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

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**Published on Mar 2, 2018**- A most beautiful proof of the Basel problem, using light.

Home page: www.3blue1brown.com/

Brought to you by you: 3b1b.co/basel-thanks

And by Brilliant: brilliant.org/3b1b

Brilliant's principles list that I referenced:

brilliant.org/principles/

Get early access and more through Patreon:

www.patreon.com/3blue1brown

The content here was based on a paper by Johan Wästlund

www.math.chalmers.se/~wastlund/Cosmic.pdf

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

usclip.net/video/p-0SOWbzUYI/video.html

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

usclip.net/video/yPl64xi_ZZA/video.html

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

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.

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:

vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

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林柏宇Hour agoOMG!!!

Siddharth Doshi10 hours agoMath is beautiful.

Helyus HD4 days agoPhy has actually smth to do with dimensions... wtf... If u think about physic and how the third dimension works... I can’t explain it, but it somehow makes all sense...

Jonathan Bryant4 days ago^{+3}11:30 what kind of math witchcraft is this?!

Rafael V.T5 days agoGordon: It's Not Basel But Basil

John Michael16 days agosmall quibble... the quote, "students are not vessels to be filled but fires to be kindled," has been paraphrased and plagiarized countless times, and attributed to everyone from ben franklin to apparently now plutarch, but the OLDEST verifiable attribution is five centuries earlier and goes to none other than the greatest teacher of all time, Socrates.

younes ayachi17 days agoOh creative

Pockets MacCartney17 days agoShow this with actual experiments.

Bigbad bith20 days agoThe maths is wonderful, but your animations are superb - how do you do that? Software?

Jack Smith21 day ago3blue1brown in all different colors

Pont Christophe21 day agoGreat animations!

Glenn Floyd21 day agoYou boring male!

Miker Shin22 days agoI'm left speechless.

David Ash23 days agoGreat video and solution! I would make the comment that, although visualizing the problem in terms of lighthouses is a great way to give a sense of physical reality to the problem, it isn't really mathematically necessary to the argument presented here. The key insight is the use of the inverse Pythagorean theorem to split one inverse square into two inverse squares. This is done repeatedly--in fact, infinitely many times--to create a series that converges to the correct series. The inverse Pythagorean can be proven fairly easily from the regular Pythagorean theorem and an observation about the area of a right triangle.

Joe Ricci23 days agoI think the truth is that the ratio of circumference to diameter does not exist on a number line because the creation of finite objects is not scientific.

For, imagine the smallest bubble that can be thought to come into existence.

No matter how small that bubble is imagined to be, lo and behold, its circumference is found to already be over three times larger.

Irrational? No.

What is irrational is believing that a finite universe created itself.

Rick W23 days agoA brilliant proof .

Saroj si24 days agoWao very nice

Farley Moab24 days agoepipolar constraint

Merveil Meok25 days agoEvery family in the land should have some of your genes. It’s hard work, I know that :)

Metastate1225 days agoRelated to the Lorenz transformation ?

Elliott Sampson26 days ago^{+3}0:40 challenge posed in 1644 first 4 digits of awnser 1.644 coincidence I think not!

tanki zoltan24 days agoJust wow.

Абдаллах Муслим26 days agowhat program has that video been done by???

neil r28 days agoi am in a v a c u u m

R S29 days ago^{+27}I want to nominate 3Blue1Brown the noble peace prize for year 2020. Thanks.

Абдаллах Муслим26 days agobecause of his wife having cheated on him it can not be))))

Nijat ShukurovMonth agoMan, what kind of pot you smoking...

ALEXANDER POLTZERMonth agoThat was awesome! Nice and wonderful work!

Juibum GeilheitMonth agoleft?

Shimul chandra dasMonth agowonderful

Gabriel ThompsonMonth ago0:04 OMG HOW DID YOU KNOW

HayJayDeeMonth agoHow much weed you have to smoke to get that wtf so increadable 👍👍👍👍👍👍👍👍

tovarischkrasnyjeshiMonth ago^{+1}This collided like a pair of trains with another intuition I have. Olber's paradox in astronomy asks why the night sky is dark if there's an infinite, homogenous universe; after all, any line drawn from your eyes will hit a star at some point. But if light is fading by inverse squares and we assume an average density (because it's homogenous) then shouldn't the amount of light one sees be approximate-able by the Basel problem? So there'd be a finite, relatively dim background radiation, easily overwhelmed by local light sources and prone to occasion dark spots. Also at some point in the explanation dark dust probably gets in the way.

Like, the big bang is supported by other evidence, but would this actually do away with olber's paradox?

Amorphous Torus26 days ago^{+1}Olber's paradox involves stars distributed roughly uniformly through infinite 3D space. The solution for the Basel problem just applies for an infinite line of lights.

The result in Olber's problem has to do with the fact that there are more stars at far distances than at close distances. So at each distance interval, the increased number of stars in that range cancels the decreased brightness of each individual star. (Think of each distance interval as a spherical shell of a fixed thickness centered around you. Area increases by the square of the distance, and individual star brightness decreases at the same rate.) Since each shell contributes the same brightness, summing over all of them gives an infinite answer.

DaeMonth agoYO LIL B HERE MY MAN

sebastian andres jara fritzMonth agothank you! I thought that mathematics was certainly boring, with this I got an idea of how important nature is and how it expresses itself.

From Chile!

Gizmoriderful YeMonth agoI liked the bright yellow lights in this video.

Mike AlexanderMonth agoThis is a really elegant & beautiful proof. Thanks!

Partha DeyMonth agoBeautifully made video

FurqanMonth agoWhy the video does not contain an Arabic translation?

abdullah almasriMonth ago^{+2}why would it? grant, the guy behind these videos isn't the one who creates translations. it's us the viewers who do this.

feel free to add arabic translations

Tanmay SahooMonth agoWhy do the squares of inverse of only the odd integers get added?

Tanmay Sahoo29 days ago^{+1}@Mike Alexander woah...thanks man. You're pretty good at explaining stuff 😊️

Mike AlexanderMonth ago^{+1}So for the final part, it's not immediately obvious what proportion of the full series of inverse squares is made up by the odd squares. However, it is fairly easy to see that we can turn the full series into the even-only series by multiplying each term by 1/(2^2). That means 1/(1^2) becomes 1/(2^2), 1/(2^2) becomes 1/(4^2), 1/(3^2) becomes 1/(6^2) etc. It follows that the even-only inverse squares sum to 1/4 of the value of the full series. That in turn implies that the odd-only inverse squares sum to 3/4 of the value of the full series. Therefore the value of the full series is 4/3 x the value of the odd-only series. We have shown that the sum of positive odd inverse squares is (PI^2)/8, it follows that the sum of all positive inverse squares is 4(PI^2)/(8*3), which equals (PI^2)/6.

Baddaboom baddabing, as they say!

Mike AlexanderMonth ago^{+1}It's because of the positions of the lighthouses around the circumference. The observer is always midway between 2 lighthouses, so if the nearest lighthouse is 1 unit away, the distance between each lighthouse is 2 units - so in one direction lighthouses are at 1,3,5,7... and in the other -1,-3,-5,-7...

The total brightness is (PI^2)/4, so by symmetry, the sum for just the positive odd inverse squares is half that, ie (PI^2)/8.

The final trick is to reason that the odd inverse squares contribute 3/4 of the full series. The video rushes this a little, took me a while to grasp, but I will try to explain...

Alex ShortMonth agodoes this only work for that particular circle?

[各式各樣魔方新手教程]泰卡Month ago17:10The 3 blue 1 brown equal 3藍1棕 haha!!

Bay XMonth agoYou should add Turkish language

Farhan Nafis RayhanMonth ago^{+1}15:39 shouldn't that add up to 0?

Tanmay SahooMonth agoNo because we add the "squares" of all the negative terms, which are positive.

Victor Silva CarmonaMonth agoRegardless of the search for the answer to the problem, do you know what you just did? Explain very well how light projects and expands in space. Which is supposed to be known, but it is hard to imagine why the night exists!

Amy KuklevaMonth ago1:45 pi creature: OH MY GOD, you put me in front of an INFINITE line of lighthouses, I'm TWO DIMENSIONAL, light falls off as 1/x, the harmonic series diverges, YOU ASSHOLE I'M BLIND NOW

GanyX PlaysMonth agoYou lost me at lighthouse

DmonitizeMonth ago^{+1}I have to say, I've seen pi in a lot of infinite series throughout my schooling but having it there was always a sort of "eh, this is just what happens" sort of thing, seeming magical.

Now its less magical in a very, very good way. I can actually visualize why pi is there and how it came about. Fantastic video, your visualizations are the toppest notch. I've seen about all good youtube math channels trying to study... this is by far the best and I'm sad that I didn't discover it earlier.

Godspeed for visual learners. Thank you good sir.

나무Month ago멍청한 내가 이해할 정도라니.. 설명오진다

C WMonth agoEVERY element, Molecule , Chemical, All have Specific Nuclear Magnetic Resonance Frequency that Determines tgeir Stare of Matter.

C WMonth agoAll energy uses Omnidirectional Wave Dispersement Naturally. Air watts motion , Radiowaves, Magnetic Resonance, Photons Electrons. The function of a vacuum and it's molecular volume determines Function of nuclear Magnetic Resonance Frequency aka what TYPE of matter it's Harmonic Transference Memory is " Quantum DNA " Why gold is not lead.

C WMonth agoPi Divided by Pi = Soundwaves

가을저녁MidnightMonth agoSo cool...

Kartik SrinivasMonth agoThis is the best possible deep thinking

Friedrich BahkMonth agoI saw this where is the stop of camera.

slayer 1Month ago14:16 realistically speaking there is no limit to a circle it is infinite it is what pie implies the circle will never be flat because it is just as it implies a circle if you would say relatively flat from our particular perspective with the games that we choose to use on that particular subject at hand that would be more correct

slayer 1Month ago^{+2}Will it be correct grammar mathematically to say that 1/4 of Pi square is similar to pi squared / 4

dat tanMonth agoSure there is nothing wrong 1/4 pi^2 = (pi^2)/4. Pi is always constant variable/ directional value mathematically say.

Eli RockenbeckMonth agoyes, but I wouldn't say that myself

Martin DLSRMonth ago^{+2}You should make a video about this using Fourier Series solution

Ayush KumarMonth agoawesome analogy

梁家河理工带学Month agogood

krakowMonth ago^{+53}12:57 that circular right angle kills me to this day

zke.ac6p2y5 days ago^{+1}Maybe he just forgot the dot inside... ? ^.^

Patrick Lerdsuwanrut5 days agoTech Made Easy the right angle symbol is drawn with a square, not a circular arc

Tech Made Easy17 days agoWhy so???

Ancap ZombieMonth agoElectric fields and light are actually the same thing, so there's no need to mention them separately as following inverse square laws.

Ancap ZombieMonth ago@Mike Alexander Well static electric fields become electromagnetic fields in the reference frame of a moving observer, with the one exception of plane-waves, which require an accelerating observer to have the additional magnetic effect.

Mike AlexanderMonth agoNot sure about that. Electromagnetic waves / radio waves are the same thing as light, but a static electric field is not, surely?

Clarabelle AlexandersonMonth agoNo one cares about that. It actually really helps to understand.

Petras LMonth agoIf schools woul have teachers like you, all kids woud understand math and geometry 110%! I have off-topic question: What software do you use to do all these magic animations? Thanks! You video is brilliant in all meanings!

Súper Saiyan 5Month agoToo long for a famous easy-to-derive series.

Chad MuseMonth ago^{+8}I've been wondering how this equation related to Geometry for more than 20 years since I first saw it in college. THANK YOU!

pratyush chauhanMonth agoHad to watch this video 3 times before I understood every bit completely...no doubts in the box!...that said...a very nice video with some new logics to work with.

Sandeep KumarMonth agoBhai please make video in hindi.

OlPaMonth agoThis video was really, really good

Not ApplicableMonth agoI just finished a trig final exam and although I understand what this dude is talking about, the whole pi squared over 6 is totally over my head. If pi is 180 degrees, I don't understand how squaring it then deviding by 6 gives the number of light houses.

Ailer AguilarMonth ago7:07 noo!

Jamshid DehghanianMonth agoNo offense but I prefer Euler's proof A LOT more! Much simpler and easier to follow.

abdullah almasriMonth ago>much simpler and easier to follow

true but the lighthouses one is more elegant ;)

TheBreakFastMonth agoDoes anyone notice that the challenge started in 1644, which are the starting four digits of the answer?