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The Hidden Architecture of the Basel Proof

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Tags: infinite-seriesgeometric-conservationinscribed-angle-theoreminverse-pythagorean-theoremthales-theoremright-trianglesinverse-square-law

In this video, we explore a geometric proof of the Basel problem:

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots=\frac{\pi^2}{6}$

Our story begins with a circle that repeatedly doubles in circumference, gradually flattening into the number line. When lights are placed at the odd integers, a hidden system of right triangles appears between consecutive circles. Together with the inverse-square law and the inverse Pythagorean identity, this geometry preserves the total observed brightness as the circle expands. By the time the circle becomes the number line, every odd integer has arrived at its final position—and the value of the Basel sum is waiting for us. This video was inspired by Grant Sanderson’s beautiful lighthouse proof for 3Blue1Brown. My goal was to explore another story hiding within that proof: the expanding circles that provide its architecture, and the way each integer appears to carry light forward to the next generation. The appendix proves a geometric fact assumed during the main argument: why the integers on consecutive circles align perfectly. CHAPTERS 0:00 Introduction 1:34 Part I — Building the Architecture 5:40 Part II — Conservation 9:07 Part III — The Hidden Geometry 11:05 Part IV — Passing the Light Forward 13:55 Part V — The Reveal 18:27 Appendix — Why Do the Integers Line Up?

Analytics

6.25 Overall score*

48 Rank

11 Votes

9 Comments

Comments

Phenomenal work! Even a computer scientist could follow:)

6.3

music when into to chapter starts, and then music fades out is highly professional

your video was WAY too slow speaking. I speed up to 1.5x speed and it still felt slow, so bumped it up to 1.75x speed, and then further to 2x speed. I suggest speeding audio up by 1.5x and adding a pitch modifier to your voice to make it deeper since it gets speed up. I dont think speeding up audio takes away from the authenticity of you speaking on camera at all. This won’t affect your score.

7:54 - “observer sees same brightness at each snapshot.” and a further elaboration plus memorability plus clarity

T-shirt at intro “light algorithm” wish i saw it up close, would of been cool at the end if you said the shirts are for sale. I’m not christian but I can appreciate the design. plus memorability

10:35 - one light becomes 2 (3,4,5 triangle). plus memorability plus clarity plus motivation (useful)

13:31 - got a little confused because those are no longer right-angle triangles. I feel this part could use more clarity.

14:55 - “flatness” had an audio bug; this issue won’t affect score.

1-9 Motivation: 4 (could have gone into practical applications beyond) Clarity: 8
Novelty: 3 nothing here is persay “new” but you did shine light on the basel problem. Memorability: 8 25/4 = 6.25

7.4

I liked the video, although I imagine it might be too technical for high school students, so I would change the target audience. Some of the images are overloaded with formulas, which don’t really help the viewer follow the video. However, if the target audience is more expert, they work perfectly well.

2.9

This is a blatant copy of the original by 3b1b. You use the exact same facts in the exact same way just maybe in a different order.

4.7

Feels very derivative of content already out there… I think I have seen a 3b1b video on this exact same topic and method. Still, it is done very well!

8.2

Not sure why, but the visualisations of the growing circles confused me quite a bit at the start. And the video felt a little bit slow at times. But overall very nice!

After watching this entry, I checked out your youtube channel and saw you’re primarily producing podcasts about hope and rationality. Although that change to the longform format for SoME took me by surprise, it made sense the more I considered it, as there’s an undeniable optimism to this video, both around the beauty of the math itself, as well as the audience’s desire to engage. I think this selects a particular audience— some folks might want to see a more grounded motivation, but the introduction is an honest and effective appeal to the aesthetic sense that pervades the video. It very accurately sets the tone and allows a viewer to decide if this is the sort of way they want to see math presented.

(NB: The only stumble in this regard might be the sentence “Some of the greatest math minds attempted, and yet failed”, which suggests that there’s going to be rather more historical focus then actually exists in the video.)

As I said, this aesthetic sense is really everywhere in the video and hard to pick out particular instances. But these two were particularly striking to me: First, the explanation of how “one inverse square becomes two inverse squares” around 10:30 is great intuitive way to lead into the next set of ideas. And second, using Thales’s Theorem was something I expected, but its second appearance was quite satisfying.

There were very few moments when I was confused, although I did pause the video a few times to try to predict where we were going next. Some of those predictions were successful, which might have made the arguments go down a little easier for me than someone watching straight through. (I don’t think this is a negative, just a comment that if you did want people to engage with the video that way, I’d advise a little more nudging to do so.)

The main video concludes strong, albeit with perhaps a bit more philosophical weightiness than works for me. (The last sentence, calling back to the generation analogy, is a cute touch.) However, I was also surprised to see that you wrap up the appendix very cleanly, since it would be quite justified to simply state the proof and sign off (the video “already ended”, after all).

8.3

I found it extremely enjoyable to see the Basel problem identity demonstrated using elementary geometry. I had also appreciated the 3b1b video on this topic, but this one has a thoroughly pleasant, “old-school” pedagogical feel. It is truly a video—or an approach—suitable for first-year university students or even high school seniors (referring to the French system); furthermore, if followed by the “classic” proof involving the factorization of the sine function, it highlights the connection between integers, geometry, and analysis.

The pacing and length are perfect, and the sequencing is excellent (for instance, saving the section on the collinearity of the points for the end was a very smart move).

I hope this “old-school” video wil get high ratings; it fosters a love for classical mathematics while highlighting the bridge to more advanced theories.

Overall I enjoyed the video and found the proof delightfully interesting. But I feel the organization of topics should have been much clearer, and it should have motivated much earlier why we’re interested in pi^2/4. It still isn’t clear to me whose proof this is - is it yours? Is it published?

More specifically: I think the intro (and thumbnail) would have benefitted greatly by showing the equation of the Basel Problem - you state that “integers, circles, and triangles” are interesting, but this just sounds like typical Euclidian Geometry. I’m watching this knowing in advance what “the Basel problem” means, but I think an unfamiliar audience wouldn’t get through the expanding circle geometry segment without first seeing the fascinating math identity.

In fact, a math equation doesn’t appear until 4:20. This equation also isn’t the Basel formula - which appears afterwards at 4:49. The history is a bit confusing to me, because it wasn’t clear that the proof you present is different than Euler’s original proof.

Unfortunately I’m giving this a lower rating because, while the proof is fascinating and the animations are excellent, as a YouTube video there were many times in the video where I would have clicked away. It would have helped me if you had reorganized the info, e.g. lead with the algebraic argument (15:24), then say “to complete the proof, we will present a geometric proof, inspired by 3b1b, that this sum is pi^2/4”.