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.
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The content here was based on a paper by Johan Wästlund
www.math.chalmers.se/~wastlund...
Check out Mathologer's video on the many cousins of the Pythagorean theorem:
• Visualising Pythagoras...
On the topic of Mathologer, he also has a nice video about the Basel problem:
• Euler's real identity ...
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/a...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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"In honor of Basel" or rather "We had to find something other to name it than 'Euler'"
Too many Euler mathematical things 😂
@@Goonercry Euler's little theorem ( ͡° ͜ʖ ͡°)
There’s a joke that mathematical discoveries are named after the second person who discovered them, because the first is always Euler.
@@JackBarlowStudios I thought that was more of a humerous fact than a joke.
@@onradioactivewaves it's mostly a joke, but euler is incredibly influential nonetheless
As a Math major,I've read a great amount of solutions to this problem, but this physicly solution amazed me most.
I'm wondering if this convergence can be easily formulated. I mean, the argument is really intuitive but saying "we let de radius tends to infinity" isn't quite enough. You must control the amount of light from the far away lighthouses in a precise way. Not so intuitive problem of summable families where the number of terms vary... However, I totally understand that the rigourous summable families aren't fit for such a video :p
@@natanielmarquis6159 My idea is to ignore all lighthouses with a distance of more than R^0.9 away from the observer. The amount of light from each scales only as O(R^(-1.8)), while the number of them scales only as O(R). As R increases, the arc we are considering becomes longer and longer but also flatter and flatter.
@Nathaniel: you can go like this: Define a_i(n) to be the light emitted by the i-th bulb on the n-th circle, and if i is too big and there is no "i-th bulb" on the n-th circle, just set a_i(n) =0. You have that lim_n a_i(n) = (1/i)^2 , and by the above geometric argument you know that, for each n, sum_(i=-\infty) ^(+infty) a_i(n) = pi^2/4. The question is then if you can commute the limit and the infinite sum. The idea is to use Tannery's Theorem to conclude you can. Note that a_i(n) is decreasing in n: the i-th bulb has a fixed distance along the n-th circle from zero, but the circles keep flattening, thus the euclidean distance from zero increases. Also, note that when the i-th bulb appears is in the topside of the circle, so it has at least a distance of a radius from zero. At each step both the number of bulbs and the radius double, so that the radius of the circle when the i-th bulb appears is proportional to i. We conclude that |a_i(n) |
the solution of euler is the most simple but very difficult to understand why?
@number25 I was wondering what you think about my thoughts about Pi and Phi: Phi depends on .5, Division = Diameter Diameter = .5 = 1 diameter = 1 degree Radius is division of diameter (division of division) Phi is diameter x3 divided by/2, 3 halves, one and a half, 1.5 Pi is diameter x2 divided by /3, 2 third, 0.75, one and half of a half Phi x3/2 Pi ×2/3 Pi 1/2, 2/3, 3/4, 4/5, 5/6... Phi 2/1, 3/2, 4/3, 5/4, 6/5... Pi is multiplication of radius Phi is multiplication of diameter Basic principle of dividing/equalizing/sharing something in equal parts (itself) and multiplying something in equal parts (by itself) Calculating with circles, squares, triangles, bars, etc... it's very interesting if you think of Greek alphabet for example (Pi and Phi) and the first person to ever have to write "numbers" to explain mathematics and wrote it as such 1 2 3 4 5 6 7 8 9 0... and wonder why it was written precisely like that and if it is a "how to" calculate anything using geometry. It's easier to see if you put a set square of 360 degrees on a picture of old TVs test pattern for example. 360degrees being 1 circle You can calculate anything that way with degrees. You can calculate "nothing" precisely in the process, too as the outside of the circle. The TV and computer invention uses the same pattern and algorithm 4:4:4 of 1234 infinitely. Using both properly is an algorithm to multiply divisions or divide multiplications infinitely. The Greek alphabet letters are ways to calculate that way for specific functions, as well. But Pi and Phi are functions. Trying to give it a value would be like trying to give a value to +,×,÷,=,/, etc...
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!
What???? Yo thats a sick coincidence Edit: I started a whole conversation just because of a mistake lol
@@ANTI_UTTP_FOR_REALsick*
or is it?@@ANTI_UTTP_FOR_REAL
@@ANTI_UTTP_FOR_REALI love a suck coincidence!
@@ANTI_UTTP_FOR_REAL i’ll suck something else
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!
I’m imagining you wheeling in a cart piled with 80 camping lanterns and placing them all over the ground while rambling through the proof and all your students just thinking their teacher is insane. 😄
What I find troubling is that how do you square a number that can't be precisely defined?
Other mathematicians: QED 3Blue1Brown: Badaboom badabing
In India, it is *HENCE PROVED*
@Tech Made Easy "quod erat demonstrandum" latin for ("that which was to be demonstrated")
@@ViratKohli-jj3wj why did you fail in semifinals?
@@kingscross4233 😂😂😂
I wonder if he got that from Beakman's World
*Pi is like an uninvited guest who shows up at every party where he isn't supposed to be*
Even Mister Bean hates Pi, for showing up at every party he goes to with his teddy bear!
Maybe a surprise guest--always welcome, especially when not expected!
"e" also
Except ... he actually IS supposed to be there, he was simply uninvited.
pi is the party host.
the part from 13:54 to the end of video really did stretch my grey matter. Here it is for slow guys like me 13:54 the fact that the lighthouses (factors) are aligned on a straight line on either side of the observer (origin) and are squared(so all negative factors are now positive), results in π²/4 = 2 (1/1² + 1/3² + 1/5² + ...) so 1/1² + 1/3² + 1/5² + ... = π²/8 15:27 the thing we want to find out is what this series is equal to : 1/1² + 1/2² + 1/3² + ... = ? in order to find that out, we need to figure out how much share each of these parts 1/1² + 1/3² + 1/5² + ... (lets call this O - for odds) and 1/2² + 1/4² + 1/6² + ... (E - for even) have in 1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... (lets call this full term as O+E) maybe 3/4 and 1/4 or 3/5 and 2/5 or whatever combination. we need to find it out. 15:40 is where you pay close attention to what he says: "now you can think of that missing series as a scaled copy of the total series that we want" implying E = some scaled copy of O+E since this is inverse Pythagoras, the denominator part in all the factors for e.g. the 2 in 1/2² or the 3 in 1/3² is nothing but the distance from the observer. if you double all denominators in O+E then you will get E. 1/(1x2)² + 1/(2x2)² + 1/(3x2)² + 1/(4x2)² + ... = 1/2² + 1/4² + 1/6² + ... proving E = some scaled copy of O+E so the earlier 1/2² = 1/4 become 1/(2x2)² = 1/16 . similarly 1/9 becomes 1/36 etc... so doubling the denominators, all factors in O+E become 1/4 of its original. therefore E has a share of 1/4 in O+E and therefore O must have a share of 3/4. or (3/4) of O+E = O But it is already known that O = π²/8 (3/4)(O+E) = π²/8 or O+E = (4/3)(π²/8) or the complete term O+E = π²/6 I must say that your idea of explaining the Basel problem using circles has indeed helped guy like me reason this answer perfectly - big thanks ! I have enjoyed all your videos - you are exceptionally brilliant
Ah, that helps me understand where the 4/3rds came from. I think the video was just a little too quick or succinct in explaining that bit, but you have filled in the implied information.
I just rewatched the last few minutes after reading this amplified explanation, and I finally actually understood and followed Grant's narration this time! Yay, finally!
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!
Math concept: [exists] Euler: “My name is involved in this.”
Eulaaaaa
@@maxwellsequation4887 Even the Martians know him
Soon may the Euler man come
JNeal134 to bring us sugar and tea and run
@@maxwellsequation4887 HI I WATCH MATH ELITE TOO
"I'm so tired of studying, guess I'll just watch some funny videos on youtube" Me 30 seconds later:
Maria Cecília This is fun
Yeah, this is really fun if you know enough to understand :)
Maria Cecília you’re not tired of studying. You’re just tired of studying the conventional stuff, the conventional way
exactly definition jajajajaja
Wisdom not necessary. They just explained everything in such fine details, all that’s needed is just some imagination.
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!
Oh, yeah, at 13:42 just expand a circle into a flat line and ignore all the geometry he just showed us to accept a handwaved answer...
its a limit.@@Taric25
@@Taric25 Okay late response, but it's not exactly just expanding the circle. Notice also how the distance between each light house is preserved in expanding the circle. As the size of the circle is doubled, the number of light houses is also doubled, preserving the distance between each light house. Form a circle x^2 + (y-r)^2 = r^2. Intuitively, as you let the circle grow to infinity, more and more of the lighthouses basically touch the x-axis (or the real number line, in this case). Also notice how, in the limit, while the majority of the lighthouses will never actually be on the axis, any values that are elevated off the x-axis would already approach a distance infinitely far away. This would mean that the value of their light is already extremely low based on their distance from the origin alone (not counting vertical distance), as the value of light from that light house would be 1/(infinity)^2, which very quickly approaches zero. This means that the values on the real number line, in the limit, can again be seen as equal to the sum of all lighthouses. Hence, in the limit, it is safe to say that infinitely expanding the circle and keeping its bottom grounded to the origin would preserve the limit.
@@WhoCares-ue5hk, extremely low does not mean zero, especially when considering an infinite sum.
@@Taric25 You're not wrong, but what I'm trying to say is that, as you increase the radius to infinity, more and more of the total length touches the axis, so we can say that as we increase the radius to an obscene amount, we can say that the x-axis is at least a good approximation, and becomes a better and better approximation as the radius increases. As you let the circle grow larger and larger, the part of the circle tangent to the line x = 0 becomes the only part of the circle that we can even "see," and so it becomes basically equivalent to x = 0 at every point.
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.
I've got a final exam to take in 10 hours and here i am watching 3B1B , best channel on KZhead IMO
Ansh Shah All the best bro
Ansh Shah same bro 😂😂
Boards??
same
math paper tomorrow lol
why these subjects are so interesting only when i'm preparing midterm exam
Procrastination
what term is at its mid point in May? just curious.
@Tech Made Easy Thank you.
Tech Made Easy No, because a) the Chinese Spring term goes from Feb to Jun b) the OP's name is Korean
Exactly man... Here i am 1 year later
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.
Excuse me, can we exchange math together, my friend?
The explanation you made at minute 2:00, is absolutely beautiful and huuuuugely intuitive... You don't get infinite bright at the origin by adding up more lights... I absolutely loved it.
I wouldn't say hugely intuitive, since in 2D, the brightness does indeed go to infinity.
this channel's quality is unmatched
I am still in high school but love watching these videos,even tough I didn’t understand 95% of what he was saying.
Im only a toddler and love watching this kind of videos
I too am in 12 . Even though I can't understand
This is magnificent... your brains are building new neuronal connections as you watch and attempt to understand... And as you become ACTIVE in using new knowledge, (take notes and as you reproduce the ideas in your own words) you are building new neuron networks. Congratulations, you have just tapped into the process of becomming more intellegent!
@@ViratKohli-jj3wj abbe hindi nahi samjhega yaha pe kisiko 😂
@Timmie Collins Same
I think this is the fourth proof I see of this, and this is certainly my top or second favourite.The other proofs I know involve Fourier series, the residue theorem for infinite sums or a Lebesgue integral. The first two weren't that easy to understand when I was studying them because I hadn't quite yet understood everything that we were using to prove this, and the Lebesgue integral was actually quite cool because even though the function used came out of nowhere, the theorems used were very explicit on what they do and then the basic integral we get didn't require much more understanding. But I learnt these 3 proofs in Uni, and they would have seemed like total garbage if I had seen them before, whereas this one is actually understandable for most people out there who are willing to listen carefully and pause the video to think about it from time to time. This is what makes this channel so great and useful. It offers new persepectives and gives everyone intuitive and clear explanations, that only require a little of motivation from the viewers. Most videos are almost self sufficient, you don't need to watch an entire series to understand the video that caught your attention, they give you a better understanding of where everything comes from but the explanations are clear enough that you can do without those additional previous videos. Truly amazing.
I'd say the most intuitive one is the one by Euler. I discovered it myself as a student learning about the Taylor series of sin(x) and cos(x). From olympiad problem solving I knew that quantities such as the sum of the inverse roots of the zeros of a polynomial could be expressed in terms of the coefficients of the polynomial. Then I wondered what if I do this with sin(x)/x as an "infinite" polynomial. Lo and behold, out comes sum 1/n^2 = pi^2/6! I was aware that I could not formally justify these manipulations, but then I found out to my surprise that this was how Euler had "solved" it. If it's good enough for Euler... 🙂
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.
0:40 challenge posed in 1644 first 4 digits of awnser 1.644 coincidence I think not!
Just wow.
Nice observation man
Next digit is 9 (for 90 years when problem was unsolved) and 34 (for 1734, year before Euler solve this problem). It can't be coincidence
1.644*9* so actually yeah
nice
15:11 "the number line is kind of like a limit of ever growing circles" - i've been thinking of a number line like this since forever, i thought i was insane, but it makes sense now
DUDE SAME this video blew my mind with that statement
It gets worse. You can think of the complex plane as the surface of an infinitely large sphere. Lines on the surface of an infinitely large sphere wouldn't just approach being parallel, they would become parallel. Now if you plot the graph Y=1/X where X approaches 0, you might think it shows that as X nears 0, Y approaches infinity. Now I know a lot of people don't like it when you say that N/0=Infinity, but screw them. I do what I want. The real interesting thing here is if you take this exact same equation, Y=1/X, but after you plot it you run it through a Circle Inversion, you get to see what happens as the Y axis approaches infinity. Now I haven't actually done this, but it seems to me that the line Y=1/X would map right through the point of ''Infinity" and come out no worse for wear on the other side. Although this would be hard to see, as the plotted line would be hugging the Y axis pretty tight as it approaches that point. So to me this seems to pretty definitively answer the question that N/0=Infinity. Some people make the argument that this can't be true, because -N/0= minus Infinity. To which I say they are the same thing. The number line's an infinite circle, travel an infinite distance and you loop right back round again. Some people think that this can't be true, because if 1/0= Infinity, and 2/0= infinity, then does 1=2? To that I say no, 2/0 = 2*(1/0) = 2*Infinity, which is twice as big as the infinity we got before. It's different. This might not seem to make sense, but if you've ever heard the solution to the problem, How do you free up infinite rooms in an infinite hotel where every room is occupied? The answer is to move every guest to an odd numbered room, leaving an infinite number of rooms now unoccupied. Some infinities are bigger than others, this is not a contradiction.
This is not how you compare infinities. 1/0 is not infinity, because division by 0 is not allowed, and infinity is not a number. 2*infinity is not a "bigger" infinity. "Infinity" is in no way shape or form a real or complex number. What you can do is write that the limit of 1/x as x approaches "is infinity", which really means that as x approaches 0 1/x grows without bound. Analysis is good, treating infinity like a real number is unacceptable.
@@isaakvandaalen3899 Your fourth paragraph contradicts your statement.
@@isaakvandaalen3899 I've always felt like the infinity of space is a circle that comes around to a singularity. Not sure why I think this but psychedelic drugs may have played a part. :)
At 16:40 you could state that since 1/4 of the TOTAL apparent brightness (B) is contributed by the sum of the inverse square of the even integers, then 3/4 of B comes from the sum of the inverse square of the odd integers, which has just been shown to = pi^2/8. Thus 3/4 of B = 3/4 x pi^2/8. So B = pi^2/6.
Thank you. Really helped to clear things out
Excellent explanation! The video is brilliant but the explanation around this part is puzzling to me. How I convinced myself of the final answer was via this: pi^2 / 8 * (1+1/4 + 1/4^2 + 1/4^3 +...)=pi^2 / 8 * 1/(1- 1/4) = pi^2 / 6.
Fantastic stuff! I am relearning applied mathematics from this excellent approach. This is much more in keeping with the way Archimededes and Newton thought about mathematical thinking in science. Feynman would love all this, I think.
Well, this approach to the Basel problem is amazing! It combines physics, geometry, and maths in the same run! The inverse Pythagoras theorem is something new to me. Will check this out further. Thanks so much for this discovery on pi day!
15:11 "The number line is kind of like a limit of ever-growing circles" MY MIND IS BLOWN
My mind exploded so hard that my round skull became straight
@Federal Bureau of Investigation - FBI ...
Believe it or not, this perspective becomes very concrete in what is called projective geometry - and it is just as useful there as it was in this proof! (Sorry for dropping in 6 years after your comment)
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.
Dude, your future is bright! Keep going, keep getting curious
I needed this alooonnngggggg time ago... Love it, maths like this just has some kind of purity to it, I can never describe the feeling when your mind come to that moment of realisation and clarity, wish I had kept up with mathmatic skills and practice, and not have to start right back at the beginning lol
the fact you can take a summation to infinity and turn it into a circle is absolutely stunning.
drkscpe No its not
P G Balagopal Warrier I don't mean the way we all learned it... dividing circles up yeah I get that but what he's done with that is astonishing.
Sorry dude i was randomly spreading negatitivity on random comments. I dont even get half of what this guy is preachin
sorry you are not summing to infinity, if you want a shorter proof check fourier series
It absolutely is. If you think about it, is a circle just an summation of infinity small points? If you took a zero dimensional point, could that infinite summation of points make any mathematical dimension? These are the questions we all should be asking.
I want to nominate 3Blue1Brown the noble peace prize for year 2020. Thanks.
because of his wife having cheated on him it can not be))))
but for mathematics FIELDS MEDAL
Абдаллах Муслим wow im ruski look Im making cringy jokes using bad English))))))) so funny right?))))
@@FiXioNxd Firstly, you don't know that that person is a Russian, secondly, what does that have to do with bad jokes?
Phobos Anomaly I think his intention was to make a joke, second, I guessed by his name hes Russian.
I' ve recently gained a passion for mathematics at the age of 27. Now that there's no pressure its lovely. Polynomials make me smile and I'm excited to be on this journey.
How is it going?
Such an explicit explanation and high quality video! Can't believe I missed this video for five years.
it's fascinating and frustrating at the same time to see how super-abstract concepts can be linked to some weird geometrical ones, like honestly wtf ?!
The universe welcomes you; enjoy your stay.
La Tortue PGM What do you think where these concepts are coming from?
La Tortue PGM This problem isn't very abstract(for a physics student) it all depends on your understanding and where you're coming from.
tbh i prefer abstract stuff, so i kinda struggle when it comes to more concrete, geometrical structures. still in high school though lol.
Roverse You can check out anytime you like, but you can never leave.
You say that wherever pi is showing up there's a circle hiding, one circle which I would die to find is the one hiding in the fact that the integral of e^-x^2 from minus infinity to plus infinity is the square root of pi. One of your best videos in my opinion by the way.
integrate over two dimensions and take the square root at the end. since you integrated the square of what you should have...
If you integrate exp(-x^2-y^2) in the Real plane, you can evaluate the integral via substitution polar coordinates, and dx dy=rho drho dtheta, then you integrate with theta from 0 to 2 pi, because 2 pi is the length of the circle with unit radius. Then here's to you pi!
The usual proof is to think of this integral as the square root of the double integral exp(-x²-y²) over the plane. To evaluate this integral, switch to polar coordinates - here's your circle!
The way you explained this is just awesome. This will remain in my brain forever.
What a beautiful video. Kudos to all the animators and of course to you for explaining the beautiful proof. The best kind of math is the kind of math that makes you tear up when you discover the truth. And this one did.
You can also use Gauss's law to approach the same solution, rather than a geometric approach. Gauss's law works since a radially symmetric field that's magnitude weakens via the inverse square law has its radius term fall out in a surface integral. This means no matter where the lighthouses are within a sphere of radius R, they can be represented by a single lighthouse of combined magnitude in its center. This also means that same combined lighthouse can be represented by equally spaced, equally lit lighthouses along its boundary. By using this law within a cylinder, and holding the "lighthouse surface density" to be 1/2, you find the surface integral to equal to π^2, and a quarter of the cylinder is π^2/4, the same result as using the geometric method. (The circle is quartered to eliminate lighthouses on the negative side of the number line, and double counting when the number line curves upwards to form the circle)
Hmm, this seems super clever, but I'm not quite sure I follow the connection between the continuous "lighthouse surface density" and the discretized case.
This surface integral works by taking the sum of the areas as the areas approach zero. However, if you hold this distance to be constant, the radius must increase to give the same result. Similar to zooming in to see individual differentials, which at that scale, would be discrete. Since I'm using a cylinder, and putting the lighthouses only on the circular boundary, the circular endcaps can be ignored.
That is indeed very clever. But I don't think it is as elegant, because you need more advanced theorems for the proof.
@Copperbotte, I'm not quite sure if I'm understanding what you're saying correctly, but I don't think Gauss' law works the way you think it does. If you have a random assortment of charges in a Gaussian surface, you can calculate the flux through the surface by assuming a lump of charge at the center, but this does NOT tell you anything about the field produced. I'm also not following your math or your explanations.
Yeah no. Gauss's Law will tell you E and V for electrostatics. But there's no way you're gonna get a pi^2 term out of a cylindrical integral unless you make the length pi or something specific like that.
Wow! This proof is so beautiful and not that complex. I was worried the channel will go down hill when I heard more people were going to join. But now I have no doubt in my mind that it's going to be GREAT! Good job Ben for the awesome video!
There is no doubt in my mind that the new additions will make the channel better.
+3Blue1brown It already is! Awesome video as always. Can you make a video about the honeycomb conjecture?
arxiv.org/abs/math/9906042 You can download the proof and I think the way you guys depict concepts is incredible so please consider it
The only thing that would make this channel worse is if the current fans start gatekeeping. Very happy to see you explicitly subverting that!
michael einhorn sadly,I believe sum of any other higher powers is impossible for a human to compute,since the extension would need higher dimensions than 3,which we are unable to properly imagine,on our own
I didn’t know this approach about series even though I learned it when I was a college student. Thank you for your explantion. I always enjoy watching your video.
Marvelous proof probably the most interesting one I have seen of the Basel problem. Used the euler product formula to prove this beautiful result but this geometric proof is brilliant. Congratulations 👏.
I've been wondering how this equation related to Geometry for more than 20 years since I first saw it in college. THANK YOU!
This reminds me of Earth. It's spherical but still feels pretty flat, even though it's size is finite.
That was the most exciting math lesson I've ever been to. Thank you for making math so fun.
Fabulous. I wish that I had been taught mathematics by someone with these kind of insights. Thank you.
So elegantly explained. Just amazing. Thanks a lot!
whats stunning here is not just the geometry behind the problem, but the effort and intellect of the 3Blue1brown Team
Even with a straight line as "a circle", I still can't draw a perfect circle.
😂
Just pretend that you drew an accurate representation of the projection of a sphere in a higher-dimension curved manifold.
@@b.clarenc9517 I'm going to pretend I know what that means.
I mean if you think about it, it doesn’t really make sense because it would have to mean -infinity and +infinity lead into each other at the top of the circle
this is a beautiful, clever approach! [though it sweeps under the rug a few questions of *convergence* -- when you say the larger and larger circles converge to the sum on the integers there is a potential pitfall: for any finite, very large circle, MOST of the circumference is *not* near the x-axis and there are many many lighthouses on that part of the circle. But this is easily fixable, you can show that the total sum contributed by them becomes negligible as they are all far from the origin and their individual contribution to the sum may be bounded by a quantity that shrinks proportionally to the square of the radius, but the quantity of them grows only linearly with the radius. This is a much slicker and accessible proof than the one I've previously shared with high school students that requires some polynomials involving trig functions and vieta's formulas.
평소에도 수학을 좋아하는데 이렇게 재미있게 설명해 주시고 한글 자막까지 달아주셔서 감사해요.
Yes
There are already so many mathematical results named after "Euler", that if they had called this "Euler's Problem" or something, it would start getting confusing...
Proof: en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler QED
This would be a problem yes.
@@SpaceyCortex Would it be Euler's problem?
The other Euler formula The other other Euler formula The other other other Euler formula
Stop Eulering
The beauty of these animations is beyond anything else on youtube
Also it clearly shows how to summon math satan
It's very beautiful. I saw that for mathematical proof only the inverse pythagorian theorem suffice. But, using lighthouses made the video more beautiful. Thank you for the video.
I keep watching this video over and over again, so rich in information and good animation
I have seen its proof by Fourier series but the way your team animated and gave physical proof is simply awesome... great work, cheers.
12:57 that circular right angle kills me to this day
Tech Made Easy the right angle symbol is drawn with a square, not a circular arc
Maybe he just forgot the dot inside... ? ^.^
my first clue into I'm not buying this.
Hh
90° is written there, square angle is used but it is not a rule neither it will affect in any way
I wasn't necessarily able to understand the proof but still i appreciate how a crazy-looking, complicated infinite product can be explained using not just brute-force math but a combination of math and intuition
Thank`s a lot for this new inspiration. Your technique might bring new insights, whether we have to improve the invese square laws for gravitation and electromagnetic forces. You go from odd numbers and extrapolate to all. But with odd numbers we have the problem, that after the prime fermat number 65537=2^16+1 , mathematician´s have not yet found a new one- and only the prime ones are constructable. I think 3*5*17*257*65537=2^32-1 is the last possible odd number you can use in physics.
Best math channel ever. Clever, original, beautiful, soothing / motivating voice... Just perfect. I've been following it since the very beginning. Every new release feels likes christmas. Please keep it on !
This was amazing! I love how geometry and algebra, while being based on completely different axioms, can represent the same concepts, and they way you switch between them is astounding.
I wish more math teachers follow in your footsteps. The fact that you actually inspire your students by trying to show off the true beauty of mathematics is far more helpful and amazing than just letting students do problems. Most of my classmates, although smart, only sought to do math in order to past entrance exams for university. But they failed to see the hidden beautiful world this subject offers. I've been researching and analysing different aspects of math all my life. And I really love what I've seen. I wish I had friends who loves math as much as me.
Wait, so pi is just the square root of 6 over each of the squares all summed together?? Oh my goodness my mind is blown.
This is wonderful! As I said in my paper, it's based on proofs by Yaglom & Yaglom, Hofbauer, and others, and I added some of my own ideas. I thought of the light sources as stars revolving around a common center of gravity, but light-houses are arguably easier to move around! :) I hope the "light-house proof" now becomes folklore, and I'm happy to have contributed to that!
again just another distraction from the truth about pi and the information contained within its code and sequences...I find it strange that after the last decade and 9000 pages of text i have written on pi i haven't had ONE single person interested in it.....lets play a game folks...lets see who knows anything about pi that isn't common knowledge.....
@Johan Wästlund you rock!!
can you share ur paper? :)
sure...let me just give you all my work@@amineaboutalib
@@thetherorist9244 well i cant find it
Mind blown. Multiple times. And I'm only half way through it.
Wish you had it now
we should have expected this, if you look at the earth from any of the points like now, it is straight, if you look from above is huge and round at many points because is a sphere not really but oval like, you can have a sphere if you cut some parts correctly, our eyes narrow it. if you look through a microscope and so on, the width of a single hair wire is like a million atoms or probably more source: kzhead.info/sun/fKqEntVxiH9sknk/bejne.html
just think how we are bending over along with the surface of the earth, seen from space our heads shoulders and so do not surpass the circular size and shape of the earth in fact they are the same and fitting with any part of it
The guy got creative using the superposition principle. Using light? He's dealing with "waves". That man studied mathematics, physics and of course, computer science. Expect combined ideas of this sort in all of his videos.
OHH ONE OF THE BEST VIDEOS ON KZhead!!!! THANK YOU SO MUCH EXCELLENT VISUALIZED!!!!
I was shocked, even one member of your Patreon animated this AND made the geogebra! thank him!
That was simply fantastic. You really show what it's like to love math. Combining light and geometry to reveal the circle in 1 + 1/4 + 1/9 + ... = pi²/6? How great is that? You sir are amazing!
I am currently pursuing UG in Mech Engg and the video gives the answers to most of my expansion related problems I've been facing since my high school days. Wow!!!!! it's beautiful.
This also beautifully shows how an apparent point charge can be distributed over a spherical surface harmonically. The incident angle always compensating for the distance. If all our physics models of point like particles instead distributed the charge over measured actual spherical size the fuzziness is the uncertainty principle. The closer you get in proximity the less of a sense you have of size. At the surface the charge takes up half the screen so although you can accurately measure the charge let’s say, the light from the original lighthouse, You can now only say that the charge lies somewhere in THAT direction, 50% of your view. Or it’s morning and I’m confused. :)
What a beautiful video. Thank you so much for sharing these extraordinary insights and your deep love of mathematics in the form of these precious videos, they are truly artistic masterpieces.
I have been doing math for a few years, and I still think that this is one of the clever stuff I have seen in a while. Feels great.
Amazing! The first thing I thought of when I saw the animations was that it kind of looked like De Moivre's formula with lightbulbs and now I wonder whether there is a hidden proof in the video. Tremendous work!
I just realized and worked out that if you use pretty much the same argument, but the starting lighthouse is θ of the way around the circle instead of π of the way around the circle, you can show that the sum of (x-πn)^-2 over all integers n is equal to csc^2(x). This essentially provides a geometric way to show that the sum of over all integers n of (x-πn)^-1 = cot(x) if you can justify interchanging the sum and the integral. This cotangent identity can be very useful when trying to find other sums as well. Great video.
The animation is so amazingly well done I am just stunned
This is beautiful man!! I wish the whole world can see and appreciate how amazing your explanations and representations in your videos are. You're showing the true beauty of maths
An amazing proof. Perhaps the best proof I've seen in a while. I really like that even a high school student could follow along with it
It makes you think that your understanding is finely tuned in until you pick up a pencil and paper and try some on your own
@Ayush Dugar I'm literally catching up on my infinite series and sequences as we speak. I'll try this after I'm done
Ofer Magen that's me!
" I really like that even a high school student could follow along with it" If only we had a President that could do that.
Steve Barnes hahaha so funny. Get a life.
stunning animation, so amazing,straightforward to explain it. astonishing beautiful
I love this channel so much, it makes math visible!
This is the best math video I've ever seen! You and Mathologer have inspired me on a consistent basis for a while now, but this video is my favorite so far.
Two 3blue1brown videos in one week? It’s a dream come true!
I was lucky enough to have a great mathematics and geometry teacher. Many questions in algebra are most fruitfully investigated when they are given a geometric interpretation.
I've lived in Basel. Grant, you make transcendent videos about math and you say the word "Basel" very differently than I learned. It kept my attention. :-D
Sir, your concepts are so crystal clear...please don't stop making these types of videos.
Watching your video for the first time and feeling myself so unfortunate that I didn't watch it until now..... Awesome work... I'm gonna recommend it to all my friends...
Did someone mention that this is also the solution to the [Riemann] zeta function when s=2 ?
That sum is the ζ(2), he just didn't mention it
That is correct.
If you don’t fully understand these videos Never give up.....Soon you will realize the intuition this channel give you
Give up guys! GIVE UP
i know this and im only 5 its so easy
@@Emperor_Shao_Kahn joined 11 years ago hmm...
I'm betting on old age tbh
I feel like there was one point which was glossed over. The entire infinite circle does not correspond to the numberline in this case (topologists agree with me) and its pretty obvious why: The lighthouses on the upper hemispheres of the ever growing circle are not corresponding to points on the nubmerline, instead their distance to the number line becomes greater and greater the more the circle grows!! Of course, this does not change the outcome because their contribution can be ignored in the limit (because again their distance to the numberline and thus to the observer becomes arbitrarily large), it's just the circle (without valuing accordin to the inverse distance to the origin) does not correpsond t the numberline!
Just read the description and he wrote the exact same thing there. Welp
That claim alarmed me too. I realized straightaway, that this was omitted for clarity, but I can see how this is not an easy and obvious thing to prove. Brushing off little and seemingly insignificant things like that is so un-mathematical. He should have said at least something like "with a caveat, see description". I bet he was just too carried away with his nice geometrical explanation and didn't notice this omission in his reasoning.
I was also slightly puzzled by a claim at 7:55 which was totally unexpected, until at 8:28 he says "Why, you might ask". Exactly, why? A little bit of forewarning would really help.
wow. I have no idea why you do these exegeses, but i'm eternally grateful for it. such eloquent and elegant explanations tap into a deep sense of beauty. Thanks Grant
If Sum(1/n^2)=π^2/6, then Sum(6/n^2)=π^2. 6/n^2 is always rational for any n € N. If it's true that the sum of two rational number is still rational, then [6/(n-2)^2 + 6/(n-1)^2]+6/n^2 keeps still rational. This means that π^2 is rational. Thus: either π^2 is rational; either there exists some rational number whose sum is irrational; either Sum(1/n^2) approaches π^2/6, but it's not the same; either there's something missing. Waiting for any feedback. Thanks.
You are correct that every partial sum of this series is rational. However, the value of a series is the limit of the partial sums, and a limit of rational numbers is not, in general, rational. For every real number, there is some sequence of rational numbers which has that real number as its limit. But almost every real number is irrational, by cardinality arguments, so there are lots of sequences of rational numbers which approach irrational numbers.
ok i'm a little past 4 and a half minutes in and i am very excited about learning how the inverse square law, which i was taught in astronomy class in college, makes TOTAL SENSE now, decades later. Jeez why didn't they explain it that way?! Thanks!
I use Pi all the time while calculating frequencies of inductor/capacitor networks, phase shift in transmission lines and lots of other stuff when designing radio frequency circuits. They're all connected to sine waves, which are just funky circles.
Till now I was just learning principles, theroms, formulas given in my book without proof I was really got angry to learn without proof but after seeing your videos I got ideas that how the these are derived and how it actually works. Now I am feeling better now.
You can get emotional with the result of such care, passion and commitment. It is a work of art.
very clever of you getting pi involved in all this mess, it all started with that small circle 2/pi diameter and the exchange of 2 lamps for one!
2 videos 1 week is this christmas all over again ?
It's Holi, my dear friend! Happy Holi! Happy Mathleting!
Is that what it's snowing?
Yes christmas is all over. Christmas is being in december time.
You are the best and you could not be happy
We can't wait another day-please, Solstice, don't be late.... [Apologies to R Bagdasarian]
This is by far the most mindblowing demonstration of whatever you call math or related to math. Thank you so much for this.
I got lost at the multiplication by 1/4 step, guess I have to brush up my algebra... but as always new ways to see things to common series and more visually demostrated to grasp why rather than its like that.... Truely as one person said below A great piece of art
I just had an increasing heart rate excitement when I clicked on the new video notification!!
2 videos in a week!
this is making me so happy
You have 314 likes for this comment. NO ONE MUST SPOIL THIS.
if and only if...
this is also used in astronany, Appaert Magnatude, to calulate distance from the observer to to the object being observered, by taking the appaernt magnatude of a type 1a supernova and using the angel of parralacx between the type 1a and the object being observe. Use that angle, plug that into the formula for angles do some path and if the distance to type 1a is correct you get the sindance from you to object you are observing. Appaerent Magnitiude is a beautiful tool.
Here’s an interesting one. There’s a skewed probability distribution called the extreme value distribution. To “normalize” it to a mean of zero and standard deviation of one, you have scale it by 6/pi^2 and shift it by gamma (Euler-Masechroni constant). With the dual exponentials at play, I can see where the gamma constant may be involved. But where does the pi come from? For reference, the integral from negative infinity to positive infinity of the following is pi^2/6. (x+gamma)^2*exp(x-exp(x))
Absolutely incredible, your videos never fail to blow my mind, keep up the good work!