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

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A most beautiful proof of the Basel problem, using light. Plushie pi creatures now available: http://3b1b.co/store More problem-driven learning: https://brilliant.org/3b1b Brilliant's principles list that I referenced: https://brilliant.org/principles/ Special thanks to patrons: https://www.patreon.com/3blue1brown http://3b1b.co/basel-thanks The content here was based on a paper by Johan Wästlund http://www.math.chalmers.se/~wastlund/Cosmic.pdf Check out Mathologer's video on the many cousins of the Pythagorean theorem: https://youtu.be/p-0SOWbzUYI On the topic of Mathologer, he also has a nice video about the Basel problem: https://youtu.be/yPl64xi_ZZA A simple Geogebra to play around with the Inverse Pythagorean Theorem argument shown here. https://ggbm.at/yPExUf7b 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. Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown ------------------ 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). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown
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Text Comments (1862)
Soohoon Choi (5 hours ago)
Beautiful video
Soohoon Choi (5 hours ago)
The number line is actually a number circle!
ilprediletto (2 days ago)
Wonderful! Oh! Wonderful!
Jan Lewandowski (2 days ago)
This is one of the best YouTube videos I've ever seen so far and I've seen much <3 It makes you think about life, my reaction is just "wow"
Wen's Channel (3 days ago)
can someone explain the 4/3 part for me plz? greatly aprreciated!
Bim (3 days ago)
Wooooooowwww....... That's craaaaaazyyyyy......
barney the fossil (3 days ago)
So this what the guy who broke a wine glass watches now
Franga Vita (3 days ago)
Then Pi is simultaneous came (from who-where) came (from who-where) came (from who-where)...
Franga Vita (3 days ago)
Why 1 + 1/4? Círculos? Yeuteamo
Alan Roberts (4 days ago)
With only 4 light houses there is only 1+1/4 visible from 8 point larger circle chord ref light house.
Alan Roberts (4 days ago)
I like Hype otters :)
Alan Roberts (4 days ago)
Gausian geometrication of circle edge. Odd number series denominator incremental.
William Brockus (5 days ago)
Isn't this how Bioshock Infinite ended?
sighthoundman (5 days ago)
Every student I've ever had would love this until the very end. What we're not proving here we're either referencing or basically saying "Not important". Everyone can follow it. But then the algebraic manipulations at the end are too many steps at once. (Maybe most especially for middle managers or salespeople. Yes, folks, you may have to deal with them some time in your career.) Just put Sum a(i) = S, so Sum a(2i) = S/4 (because of the inverse square law), therefore Sum a(2i+1) = 3S/4 (because of subtraction), and so forth (and you don't even have to mention convergence, or even worse, absolute convergence). (I'm putting the ugly, not-even-quite-ASCII version in the comments because I'm sure you can handle it, buy you have nice graphic artists for the video and the rest of us have to for our users as well.) It only adds a minute and it makes a smoother transition for the user and it's just one nice little "you're not stupid" touch that makes mathematicians and scientists seem friendlier.
Laurelindo (5 days ago)
Euler solved it? Wow, I am so surprised.
theRealPlaidRabbit (5 days ago)
So: the Leibniz Series (1 - 1/3 + 1/5 - 1/7...) converges to pi/4. If you square each of the terms, the sum 1 + 1/9 + 1/25 + 1/49... converges to pi^2/4. Square each of the terms of an infinite sum, and the result is the square of the original sum (times 4). This I would not have expected!
Jay M (6 days ago)
*takes one look at tumbnail* Riemann Zeta, my old nemesis, we meet again!
Toby Garcia (6 days ago)
18:27 - can someone explain why the brilliant.org example is incorrect?
Stan G (7 days ago)
Good old "Oiler"
coryj1990 (7 days ago)
8:58 I wasn't buckled up.
Raja Shahja (8 days ago)
B S !
Himanshu Samal (9 days ago)
Are you a Magician? #MathMagician👏
dAvrilthebear (10 days ago)
The last part was so hard, but it was all great!
HITESH ESWARALAL SAHU (11 days ago)
Amazing
samar banodha (11 days ago)
Hey ! Taking the lighthouse on the x-y plane is more like Taking components of a vector on x-y plane!
Saurabh Negi (11 days ago)
this one really blew my mind
Steeve Cantave (13 days ago)
The beginning of the video is what interests me the most, and it's the part about pi not being thought of as originating from circles. I wish more people would approach pi from a different perspective.
Rainer Buechse (13 days ago)
Thanks, thanks, thanks!!!
Tsavorite Prince (14 days ago)
I actually see as pi as this sort of thing that appears in math sometimes which just happens to be a circle constant, meaning there’s always going to be something related to circles somewhere.
Chinisa Innukshopa (14 days ago)
I remember bumping into pi^2 (or more accurately pi^2/2) as multiplier for the 4-content of a unit (4-)sphere [A=pi x r^2, V=4 x pi x r^3/3, C4=pi^2 x r^4/2, C5=8 x pi^2 x r^5/15, et cetera, ad nauseum, ad puke-your-guts-out-ium]
Modar Alnajjar (14 days ago)
Please translate this to arabic
조성현 (15 days ago)
오...
Sanket Mhaske (15 days ago)
Space-time curvature makes more sense now. Mind=Blown!
Jorge Eduardo (16 days ago)
Pi is the key to go to other dimension.
Kyle Countryman (16 days ago)
make to heaven foe ues pthyongon
nipun malik (16 days ago)
projection component may be used
Paulo Constantino (17 days ago)
ppl thnk jst bcse pi is to do wth crcls tht it cnt b anywhre els. mstak.
Vidyanand Wagh (18 days ago)
At the beginning you said, "You've never had the experience of your heart rate increasing in excitement, while you were imagining an infinitely large lake with lighthouses around it. Well, if you feel anything like I do about math, That is gonna change by the end of this video." And that does. I literally had goosebumps at the end!
Patar Nababan (19 days ago)
I think the idea of light house is as difficult as the rest of ideas involved in Basel program for a layman. For a mathematician, no light house idea is needed.
Glen Petherick (19 days ago)
7:34 "EXTERMINATE".....
Nouta Albert Einstein (19 days ago)
I beleive thisath problem can be solved by another method,this one of light houses really involves many considerations...I think we could use the Riemann sums formula at the limit where n turns to infinity to get the answer...
gosuF7d (19 days ago)
can you also prove in a geometrical way that integration of sinc function is pi?
paul barnett (20 days ago)
Why is the menorah hidden in Pi along with the Name of God?
Sung Woo Hwang (20 days ago)
this is beautiful proof.
Paul-Stelian Olaru (20 days ago)
What is the proof that the sum converges? (because you are using the fact that it converges in order to find the value)
Tarquin The Rotter (20 days ago)
Very good! Thank you.
Albert Abuzarov (21 days ago)
I'm so dumb...
Naimul Haq (22 days ago)
Madhava was the first to sum infinite series. The Arabs improvised upon it and a book written by the Abbasid mathematicians was preserved in the library at Alhambra. Books preserved in the library were translated by the Europeans and taught Kepler, Copernicus, Galileo, Descartes, Fermat, Leibniz, Newton, Euler etc., freeing the Europeans from the dark ages and into enlightenment, creating the modern age. How the white Europeans paid back the debt with Islamophobia.
Naimul Haq (22 days ago)
Simply magnificent, breath taking and spellbinding. Justifies Max Tegmark;s remark, that 'physical reality have mathematical structure' and 'a priori' nature of mathematical laws, rules, etc., implying the 'mind of god', or the mind of Ramanujan.
Rachim Baskin (22 days ago)
God damn! Thank you.
Michael Law (22 days ago)
Great, loved it!
ajaikumar nadar (23 days ago)
can anyone explain me why did he flip that 3/4 to 4/3 ?
Stone Zhong (23 days ago)
I read another "proof" when I was in high school, forgot which book, the idea is, sin(x) has root of 0, pi, -pi, 2pi, -2pi, 3pi, -3pi,etc if we consider sin(x) is polynomial, I guess it will be sin(x) = x(1 - x/pi)(1 + x/pi)(1-x/2pi)(1+2pi)(1-x/3pi)(1+x/3pi)... so sin(x) = x(1 - x^2/pi^2)(1 - x^2/4pi^2)(1 - x^2/9pi^2)... sin(x) / x = (1 - x^2/pi^2)(1 - x^2/4pi^2)(1 - x^2/9pi^2)... = 1 - x^2(1/pi^2 + 1/4pi^2 + 1/9pi^2 ...)+... so (x-sin(x))/x^3 = (1 + 1/4 + 1/9 + ...) / pi^2 with, (x - sin(x)) / x^3 apply L'Hospital's Rule, when x approaches 0 ==> 1 - cos(x) / 6x^2 ==> sin(x) / 6x ==> cos(x) / 6, it is 1/6 when x approaches 0 so (1 + 1/4 + 1/9 + ...) / pi^2 is 1/6
Arnab Dutta (25 days ago)
How do you mathematicians approach and solve such problems?
evil brat (25 days ago)
I feel I have to inject this thought - Why is Pi Here? Pi[e] is here & why squared because it is Not in the Refridge and squared cause someone thought it would look cool Not round.
wonderful nepal (26 days ago)
Great video
Jake Pettibone (27 days ago)
changing subject a little, if each circle is 360degrees and each degree has its own 360degree on a sphere.how would I write that?...tbone
Marouane Talaa (27 days ago)
Can anyone explain to me that step at 15:52 from odd fractions to all integers fraction pleaase
iam nutzzz (27 days ago)
hey you can extend the list by adding newtons gravity eqation shown at 5:10 time and thank you soo much it made me to not hate maths youare agood teacher!!! thks
yangjie xu (28 days ago)
amazing method to prove this question ! Thanks for the contribution
David K (29 days ago)
you guys really need to find another narrator. this 20 minute video could have been 8-10, and not annoying to listen to.
Thomas Moustas (29 days ago)
Amazing video
Vandana Magar (29 days ago)
Hey I didn't understand the part where you add the even inverse squares....plz help
An Giang Gia (30 days ago)
Wait, why am I here? ( ͡◉ ͜ʖ ͡◉)
Ankit Tiwari (30 days ago)
He is really genius
Simbion12 (1 month ago)
I don't quite get the part where the distance from the observer to a light post is approximated by the length of the arc from the observer to the light post. I get it why it works for "close enough" light posts, but for example the light post that is opposite the observer cannot be approximated that way as you can't really approximate a line with a semi-circle and the argument as the number of points gets infinite doesn't really work as there will always be points opposite the observer and this approximation works only for finite number of the terms in the series. Can someone please help me understand that last part of the proof as it seems pretty cool.
A.M. Fortas (1 month ago)
I am not in the very slightest mathematically inclined. But heck, sir, I understood that !! :)
Leeso nneville (1 month ago)
This is what we had to do at Papa John's to cut the pizza so glad I don't work there anymore
MrTiti (1 month ago)
well, i appreciate a lot of work and ideas but the end is a bit frickeld and added. you didn not use your transaltion to common sense anymore, but helped yourself out with some tricks please be aware that systematical thinking is the base for our society and i highly like you to be excellent and coming out of all our non-common sense
chunfeng zhang (1 month ago)
too trivial
chunfeng zhang (1 month ago)
great
Akaleti 47 (1 month ago)
I had to pause and rewind multiple times to understand this. But, it was worth it. It's so beautiful. You are awesome.
Minute PCM (1 month ago)
awesome
phil durre (1 month ago)
awesomev video
Bhargav Kartik (1 month ago)
Absolute brilliance
42_answer (1 month ago)
Just realised, Mathologer was my professor...
Gabtube252 (1 month ago)
now I'll be the coolest kid in my cal class
83vbond (1 month ago)
I subscribe to a lot of great channels, but you are the first one I am going to support on Patreon. You are a gift to the human race. Thanks for illuminating all of us.
83vbond (1 month ago)
I am quite a bit into photography (have had work featured in exhibitions etc), and knew about the inverse-square light falloff, but never thought deeply about it. The way you showed the expanding square grids illuminated my mind. Thank you!
Santiago Flores (1 month ago)
You are an outstanding explainer! One of the best I have ever encounter. Congrats!!
A Prkr (1 month ago)
Interesting mathematical coincidence, but (π^2)/6 ~ 1.645 is also the critical value for a hypothesis test with significance of α = 0.10. Any idea if that's just a coincidence or related to the properties of the area under the normal distribution itself?
Shreyas Atre (1 month ago)
Very Beautiful. Thank you.
DV8 (1 month ago)
But light is measured in lums ...
Arnon Tillmann (1 month ago)
So, the sum of 2 "superpositions" gives you 1 particle equivalent position measurement. And if you measure one you will know the position of the other lighthouse. Tell me more...
Tony Ng (1 month ago)
It is really interesting but I lost my way. My brain isn't smart.
J Walker (1 month ago)
Thanks for posting I enjoyed watching this. Wow blowing my mind actually. Radio signal is on the electromagnetic spectrum so it's essentially light in my opinion. this is really off topic but wouldn't the inverse square law of light prove/disprove flat earth? They assume the sun reaches a vanishing point. If someone get's where I'm going with this...
Venkatesh babu (1 month ago)
All ratios are specific numbers. Mostly all dimensions work with ratios. Phi is a specific ratio. To understand Phi if you take a line and divide by specific ratio Phi it gives what is called a circular curve like all ratios and proportions. Squaring means surface dimensional ratio or planar ratio. Somewhat like if you take a stretched sheet and take Phi square ratios you get what is a circular surface. The curves of these ratios in number line follows what is called a switch pattern.
Venkatesh babu (1 month ago)
The ratio of time to that of mass is 3×10^90000. The ratio of time to length 4×10^1600000000.
Venkatesh babu (1 month ago)
Mostly surface patterns on number line looks like a switched patterns of weaving
hanspeter2210 (1 month ago)
easy. Just make an educated guess for a function. Rectengular function. Do a fourier analysis. sin cancels out, remaining cos terms. cos terms always 1, with a factor of 1/n² so the infinite sum of 1/n² converges against the rectengular funktion.
Nhoj Strebor (1 month ago)
The Key to understanding Math is understanding it's language.
Mickey Bitsko (1 month ago)
Pi aren't square. Pi are round. Cornbread are square.
Tejasvi Sharma (1 month ago)
My god. This is the most beautiful thing I have ever seen.
Jiawei Sun (1 month ago)
16:15 why that equals to doubling the distance of every lighthouse? for the one at position 1, it is, but for others, they aren't.
Charles Goldbach (1 month ago)
one of the best math videos
山登 (1 month ago)
早口なんだな。ほとんど聞き取れ無かった。
Simply Connected (1 month ago)
Absolutely amazing ! This is genius surely ? Great mathematicians like Euler took many years to obtain a proof of this. It is one of the highlights of a first course in complex analysis, as an application of the Residue Theorem. I figured out my own geometric proof of Euler's Rotation Theorem, for which I couldn't understand any of the proofs I found online - I must post it someplace (hopefully there is no error in it!). I also have to say this is a very fine explanation, though some details I still have to think through, I still recognize from your explanation the elegance of this proof. The one tricky part though I think will be the limiting case for the similar triangles you mentioned. I worked out the ratio of the cosines of these angles is :- (1 - k) sqroot ( ( (ksin A)^2 + (cos A) ^2 ) / ( (ksinA)^2 + ((1 - k)cosA)^2 ) ) where k is the ratio of the smaller triangle to the larger, and A is the angle of the hypotenuse to the horizontal. This tends to 1 as k tends to 0, as claimed. However the problem is light intensity depends on SOLID angle, not on planar angle - so something more needs to be said here. One of the things I dislike about sketchy arguments is details are glossed over, so we could be hiding absurdities - like every derivative is always 1 - BECAUSE, Delta x approx equals 0, Delta y approx equals 0 - thus Delta x approx equals Delta y - HENCE Delta y / Delta x approx equals 1 - AND in the limit as Delta x tends to 0, it thus tends to 1, HENCE etc. Yes its nice to try and make it more accessible - but don't forget the basics ! An infinite radius circle is like an infinite straight line because it has ZERO curvature. Its a tricky concept - as we think the circle must come back on itself. But if the curvature has fallen to zero it never does that ! To me this is a heuristic proof, not a rigorous mathematical proof - though it is truly excellent, and very cleverly conceived.
龍烏 (1 month ago)
awesome!
Kuldip Gohil (1 month ago)
Sorry not got satisfaction solution need maths solution
medexamtoolsdotcom (1 month ago)
Pac man at 13:18. This thing put me to sleep. It's easier to see why the inverse squares add to pi^2 over 6 if you just find the fouier series for a damn squarewave. That makes so much less of a mess. You're welcome.
훼더 (1 month ago)
Damn this is amazing
Wilmer Day (1 month ago)
Flat earth theory , make a circle endlessly flat hahahahahahhahha!

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