Summer of Math Exposition

It is a highly techincal video with good concepts, but feels to long without any hooks or story. The material in it is great.

For a topic that is rarely covered, this video did really well at introducing it and showcasing some of the fascinating results that can arise from it. The animation, script, and math content were all really good. There were even connections drawn to other areas of math, such as elementary symmetric polynomials and conic sections. I am definitely interested in seeing more projects from your channel in the future.

Some areas of improvement:

• I would refrain from starting your video with a very long disclaimer about how the viewer is unprepared to watch the video
• There is a LOT of algebra… I was expecting (from the disclaimer) that we would be talking about some very high level stuff, and I didn’t see that
• There are also too many examples; if your objective is to get to the 5 point case, probably just jump there earlier
• Animations, especially for the parameterization of the folium you start with would go very far! You can even do them in Desmos if you want

I think there’s a way for this video to be interesting, but at present, it’s missing the motivation that gets me excited about the folium and then hits me with a bunch of calculations and I just zoned out ^_^;;7

Oh no! This video is forcing people to go and explore some cool mathematics! What an appalling tragedy. :)

You sound passionate, which is always great. I’d much rather learn from someone who really cares about what he’s telling us.

Excellent video, particularly for a first attempt. These are some cool and surprising results. I’ve never seen this before, and I’d never have come up with it.

But that raises the question of motivation. I can follow all the steps, but I don’t know where they come from. How would someone think to construct this combination of four or five points, especially when three doesn’t have any nice properties? And why would they think to connect that to the conic? Help the audience build that intuition.

Wow, first video? Incredible. I can’t treat it as a first for scoring, but I will adapt my comments. The video is a great example of a teaching maths video with the intro providing level, examples, equations, visuals, gaps to allow viewers to pause and work out the answer themselves. I think some motivation at the beginning would be good. The development and ending draw together so many ideas that it would be appreciated to have some idea of those at the beginning. Overall, I’m going to subscribe.

I was a little confused about why this was an interesting problem, I know math can sometimes be just about solving problems, but there seems to be no motivation for this one. Without a broader context this seems like a problem in isolation with no connection to anything. Also, why is the 3 non communitive while all the others seem to be, this seems like an interesting point that might apply to other areas of math. Also, the novel result came out of nowhere, there could have been a bit of buildup to that. Visuals were good, though at times a bit messy, but that is to be expected for what you were showing. Video was engaging and easy enough to follow. Thanks for uploading.

This was neat! Totally believe that you are an experienced teacher! Even though the presentation is very simple, it is more than enough. You could leverage the medium of video a bit more by showing how, say, certain coefficients or coordinates change when parameters change. Just to give the viewer a clearer connection between the things that are connected. But that’s a minuscule point. The only real downsides of the video are 1. that you often speak for a long time while nothing moves (which is irrelevant to the explanation but makes it a bit harder to keep focussed while watching) and that you speak over an empty screen quite a few times and 2. that the whole video lacks a bit of motivation. But yeah, love me some good projective geometry.

I really like the animation at the end. Interspersing more graphics throughout the video would help to clarify the content. The content could be broken into separate videos to allow more time for explanation while keeping the video at a reasonable length.

Overall this is a clear and engaging video. Your teaching skills really come through. The animations are great and make the ideas clear.

Minor point about the first several seconds: For engagement and to immediately orient the viewer, it would probably be better to show the graph of the folium rather than blank space. Also, a few times later in the video, the screen goes blank while the voiceover continues. This makes the viewer feel slightly abandoned, since the visuals provide such a good anchor throughout the rest of the video.

You spend about 30 seconds on background without saying specifically what the background is, so you could cut this or move it a little later to get the viewer to the main topic sooner.

The discussion of the folium product of 2 points is great. My only suggestion is that it felt like the equations f(t_i, t_j) = … and u_{ij} = u(f(t_i, t_j)) were setting us up to then get a formula for u_{ij} directly in terms of t_i and t_j, and you stopped just short of that.

The transition to non-associativity is great. This is a natural question.

However, the transition to 4 points, “A potentially fruitful question to ask would be ‘Is the folium product of 4 points well-defined?’”, is unmotivated. We shouldn’t expect associativity for 4 points if there isn’t associativity for 3 points. Also, “A potentially fruitful question” sounds disingenuous because it suggests the viewer should see it’s an interesting question without the benefit of already knowing the answer. Something like this would be more straightforward: “But there’s a surprise when we look at products of 4 points.”

“Say we join u_1 and u_2. That forces u_3 and u_4 to be joined together.” Here you assume the bracketing (u_i . u_j) . (u_k . u_l) without establishing why. You just showed that other bracketings can change the result, so why are we ignoring the other 4 possible bracketings? Also, non-associativity means it’s ambiguous to write u_1 . u_2 . u_3 . u_4 without brackets.

In the discussion of 4 points… Since the previous main point was about non-associativity with 3 points, the viewer thinks we’re still asking about associativity, but now with 4 points. So what we get is an apparent muddying of associativity vs commutativity, since instead of associativity you explore a sort of commutativity by swapping u_2 and u_3. It’s not clear why we’re doing this or where it’s taking us.

You reference elliptic curves as if it’s an explanation, but I’m familiar with elliptic curves and don’t see what you’re getting at.

It would be great to give more of the history. What did Descartes do? Is all this recent or centuries old? At the end, you say this result is probably novel. But it would be better to say this earlier rather than retroactively, because now the viewer isn’t completely clear about how much of the video to retroactively apply novelty to.

I pointed out several small issues, but overall the video is great, and I look forward to seeing more from you.

This is a great video, especially for the very first video on your channel. The visuals were simple, yet powerful. What a beautiful discovery of the connection between conics and the Folium! The idea of product of two points on the Folium was fascinating and yielded fruitful results. Well done! It was also well-narrated with a good voice.

As for constructive criticism (to help with future videos):

*You could eliminate the introduction where you talk about the necessary math background required. The first 30 seconds are extremely important for getting a viewer’s attention, so you don’t want to waste it. For example, you could maybe begin with the visual you used at the end of the video and ask the viewer a question about the connection between points on the curve and a conic. So a stronger intro that grabs attention would be helpful. (Note: The YouTube algorithm will “reward” and “punish” videos based audience retention, and this is most important at the beginning of the video.)

This was a phenomenal video; I found the animations to be spot on and the content interesting. I truly enjoyed how it truly felt like I was learning in the video (for example, working through the constants with elementary symmetric polynomials and having the viewer do it as an exercise the second time). Moreover, this video felt very novel, not a topic I have seen discussed often before. The only comment I can give for improvement might be a bit of a story or motivation as to why this is an interesting question to tackle beyond the beauty of the math because I believe that for a wider audience beauty of math is not a strong enough motivation.

I’m about 6 minutes into this video, and while I am finding it somewhat interesting, I am somewhat puzzled as to motivation. The video just got started by saying we’re going to talk about some properties of the folium of Descartes. My response is, well, okay, but why? It may be implicitly interesting to the creator, but it is not necessarily interesting to the viewer, at least not without some type of motivation.

The creator seems pretty knowledgeable, but this presentation is really just a lecture with pictures. The power of video is never really taken advantage of. Unfortunately, this makes the video less valuable, in my opinion, than one that actually uses video to demonstrate things that can’t be demonstrated with static images.

Love the exploration of the shape, the concept of connecting the product to the geometric construction. The animations are simple but pretty. The only thing I think is missing is some more dynamic movements, similar to the ones in the very end. Either way, great video.

Pleasant video that was pretty easy to follow and to explore a simple idea on projections of simple curves. I liked that you encouraged people to go through the calculations to really help to get the intuition for the algebra in their mind and I was found the intersection with conics very interesting at the end. I think this video can be tagged as suitable for high schoolers, as it was explained quite well. I think a small edge case that needs to be addressed is what happens when the folium product is not well defined (e.g one of the points is on the cusp)