Summer of Math Exposition

I liked this one. It had nice animation.

The wall does the work: no water is added or lost, so the curve is forced, not drawn. The limit formula enters as a definition and comes back as a result. One gap: the red parts shrink on screen, but nothing says why their total goes to zero. One line, and the gap closes.

Very nice explanation

The music is a bit too loud/clashing with your voice over

Very great video, visuals are great, and the topic is really easily visualizable and interesting. One suggestion I would like to make is that it takes a bit too long to get to the “point” of the video.

It was an novel, interesting depiction of relating area with the limit formula for e.

The introductory premise took a while to be established which I would have preferred being done immediately as I watch the video. (Something like instead of talking about pi and other numbers which felt a bit disconnected, it would have worked better if it started with the hook of asking the main premise directly.)

The background music is nice, but I feel it’s too disconnected from the math so it seems like it’s playing in parallel to the video instead of in the background. Maybe lowering volume during the math portion could work better.

I think that was an amazing video! It had a great pacing! It had a very nicely motivated question that it pursued. And you chased it down beautifully! I think the visuals were great!

I wish you would have shown and explicitly said the length $x_n$ IS the value of $e$ and how it formed a multiplication tower of $(1 + \frac{1}{n})$.

Other than that, oh! It would be wonderful to actually trade the thing out into an actual vessel and make the vessel stand horizontally, like a boot! That way you could actually physically measure the height of the fluid and compute the value of $e$ physically! Very nice physical model to compute $e$. I wonder how one would trace the exact $frac{1}{x}$ curve, but sure.

Thank you so much!

Overall, a nice entry to SoME. Aquarium analogy is a little bit weak, but better than not having anything.

Nice explanation. Also - If the mathematics department asks for an aquarium, REFUSE!!

Strong SOME this year! Love the visuals with movements. The short format with the simple goal of helping understand the limit geometrically feels like it should be the reason for a video. Well introduced, well finished (with a question). I think this video ticks almost all boxes. If I had to find something to improve I would have to say that the learning point was single and it could have been developed with more history or surrounding information. But I’m being picky. Would like to see many more like this.

Very nice and clear explanation. The visuals pair nicely with the algebra.

This is a very nice video. The creator did a wonderful job with the argument, both logically and graphically. I thought the pacing was just right, so that I could follow each statement, but it wasn’t too rushed, like most videos. If I were to try to provide some constructive criticism, I would say the music was perhaps a touch too loud, and the quality of the graphics could have been better (the equations and graphs are pixelated, which can easily be fixed by telling manim to compile in high-resolution). But these are somewhat minor quibbles.

The other criticism I could make is that the topic is fairly well known, so while I enjoyed the video—and I did learn something—I didn’t really learn anything that I didn’t already know. Again, this is a somewhat minor criticism, and I also understand that a “simpler” topic is probably the best way to start when you are making educational videos. So I’m not holding against the creator.

Some good ideas here — I liked the introduction and question about pi and e. Felt motivating and interesting as a question to someone new to the topic.

I would suggest keep working on this, or taking it forward to another topic. I’d consider losing the aquarium idea. I wonder if it feels like something notable to you, but won’t maybe resonate with a non-you audience and there may be a simpler way to present that specific part that does not involve the aquarium concept. In fact, maybe even just deleting that part would make an improvement.

Worth lingering more on the rectangle part too — a lot of stuff to juggle there mentally.

Definitely keep making videos — this was overall a good one, and I’d encourage you to continue.

Really nice! This is something I’ve never seen before, and I don’t think I’ve ever seen any visual intuition for the value of e. The animation was simple and direct; it got the point across without any unnecessary distractions and the narration was paced perfectly. It’s common for people to speed up when the explanation gets complicated, and this can leave viewers confused, but you did not fall victim to this.

Great job.

Nice video, plus nice visualizations. It was just a bit underwhelming for the video, since it was just the integral explained without using it. I was expecting a more geometrical approach based on how you started the video

The video can be improved in many ways. First, the “motivation” behind defining e in a different manner was not very motivating. The author claims that the traditional limit definiton of e isn’t “satisfying,” which does not make much sense. The aquarium example was decent in that it demonstrated why the area had to be equal, but the proof that followed heavily deviated from this idea and was hard to follow in general due to its lack of clarity. The video itself was not that memorable, either.

I like this. I think I liked it a bit more than the intended audience (which seems to be pre-calculus students: if you’ve seen calculus the talking-around of the integral drags a bit, and and if you’ve not seen ‘e’ before then your emotional attachment to the motivation will be pretty weak.) This is mostly because the “physical” model didn’t really fit the computation that well. We just keep breaking this aquarium ;-; In all seriousness, it has a bit of “math class real-world” vibes so common in HS math books, that is likely to turn the intended audience off.

That said, the explanations are lucid and the animation complements them well. In particular, the section starting at 4:40 has excellent pacing, in my opinion. I also think this was the hardest bit to get right, even more than the “integral” definition, so was surprised and impressed that you handled it so well. Incidentally, I’ve never seen this explanation before, and I definitely think I’ll use it in the future :)

(A minor gripe: the background music was a bit too loud for me.)

Not bad, though I feel there isn’t a real connection between the aquarium shown at the start (where the water level is horizontal) and the setup used subsequently.

On the other hand, I think it’s a good demonstration of the equality e = lim(1+1/n)^n.

So, it’s definitely suitable for use in class.

This is a nice way to view e, but is somewhat unmotivated (why do we care about this aquarium?) and seems to just be pulled out of calculus.

Cool demonstration. I would have said something near the beginning that we’re talking about “two-dimensional water” that has area instead of volume, just to make that clear. It also wasn’t clear “how” we were moving the water. At first the curve represents the corner of the tank as we vary the width, then it becomes a container of water itself. So the movement of he water under the curve felt like an abstract operation. I’ve never seen this particular demonstration of the inttegral of 1/x = ln(x), so the final limit argument was pretty cool. I thought this was short and well paced, and your narration sounded natural. I’d dial back the background music just a bit.

This was a very good video explaining a visual way to understand what e really is. I love how you used the example with a aquarium. One detail you could have added to your video is why the area under 1/x from 1 to e is 1, instead of just giving the viewers the definition. The most engaging part of your video is when you said “Is there a simple, visual, geometric definition for the constant e?”. It made me pause and ponder what you said and try to find a simple geometric definition. I liked how you ended the video with a problem to think about.

I thought this was a good way to motivate the limit definition of e geometrically.

The animation works really well, and made it very easy to understand your entire argument.

For me, the main limitation was simply the scope of the mathematical payoff. The geometric relationship between the area under 1/x and the limit definition of e was already fairly familiar to me, so the main new element was the aquarium visualization itself. I can definitely see this being a very effective and memorable introduction for someone encountering these ideas for the first time, but personally I didn’t find the underlying mathematical connection especially surprising/novel. Would have loved if you expanded the scope more - maybe elaborate more on the question you pose in the end or something else.

Overall, though, I thought it was a good, well-paced, explanation!

Effortful but not insightful.

I loved the visuals in the video. You used them to great effect with your descriptions. And I thought that the pacing was great once it got started. I find a lot of videos in this format go through formulas very quickly, but yours was just the right pace. I was thinking of suggesting including descriptions of how the formula for e you used came from the integral of 1/x from 1 to e, or how you could use the trapezium rule could be used instead of the approximation you used, but you kept it focused on a single explanation, which I like.

I felt like the video was a little slow to start, I know that you wanted to motivate the rest of the video by talking about how constants appear in mathematics, but I was skipping through it to get to the content.

And I think the only thing I would have liked to see was if you could use the aquarium in real life? Is there a straightforward way to construct the aquarium? I have seen real life demonstrations of Pythagoras’ theorem using water, where the water from the large square leaks out and fills the two smaller squares.