Early view of Basel video

You should be proud. Ben did an outstanding job animating and narrating this video.

I'm impressed the "completed fixed" version came out so quickly after the beta version.

seerpea

Thanks! Indeed, receiving a bit of feedback from the patrons on the first draft was very helpful.

3blue1brown

Hey, just wanted to mention that the final version of the video turned out really great! I especially liked the extra attention given to the inverse-square law explanation, a beautiful little nugget for those of us who don't know anything about physics :) Really, one of your best videos to date. Really happy the collaboration worked so well!

Edan Maor

Thanks! There is always a risk of losing something in going form 1 to N, but I think the key is to find well-aligned people.

3blue1brown

I'm really happy with how well this collaboration turned out. I know there's a lot to worry about when bringing in new people to one's pet project. Of course I only have my own opinion to offer, but I think this is a great success!

Max Goldstein

We may or may not put out a sort of appendix to address this point. You can see the paper linked in the description for the full details, and the ones that *it* references also have slightly different framings for why the limit works out nicely.

3blue1brown

this is a very neat solution o the basel problem. Though I think especially with these geometrical proofs it is very important to show why taking the limit here is valid, why for example a lighthouse on the "opposite" side doesn't ruin the equality. I know you don't want to confuse with formality, but I think an epsilon argument might have been nice here. Especially since it helps to build intuition in what way a infinitely large circle is a line and in which dimensions this is valid

ChalkyChalkson

The animation at around 5:20-5:30 is a bit jittery.

Kevin McCurdy

Thanks for the feedback! I think the new version should help to alleviate these concerns.

3blue1brown

One thing that could have made my understanding a bit easier would be to mention *why* the angle between the observer and the diameter that connects the light house and the top of the smaller circle is 90deg (inscribed angles again)

Lionel Pöffel

Loved the vid. Only the comments here made it clearer to me though that jumping back and forth between 2d and 3d requires the insight that the observer and the light houses are located in a 3d world all the time but we're looking at them from the top for most of the video.

Lionel Pöffel

Just two minor visual details around 13:30. 1. The arc of theta could be a little smoother. 2. The point on the circle where the line segments meet go outside the circle. Perhaps making the circle thicker and drawing it over the lines would make it look better.

I hate to nitpick (and this video is as great as they always are) - but <a href="http://www.bbc.co.uk/blogs/magazinemonitor/2011/09/how_to_say_baselbalebaslebasil.shtml" rel="nofollow noopener" target="_blank">http://www.bbc.co.uk/blogs/magazinemonitor/2011/09/how_to_say_baselbalebaslebasil.shtml</a> Euler was German speaking, so you really want to say "Baa-zuhl". (and I just noticed I'm not the first one on this. Oh well). (Liked the inverse Pythagoran Proof)

Hmm, I'll try phrasing this to be clearer. The evens plus the odds give us the whole thing. So if the evens make up 1/4 the whole sum, the odds must make up the remaining 3/4 of that sum.

3blue1brown

Hmm, good point, I'll see if I can work something in there.

3blue1brown

Wow, this was awesome!

This is a classic. Looking forward to sharing with my friend who studies the zeta function (when you post to YouTube of course). What I really enjoyed about this solution (which I've never seen before!) is that I didn't even see the punchline coming till the very end. It was never clear to me until the last minute how the circular arrangement had anything to do directly with the infinite series, because the distances to the observer in the circular arrangement seemed intractably difficult to calculate. Love it. I'm also happy to say that none of the nitpicks you and others noticed were noticed by me. Thus, the video was perfect as far as I was concerned.

Jacob Mirra

I don't really understand where the 3 comes from in the 3/4 factor. Edit ok I've been rewatching it. So if you move all the numbers to the even numbers only in 1 to 1 correspondence, brightness goes as 1/4 since every lighthouse moves twice as far away. I can see that moving lighthouse 2 from the blue line scales in intensity with 3/4 when moved onto the yellow line to lighthouse at position 3, but how is this true for higher numbers? Since when doing 1-&gt;2, 2-&gt;4, 3-&gt;6, you always have a "times 2" relationship, but when 1-&gt;1, 2-&gt;3, 3-&gt;5, 4-&gt;7, you don't always have a "times 3/2" relationship which would give x3/4 intensity.

Ah, just noticed Courtney Hilton already pointed this out.

Awesome!!! One minor gripe though. It is more accurate to pronounce Basel as "Baa-zel".

Hi, as always, great video. Though I feel like the crucial step of making the circle (lake) infinitely big is not explained enough: it doesn't seem immediately obvious to me that the distance of the lighthouses from the observer is the same as the their distance along the circle circumference. It does make intuitive sense, as the lighthouses across the infinite pond are too far away to have any effect, so we can probably ignore the distance difference (straight versus along the circumference), but that's not really a robust argument. I think there should be some discussion about what it really means to take a limit of the circle size as it approaches infinity and why it allows us to simplify the notion of distance for both nearby lighthouses and those across the pond.

Petr Čertík

This took me way to long to get.

3blue1brown

Euler's proof was very different. Mathologer has covered it pretty well, so perhaps I will link to him.

3blue1brown

Ooh, good catch. This is why I show you these things :).

3blue1brown

Hmm, maybe it should be better emphasized that the inverse square law is a 3-dimensional phenomenon (and hence applies to a world where screens are 2d). In flatland, it would not be an inverse square law, but just an inverse law. Either way, the point is simply to give a physical representation to idea of adding 1+1/4+1/9+..., so the point is to make that physical representation be whatever it has to be, then to try to understand the rules of how we can manipulate it.

3blue1brown

i'm confused about the part of the video that explains why moving the screen/retina to two times further away, makes the "brightness" reduce to 1/4 of what it was before. in the video, it says that we need to think of our screen as a 2d screen with length and width, instead of a 1d line. but why is that? it feels like "cheating"; off course 2d stuff seems to square things, just like drawing a line, measuring its length, and then extending that into a 2d square squares the length (which is now an area). why can't we see how much the "brightness"/angle changes when we move a 1d screen away, by staying in 1d? or if you insist on making it 2d and then finding out that moving the screen away so it's twice as far, will cause the brightness to go to 1/4 of what it was before, then why do i believe that this is still true when asking the original question (ie, about a 1d screen)?

David Nijjar

This is unbelievably good ! The (your) creativity in these videos is so impressing.

Is this how Euler proved it? If not, could you give us an endnote mentioning his method?

Really neat video! One small nit - from 5:03 to 5:18 or so, when the larger squares are shown, shouldn't the rays of light continue to touch the borders of the smaller squares? It seems like the proportions or distances might be off a bit

This is great Grant (and now team)!... quick tiny thing, I'm pretty sure Basel is pronounced more like 'Baaasel' rather than basil (like the herb) :)

Is Thyme next?

seerpea

That’s pretty awesome.