Summer of Math Exposition

Great choice for a video. Simple, compelling, and self contained.

I enjoy the video very much!

Personally, I think the first figure where there’re 3 sets of 4 points should be drawn separately into 3 figures instead:

• the randomly 4 points, yielding 6 slopes;
• the squared 4 points, yielding 4 slopes; and finally
• the collinear 3 points with another point not on the line, this yield 4 slopes also. I think this will hook the audience to “follow” the problem and try to come up with solutions themself. Even before you have shown all 3 sets of slopes and encourage the audience to “pause, grab pen and paper”.

And maybe you need to clearly put a “restriction” to the problem??? Apparently, the trivial solution where all points are collinear gave only 1 slope. I think this should be drawn as another figure to emphasize that we will NOT consider this as a solution.

I also think you shouldn’t jump to section 2 with examples for n in {4…9} too fast. You should tease the audience a little bit more, like, show the case of n=6 next since this is another even number, which gave audience an impression that the answer should looks like n slopes… The, broke audience expectation by showing the case n=5 which result in only n-1 slopes! The element of surprise will make storytelling more compelling and memorable, imho.

The 3rd section where you introduce problem formulation and transformation of the problem (from Points -> Permutations), the name mentioned (e.g., J E Goodman) should have year of the publication attached. This will allow easier search of the work later (for one who interest). It will also hints why the problem was hard and/or why should it be interesting.

Around the end of section 4, where you show 3 sets of 4 points. I think there should be vertical lines (dashed) to separate each set of points. Also the animation of “project in every direction” is hard to follow… Maybe try:

1. Draw a projected line L, where every point will be projected onto this.
2. For each point p in P, draw a line from perpendicularly to L. Also draw point p’ on L, which is a proejction of p.
3. Don’t draw many “projection” like the video is now doing, instead:
4. Rotate the line L and P’ (which is the set of all projected point p’). Rotate it not too fast so the audience and inspect that the order of points in P’ are “swapping”. Essentially, this is the animation the you’ve done at the start of section 5, but fix the main points and rotate the projected points instead.

The animation in section 5 is still crucial. It just that if the animation in section 4 has very similar structure, it will make transition from section 4 to 5 more smoother.

Overall, I think the problem is interesting! The answer is quite “easy” to guess (I think even highschool students can conjecture this): 2*floor((n-1)/2) slopes. But the proof is moderately hard and linked to many interesting area in mathematics.

Motivation The introduction was good. Got my attention

Clarity Well stated

Novelty This is a new problem that I hadn’t thought of.

Memorability This was a problem I actually understood. Nice video!

Very well presented and explained. I haven’t seen this problem before.

Good presentation, sometimes a bit fast but I guess people can pausr

Really approachable introduction to what feels like a “real” problem in finite geometry. The visualization of the permutations’ relationship to the physical slopes was wonderfully intuitive and well-animated

Would benefit from some tighter editing. Also, it would be nice to know why the arbitrary “not ALL colinear but some colinear is allowed” rule.

I am biased, as a combinatorialist. Fortunately for you, it is in your favor :P

You don’t need me to tell you this, but you have a very clean and well-practiced aesthetic, both graphically and orally, that it’s hard not to like. The jump to partially-digital graphics was very unexpected and quite delightful once the rotation began. Truthfully, I think that little animation will stick with me long after I remember what it was useful for! But I also liked that they didn’t stick around longer than needed; your whiteboard format was quite charming. Finally, the outline at the beginning was probably critical to my enjoyment of this exposition. It’s a small signpost but it segments the video very nicely, and it speaks to the general deftness of this script.

My only critique is that the final section (after the classification of moves) was a bit slick for my taste, in the specific sense that without realizing it for a while, I didn’t understand what the indices on the k_i were doing. That was because the diagrams on the previous slide had accidentally given me the idea (which you did not actually say) that we should perform a single crossing move of maximum order— that does, after all, flip all the points to the “correct” side (at least in the even case), which is what we want. I only realized the issue when I remembered that you’d given two different optimal configurations at the beginning of the video, and, wondering how that was consistent, realized that in my maximal flip, all the points would be collinear. Then I was able to make sense of your commentary relating the k_i to n (which I do think should have been written somewhere, not just gestured at). So this is a somewhat… forehanded critique(?) in that you actually gave me all the tools in the video to resolve my concern myself— but it did take a fair bit of effort that might have been avoided if the presentation had a bit more grit.

To this specific complaint there is a straightforward fix: after the proof, show one of each type of configuration following the constraints you describe. Now, I suspect that folks with less combinatorics background than me will have more pressing suggestions, but if those suggestions are like “do more examples”, this would be an example I’d recommend.

I liked the topic a lot. Execution also was great. One thing i would change is mentioning that the moves form a loop instead of saying by very similar logic at 11:41. Overall was pleasantly surprised by the proof and the quality was amzing :)

Ah, love me a good projective geometry problem! And your presentation style is great! Overall, maybe a bit slow and monotone, though.

The constraints weren’t really clear or written down. Lots of versions of this problem exist which was not even mentioned. It was a bit too slow. The transformation from geometry to combinatorics was good.

this video was slow paced for me hence it was easy to keep up

This was interesting, and the explanation was well-executed, but it wasn’t entirely clear to me where the idea to turn this problem into permutations comes from. It feels more or less handed down from on high that we’re just supposed to do this, and of course an answer does result, but I’m not sure why we were supposed to perform that specific technique instead of any number of alternative approaches.

Two other minor grievances:

• The sudden appearance of animation on an otherwise hand-drawn board at 3:40 felt a little jarring to me stylistically. Of course it doesn’t impact the explanation, but I would have committed to the whiteboard aesthetic here
• The video seems to be frozen from 9:34 to 9:55 while you seem to have intended to be pointing at other things

Really enjoyed this - great job!

Can’t honestly say I followed every step in the thought process, but loved the permutation explanation - felt like a Radon Transform type idea, taking a snapshot from every angle?

Feels like a problem that took an enormous amount of investigation to reach such an elegant conclusion. Really appreciated how slowly the narrator took things too!

This video surprised me with tackling a surprisingly complex problem through an elegant solution. The starting intuition works really well in hooking people in before formalizing the general problem and you deliver a clear explanation that is easy to follow. Some of the visual pacing and engagement could still be improved which mostly comes from the static nature of a directly-on-the-board style math explainer, but I would say it’s above average compared to most. Excellent job!

This was good! I think I was a bit confused at times about how the geometric rules carried over into restrictions on the permutation moves, but it’s a really cool problem, and I really enjoyed learning about the clever insights.

beautiful!

you should have emfisize more that you are looking for a lower bound since you have a costruction. so it is OK not to prove tha tyour conditions are sufitiant. also you swiped somthig under the carpet since you “did not” use the “not on tyhe same line ” asumption.

You can also improve the tqunical qualty, but other than that it is perfect!

It’s a neat little exposition into a question I might have seen on a math forum. When you first described the question I didn’t have any idea how to approach it; I thought perhaps analytical geometry could put some constraints on slopes between different points. I hate analytical geometry and love algebra, so I was pleasantly surprised that it was a group theory proof!

It is a nicely explained proof, the exposition goes at a nice pace, everything is explained in the correct order, and there is no chaff that can be eliminated. I really find nothing wrong with it (other than a bit more exposition or examples could have explained the key steps of counting the outside and touching moves, but that wasn’t too big a hurdle to jump, and watching it a second time was enough for me to figure it out).

That being said, that’s all it is, a nice little exposition. I don’t see anything that makes it pop, other than the actual proof which comes from the book. So I rated it slightly higher than average (because I have seen some CRAAAP), but I couldn’t give it much more than that.

I like the video. Good choice of topic - easy to understand. I liked referencing the book at the end. I liked the whiteboard style.

Explaining the types of move “Crossing”, “Touching” etc, would have been great to see some animated examples!

Also the summary, would’ve been good to see some animated examples of the original geometric formulation.

Great introduction to the problem with an intuitive motivation behind the combinatorial step. There was little visual intuition behind the rotating the points and projecting them onto a line.

Very nice! This kind of very specific topic is really fun, and it also shows just how much ingenuity is needed to solve a seemingly “simple” question.

A small suggestion: from the 7:30 mark, you might want to “slow down” a bit—or at least keep illustrating the equivalence between rotations and permutations. It’s not easy to follow!

I really appreciate the economy of the animation; it’s very simple and understated—perfectly suited to the topic.

Genuinely fascinating, though the part about the move sequences was a bit hard to follow