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

  • 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:
    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:
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Comments • 2 383

  • 林柏宇
    林柏宇 Hour ago


  • Siddharth Doshi
    Siddharth Doshi 10 hours ago

    Math is beautiful.

  • Helyus HD
    Helyus HD 4 days ago

    Phy 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 Bryant
    Jonathan Bryant 4 days ago +3

    11:30 what kind of math witchcraft is this?!

  • Rafael V.T
    Rafael V.T 5 days ago

    Gordon: It's Not Basel But Basil

  • John Michael
    John Michael 16 days ago

    small 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 ayachi
    younes ayachi 17 days ago

    Oh creative

  • Pockets MacCartney
    Pockets MacCartney 17 days ago

    Show this with actual experiments.

  • Bigbad bith
    Bigbad bith 20 days ago

    The maths is wonderful, but your animations are superb - how do you do that? Software?

  • Jack Smith
    Jack Smith 21 day ago

    3blue1brown in all different colors

  • Pont Christophe
    Pont Christophe 21 day ago

    Great animations!

  • Glenn Floyd
    Glenn Floyd 21 day ago

    You boring male!

  • Miker Shin
    Miker Shin 22 days ago

    I'm left speechless.

  • David Ash
    David Ash 23 days ago

    Great 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 Ricci
    Joe Ricci 23 days ago

    I 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 W
    Rick W 23 days ago

    A brilliant proof .

  • Saroj si
    Saroj si 24 days ago

    Wao very nice

  • Farley Moab
    Farley Moab 24 days ago

    epipolar constraint

  • Merveil Meok
    Merveil Meok 25 days ago

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

  • Metastate12
    Metastate12 25 days ago

    Related to the Lorenz transformation ?

  • Elliott Sampson
    Elliott Sampson 26 days ago +3

    0:40 challenge posed in 1644 first 4 digits of awnser 1.644 coincidence I think not!

  • Абдаллах Муслим

    what program has that video been done by???

  • neil r
    neil r 28 days ago

    i am in a v a c u u m

  • R S
    R S 29 days ago +27

    I want to nominate 3Blue1Brown the noble peace prize for year 2020. Thanks.

  • Nijat Shukurov
    Nijat Shukurov Month ago

    Man, what kind of pot you smoking...


    That was awesome! Nice and wonderful work!

  • Juibum Geilheit
    Juibum Geilheit Month ago


  • Shimul chandra das
    Shimul chandra das Month ago


  • Gabriel Thompson
    Gabriel Thompson Month ago


  • HayJayDee
    HayJayDee Month ago

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

  • tovarischkrasnyjeshi
    tovarischkrasnyjeshi Month 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 Torus
      Amorphous Torus 26 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.

  • Dae
    Dae Month ago


  • sebastian andres jara fritz

    thank 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 Ye
    Gizmoriderful Ye Month ago

    I liked the bright yellow lights in this video.

  • Mike Alexander
    Mike Alexander Month ago

    This is a really elegant & beautiful proof. Thanks!

  • Partha Dey
    Partha Dey Month ago

    Beautifully made video

  • Furqan
    Furqan Month ago

    Why the video does not contain an Arabic translation?

    • abdullah almasri
      abdullah almasri Month 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 Sahoo
    Tanmay Sahoo Month ago

    Why do the squares of inverse of only the odd integers get added?

    • Tanmay Sahoo
      Tanmay Sahoo 29 days ago +1

      @Mike Alexander woah...thanks man. You're pretty good at explaining stuff 😊️

    • Mike Alexander
      Mike Alexander Month 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 Alexander
      Mike Alexander Month 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 Short
    Alex Short Month ago

    does this only work for that particular circle?

  • [各式各樣魔方新手教程]泰卡

    17:10The 3 blue 1 brown equal 3藍1棕 haha!!

  • Bay X
    Bay X Month ago

    You should add Turkish language

  • Farhan Nafis Rayhan
    Farhan Nafis Rayhan Month ago +1

    15:39 shouldn't that add up to 0?

    • Tanmay Sahoo
      Tanmay Sahoo Month ago

      No because we add the "squares" of all the negative terms, which are positive.

  • Victor Silva Carmona

    Regardless 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 Kukleva
    Amy Kukleva Month ago

    1:45 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 Plays
    GanyX Plays Month ago

    You lost me at lighthouse

  • Dmonitize
    Dmonitize Month 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.

  • 나무
    나무 Month ago

    멍청한 내가 이해할 정도라니.. 설명오진다

  • C W
    C W Month ago

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

  • C W
    C W Month ago

    All 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 W
    C W Month ago

    Pi Divided by Pi = Soundwaves

  • 가을저녁Midnight

    So cool...

  • Kartik Srinivas
    Kartik Srinivas Month ago

    This is the best possible deep thinking

  • Friedrich Bahk
    Friedrich Bahk Month ago

    I saw this where is the stop of camera.

  • slayer 1
    slayer 1 Month ago

    14:16 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 1
    slayer 1 Month ago +2

    Will it be correct grammar mathematically to say that 1/4 of Pi square is similar to pi squared / 4

    • dat tan
      dat tan Month ago

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

    • Eli Rockenbeck
      Eli Rockenbeck Month ago

      yes, but I wouldn't say that myself

  • Martin DLSR
    Martin DLSR Month ago +2

    You should make a video about this using Fourier Series solution

  • Ayush Kumar
    Ayush Kumar Month ago

    awesome analogy

  • 梁家河理工带学


  • krakow
    krakow Month ago +53

    12:57 that circular right angle kills me to this day

  • Ancap Zombie
    Ancap Zombie Month ago

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

    • Ancap Zombie
      Ancap Zombie Month 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 Alexander
      Mike Alexander Month ago

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

    • Clarabelle Alexanderson
      Clarabelle Alexanderson Month ago

      No one cares about that. It actually really helps to understand.

  • Petras L
    Petras L Month ago

    If 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 5
    Súper Saiyan 5 Month ago

    Too long for a famous easy-to-derive series.

  • Chad Muse
    Chad Muse Month 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 chauhan
    pratyush chauhan Month ago

    Had 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 Kumar
    Sandeep Kumar Month ago

    Bhai please make video in hindi.

  • OlPa
    OlPa Month ago

    This video was really, really good

  • Not Applicable
    Not Applicable Month ago

    I 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 Aguilar
    Ailer Aguilar Month ago

    7:07 noo!

  • Jamshid Dehghanian
    Jamshid Dehghanian Month ago

    No offense but I prefer Euler's proof A LOT more! Much simpler and easier to follow.

    • abdullah almasri
      abdullah almasri Month ago

      >much simpler and easier to follow

      true but the lighthouses one is more elegant ;)

  • TheBreakFast
    TheBreakFast Month ago

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