Why is pi here? And why is it squared? A geometric answer to the Basel problem

A most beautiful proof of the Basel problem, using light.
An equally valuable form of support is to simply share some of the videos.

Brilliant's principles list that I referenced:

Get early access and more through Patreon:

The content here was based on a paper by Johan Wästlund

Check out Mathologer's video on the many cousins of the Pythagorean theorem:

On the topic of Mathologer, he also has a nice video about the Basel problem:

A simple Geogebra to play around with the Inverse Pythagorean Theorem argument shown here.

Some of you may be concerned about the final step here where we said the circle approaches a line. What about all the lighthouses on the far end? Well, a more careful calculation will show that the contributions from those lights become more negligible. In fact, the contributions from almost all lights become negligible. For the ambitious among you, see this paper for full details.

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

Music by Vincent Rubinetti:

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

------------------

3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[SherinFunmes]

*Pi is like an uninvited guest who shows up at every party where he isn't supposed to be*

[henryg.]

Other mathematicians: QED
3Blue1Brown: Badaboom badabing

[qylzwyz]

As a high school math teacher teaching calculus, this channel has provided wonderful intuitions about how to teach calculus to students in a wonderful way. The essence of calculus will be delivered to students in an interesting way thanks to all people who helped to make this video!

[s_feles_]

I love the proof, but what I also find surprising is how the first four digits of π^2/6 are 1.644, like the year 1644 when the problem was first posed!

[number-kvpx]

As a Math major, I've read a great amount of solutions to this problem, but this physicly solution amazed me most.

[battleclan]

"In honor of Basel" or rather "We had to find something other to name it than 'Euler'"

[yds]

This is amazing! I have a PhD in physics, and I've never seen this proof. It's probably the best intuitive proof for this theorem!

[WilliamFord]

Math concept: [exists]
Euler: “My name is involved in this.”

[mariaceciliafp]

"I'm so tired of studying, guess I'll just watch some funny videos on youtube"
Me 30 seconds later:

[SludgeFuZZ]

That was absolutely beautiful. I must admit that I would not have questioned why pi is squared, but I can honestly say that I really enjoyed the answer.

[mushtaqrasool]

Professor gave us an insight of not only Mathematics but also Physics! Just shows how good of a teacher you are. Thanks for all of this.

[funkycude]

I've got a final exam to take in 10 hours and here i am watching 3B1B, best channel on YouTube IMO

[TwoForFlinchin]

this channel's quality is unmatched

[EvilDudeLOL]

This is incredible. So intuitive that, as a 14 year old kid with not very wide knowledge of calculus, I could understand it all. Splendid explanation– such characteristics are very rare. Thanks a lot, 3b1b, for this absolute masterpiece.

[gtb]

Unbelievably good :) I remember asking this same question in college, when I first saw this sum in a Fourier series class, and getting answers based on complex analysis :) This is so beautiful, thank you very much for posting this and providing fantastic insight.

[megablademe]

I am still in high school but love watching these videos, even tough I didn’t understand 95% of what he was saying.

[sebastianbg]

The first time you watch a 3b1b video you are puzzled by the new perspective it gives to the most common math problems. Then you incorporate that perspective into the way you solve problems (believing that you already understand everything). Then you watch the video again and new doors open, it's amazing how much ability you have to share knowledge!

[whogashaga]

Such an explicit explanation and high quality video!
Can't believe I missed this video for five years.

[vaibhavgupta]

The way you explained this is just awesome. This will remain in my brain forever.

[sstsvmq]

why these subjects are so interesting only when i'm preparing midterm exam