered on distributions of factors, I might take $mu$ and $nu$ to have steady distributions. This is able to permit me to imitate the proof concept in Determine 5 of the paper, however keep away from annoying combinatorial arguments by as a substitute passing by way of the intermediate worth theorem. After writing out the argument, it was nonetheless too technical in comparison with the 3-paragraph proof of our impossibility theorem, so we determined to go together with 1.

Half 2. The end result

Within the paper, we offer definitions of

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

earlier than stating the primary end result:

Theorem. Given $delta,gamma,alpha,beta,k$ , there exists a distribution of voters such that each partition into $k$ districts violates no less than one among (1), (2), and (3).

This presentation of the end result was notably related for the reason that effectivity hole metric was the topic of a SCOTUS case on the time. Nonetheless, the effectivity hole is just not terribly intuitive. Additionally, in our proof of the end result, the second paragraph concludes that (1) and (2) collectively power each district to be received by the bulk get together, whereas the third paragraph simply explains how this violates (3). This implies we are able to concurrently make clear the end result and simplify its proof by changing (3) with the extra intuitive requirement

(3′) A minority get together with no less than $1/2-epsilon$ of the vote will get no less than one seat.

Subsequent, the proof doesn’t use the total energy of (1), however slightly, that every district incorporates no less than a fraction of the voters. Additionally, the argument focuses on an arbitrary district, which results in one more enchancment within the theorem assertion:

Theorem. Given $epsilon,delta,gamma$ , there exists a distribution of voters during which the minority get together has no less than $1/2-epsilon$ of the vote, however any district with no less than $delta$ of the voters that has a Polsby-Popper rating of no less than $gamma$ is essentially received by the bulk get together.

Whereas this model of the result’s a lot clearer than the unique, it’s nonetheless not good. The assertion would have extra shock worth if it took the shape “honest districts should be ugly.” Additionally, the second paragraph of the proof makes use of a number of rounds of “it suffices to point out”, which signifies that we’ll get a less complicated argument after contraposition. This implies one more formulation:

Theorem. Given $epsilon,delta,gamma$ , there exists a distribution of voters during which the minority get together has no less than $1/2-epsilon$ of the vote, however each majority-minority district with no less than $delta$ of the voters has a Polsby-Popper rating lower than $gamma$ .

Apparently, the notion of “majority-minority district” might be utilized not solely to the political minority, but in addition to any racial minority. This utility was not evident within the unique formulation of the impossibility theorem for the reason that effectivity hole is essentially a partisan metric. For exposition causes, the majority of the video takes $epsilon=0.1$ , $delta=0.15$ , and $gamma=0.01$ with out affecting the concept of the proof.

Half 3. The proof

The above adjustments to the primary end result already give huge simplifications to the proof. We made two further modifications to optimize for video.

First, our unique proof doesn’t explicitly use the instinct captured in Determine 1 of the paper. Within the determine, we take a grid of squares and distribute 5 majority voters and 4 minority voters in every sq.. For such an association, the boundary of a majority-minority district must rigorously cut up “minimize squares” in order to mixture minority inhabitants and make up for losses within the “inside squares”. That is implicitly used within the first show of the proof, however for the video, we made this express: At the least a fifth of the squares that intersect the district should be minimize squares. This isolates how being majority-minority impacts the geometry of the district. Additionally, for the reason that complete variety of squares corresponds to space and the minimize squares correspond to perimeter, this naturally motivates the examine of isoperimetry.

Subsequent, we wished to sure the variety of minimize squares by way of the perimeter. Our unique proof used an epsilon-net-type argument, however we wished constants that weren’t so ugly. We investigated tight estimates from integral geometry, however we didn’t have any luck discovering a slick proof of those. As an alternative, we discovered a sure with a clear fixed together with a satisfying proof that makes use of the probabilistic methodology. We have been closely impressed by Buffon’s noodle, and I’m kinda shocked we pulled it off.

I didn’t anticipate that optimizing for exposition would have such a profound affect on my depth of understanding of my very own analysis, however right here we’re. I suppose I ought to have anticipated this contemplating how a lot I study after I educate.

Half 4. The content material

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

For the video, I used to be pleasantly shocked by the aptitude of all of the software program that got here with my MacBook Air. After storyboarding on Google Slides, I made animations in Keynote. Then I wrote a script and recorded utilizing GarageBand. I’m not the perfect voice actor, so this took a number of takes and many enhancing. Then I used the Screenshot app to file 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 simply the animations, partly as a result of I used to be concurrently studying what all was doable. Earlier than beginning, we toyed with the concept of utilizing Manim, however we didn’t know what the training curve could be, so we went with a extra acquainted choice. I assume Manim would give me extra management of varied parts a la LaTeX versus Microsoft Phrase.

Whereas I cooked up the video, John put the web app collectively. He would ship me iterations, and we’d talk about prospects. He appeared to get pleasure from fixing the underlying programming puzzles whereas injecting a dose of artistry.

We shared a draft of our content material to a couple individuals for suggestions and let it sit for every week earlier than making the ultimate edits. A few of the pacing was a bit off within the preliminary model, however I used to be too shut to note, so it was good to get outdoors opinions.

Half 5. The algorithm

The content material was finalized by August 1, however we nonetheless had work to do. Not solely do we would like our video to be aggressive within the SoME3 contest, we would like it to carry out effectively within the broader YouTube ecosystem. After loads of analysis, it grew to become clear that individuals received’t watch our video except they click on on it. In the meantime, the one information that informs this resolution is our title and thumbnail. So these should be optimized.

First, the title ought to current a curiosity hole: give simply sufficient data to entice the would-be viewer. Additionally, the title shouldn’t be for much longer than 50 characters, since in any other case YouTube cuts it off and shows an ellipsis on the finish. This led us to

Why You Need Voting Districts To Be Ugly

since this contradicts typical knowledge. We thought the title would catch the attention higher with numbers, so we determined to make a reference to the Polsby-Popper rating:

Why You Need Voting Districts To Be 4% Fairly

Certainly, the primary character of our story (IL-4) has a Polsby-Popper rating of 0.04. Then we bought a tip that this may very well be interpreted as “no less than 4% fairly”, so we went with

Why You Need Voting Districts To Be Solely 4% Fairly

notably, avoiding a cut up infinitive. We additionally added the hashtag #SoME3 to draw the 3Blue1Brown viewership.

The primary draft of our thumbnail was eye-catching:

however we didn’t like all of the textual content, and the background was too darkish to be satisfactory as a map when zoomed out. So then we went with

We favored that the inexperienced textual content contrasts with the sunshine background, however when the image is small, it’s hardly legible. So we flipped by way of a bunch of profitable thumbnails and got here up with this resolution:

At this level, we’re fairly proud of the title and thumbnail, however they nonetheless may change if we study extra about what works. YouTube’s algorithm is fairly mysterious, and that’s factor (it retains undesirable gamesmanship at bay). For instance, I don’t know why, however our video is just not exhibiting up on the hashtag landing page.

Any pointers are welcome!

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