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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    If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
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Comments • 2 528

  • Kirk Hodges
    Kirk Hodges 15 hours ago

    Dude, don't show tau in your visual for pi (1:30) and think tauist won't go nuts.

  • Hoiman Siu
    Hoiman Siu 16 hours ago

    Holy moly thats so creative

  • Lochlan Smith
    Lochlan Smith Day ago

    11:31 he almost summoned satan for a moment there.

  • karolak kolo
    karolak kolo 4 days ago

    My favorite 3blue1brown video!
    I'm rewatching it for the 4th time!

  • Sondre Grøneng
    Sondre Grøneng 4 days ago

    Now, it might be important to note for anyone considering building an infinitely large circular lake with evenly spaced, perfectly identical, light houses, that this would only work in a vacuum with no light pollution.

  • ccdsah
    ccdsah 4 days ago

    So much easier with calculus

  • Imran Shishir
    Imran Shishir 7 days ago +1

    And my brain just shattered into a infinite number of pieces.

  • Sherin Funmes
    Sherin Funmes 7 days ago +43

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

    • surfer855
      surfer855 4 days ago

      Even Mister Bean hates Pi, for showing up at every party he goes with his teddy bear!

  • Riccardo Foschi
    Riccardo Foschi 8 days ago

    Tha quality of your videos is remarkable! You make hard math problem easy even for me that I do not have a mathematical background! Thanks for your great work!

  • Yo wut
    Yo wut 9 days ago

    What. I googled 'basel pie'

  • Carlos Yuichi
    Carlos Yuichi 9 days ago +1

    3blue1brow:YOU GET A PI, YOU GET A PI TOO, *E V E R Y O N E* GET A PI

  • Muhammed Ali
    Muhammed Ali 10 days ago

    Everyone animation..blowing my mind

  • Claylo
    Claylo 10 days ago +1

    6:02 On the left? That seems to be right...

  • DavidGP
    DavidGP 11 days ago +2

    Mindblowing stuff, presented by the most soothing voice imaginable. Awesome

  • marwan aljohary
    marwan aljohary 12 days ago +3

    That was the most exciting math lesson I've ever been to.
    Thank you for making math so fun.

  • j s
    j s 12 days ago

    So confusing

  • Dark Knight
    Dark Knight 12 days ago

    what software is he using to visualise this?

  • Federico Masini
    Federico Masini 13 days ago

    This is so awesome!

  • Alex Homer
    Alex Homer 13 days ago

    someone do some math on my symbol... : )( : triangulating circles with squares, haha

    • Alex Homer
      Alex Homer 13 days ago

      always wondered why I had an affinity for St. BASILS cathedral

  • Toni Scott
    Toni Scott 14 days ago

    try explaining in the beginning.... unless you don't want ppl to know why the light is 1/4 more...if you are smart you can explain the why...

  • Tilo Droid
    Tilo Droid 14 days ago

    I don't get the "multiply by 3/4ths" thing

  • bazoo513
    bazoo513 16 days ago +1

    Risking to be accused of using a pun, this is brilliant!

  • Loveena Saran
    Loveena Saran 19 days ago

    How is that the apparent brightness on that point is 1/d^2 for the circle ??🤔

  • Ornithocowian King
    Ornithocowian King 20 days ago

    Dear Ben: amazing video my dude

  • gk10002000
    gk10002000 20 days ago

    It is always interesting to see how some underlying things have a physical basis

  • S0L4ZUL
    S0L4ZUL 21 day ago

    Can you find relationships of these rectangles mentioned in the video (in link below) to circle(s)?

  • Pratik Joshi
    Pratik Joshi 21 day ago

    Can you please explain why the sum of all the natural numbers is -1/12 as per Ramanujan?

  • Jai Carlson
    Jai Carlson 22 days ago +5

    that settles it. mathematicians are witches, and must be burned at the stake.

  • Kenneth Garringer
    Kenneth Garringer 22 days ago

    Vexed causes cancerous

  • Kenneth Garringer
    Kenneth Garringer 22 days ago

    Guide lines taxes

  • Kenneth Garringer
    Kenneth Garringer 22 days ago


  • Kenneth Garringer
    Kenneth Garringer 22 days ago

    You're right cost money

  • Kenneth Garringer
    Kenneth Garringer 22 days ago


  • Kenneth Garringer
    Kenneth Garringer 22 days ago

    Ground ing

  • Kenneth Garringer
    Kenneth Garringer 22 days ago


  • Kenneth Garringer
    Kenneth Garringer 22 days ago

    And cutting down a lot of Forrest s

  • Kenneth Garringer
    Kenneth Garringer 22 days ago


  • Kenneth Garringer
    Kenneth Garringer 22 days ago

    In our township mean s higher engery bills.

  • gerardo santana trujillo

    Very nice, thanks a lot!

    PASUNURU SAI VINEETH 25 days ago

    0:53 As a Fedfan, I am so proud that Euler is also from Basel 😇

  • Sabertooth Turtle
    Sabertooth Turtle 28 days ago

    Am I the only one who noticed at 0:53, that the physicist who posed the mathematical problem did so in year 1644 and the answer to his problem was discovered almost a hundred years later, the answer was 1.644. Spooky.

  • 葉修誠
    葉修誠 29 days ago


  • Lim Pro
    Lim Pro 29 days ago +1


  • Deyu Zhang
    Deyu Zhang Month ago

    Thank you! Your class is also so hot in china! Can you open a pay way in China we want to support you

  • surfer855
    surfer855 Month ago

    Very good!

  • Abhishek Vankit
    Abhishek Vankit Month ago +1

    So there's good reason why everyone is not great at math

  • anelepercon YT jr.
    anelepercon YT jr. Month ago +1

    0:45 Answer: *1.644* 984... Proposed in *1644*

  • james van
    james van Month ago

    Although I didn't understand it, I still want to praise it.

  • Gwebanget Forever
    Gwebanget Forever Month ago

    A=Pi.r^2 is okay. But C doesn't equals 2Pi. To prove it, please examine 15° or below, on triangle's base line [ 2sin(X/2) ], arc line [ (X/360)× 2Pi ] and 'neck' [ (1-cos (X/2) ].

  • David Terr
    David Terr Month ago +1

    Wow, this was a very enlightening derivation of Euler's formula! (pardon the pun!) I never imagined this formula was so closely related to circular geometry. I wonder if it's possible to devise similar geometric proofs to the formulas for zeta(2n) in general.

    • gk10002000
      gk10002000 20 days ago

      Unfortunately the complex number domain used in such things and in Fourier Series and stuff becomes a stretch in the real physical world

  • Dustin Gunnells
    Dustin Gunnells Month ago +1

    You are Appreciated!

  • Fenil Kamdar
    Fenil Kamdar Month ago +6

    After every amination
    Me: Wowww!!!

  • Konstantin Kouptsov
    Konstantin Kouptsov Month ago +1

    I cannot say I am convinced.

  • zhang东
    zhang东 Month ago


  • xx ab action xx
    xx ab action xx Month ago +1

    ???, shouldn't the radius be 0.5 at 1:31

  • jhony angarita
    jhony angarita Month ago

    Lo único que hay que entender es que curva o recta es la forma de la línea
    Una línea curva no es una línea recta ni está compuesta por rectas el
    Concepto de pi está mal enfocado
    Procedimientos como algoritmo están elaborados para hallar los lados de polígonos
    Una circunferencia es una curva cerrada

  • Uni QuE
    Uni QuE Month ago

    Can you spot the mistake?

  • Aaron Farish
    Aaron Farish Month ago

    Ok I get it

  • Ivan Zrinski
    Ivan Zrinski Month ago +1

    Best Teacher ever!