But HOW did Euler do it?! A BEAUTIFUL Solution to the FAMOUS Basel Problem!

[@rob6129]

That last part is mind blowing. Makes me think that Euler just played around with the sin function and that his result was "just" a byproduct of his experimentation. Really amazing

[@vergorance]

fun fact: euler calculated the first 16 digits of pi^2/6, before developing his actual proof.

[@bobus_mogus]

Teachers: only first graders can produce unrigorous proofs Euler: hold my π²/6

[@dsm5d723]

I must promote the chap I chatted with who really did what he said he did, just like me. A fucking genius. Pi IS 3,16 or 16/9. He finished the history of human astrological observation and fixed so many things we take as true. I am rewriting math and physics for RATIONALITY in both as well as language. Stay tuned, and as long as channel metrics decide truth, get ready for aliens.

[@ryanjagpal9457]

@@dsm5d723 Pi is 3.141592653 and 16/9 is 1.78

[@dsm5d723]

@@ryanjagpal9457 It is his thing. i will give his screen name. He does have the perfect explanation of pre-literate astronomical observation. It is the obverse of my thing, Dimensional Gauge Symmetry. Three rational "irrational" gauges, e (c), sq2 and Pi. I was working with the ERRORS of modern math, and I did find them. Add the three numbers to the third decimal and the first 1 from the prime sequence and you get a mathematical model of a dynamic dipole, plusy dynamical friction embed in the Euclidean Plane. Math with out a dynamic explanation of physics is not measuring anything but math. 1+0.577+1.414+3.141=6.132 And I am working on the last bit of rationality in the modern paradigm, and it has to do with the resolution hidden in the "irrational" decimal expansion of these three numbers. Repeating is not understanding. I finished Einstein and Poincare with the Tesla Identity Matrix Determinant. gab.com/23andMe24andYou/posts/105477983888996278

[@dsm5d723]

@@ryanjagpal9457 DysonTorus Tesla Code is 3r 6r and 9r 5 days ago Ed leedskalnin. Coral castle π is 3.16r Tau is 6.3r Everything is out by 1.1 1.1s 66.6s 66.6m 22.2h 333.3days We are in 2222/3 Egyptians used 3.16 I recurred it and tested manually. If there's missing math then there's missing km2 of earth Eratosthenes was 10deg adrift lattitude as the magnetic equator is the real equator. DysonTorus Tesla Code is 3r 6r and 9r 4 days ago @DSM 5D best comment I've ever heard. Thanks buddy! Now find the sq root of 10. 3.16227766. That's just using my phone. 3.14159 is the error to hide 47m km2 in the south and is hidden in the north, or just missing full stop. 111.1 km as everything is a ratio of 1:1.1r mostly in my model. 10deg passed np from UK is let's say 111.1km, that offsets everything. Bit if there's 26666.66r km from true south to true north as 6666.66 is deci more than the famous 666 or 666r. It's all in plain site. The moon is the ruler. Look up first use of lunar calendar. It's way older than we've been worshipping the sun. Base the whole geo model onto the moonpole or monopol'y' as I call it. Gets really interesting. Please sub as all my videos are going into one amazing presentation noman has ever thought of since 4236BC DysonTorus Tesla Code is 3r 6r and 9r 3 days ago @DSM 5D Topman. I like Ur style. Babylonians. It's all about time line. 60 didn't fit in with lunar. Sumerians were 3000bc. Egyptians used lunar before 4236bc and Scotland found evidence of the lunar calendar 8000bc at least. So 60 base started 4236 by Egyptians. 365 calendar was 4236bc as well. I claim they could never figure it out. Without studying the world, they would have never have know the full path of the moon. We do. We should be using a 100 based system for time which is navigation. Because we don't, I have proven 10deg x 4 is missing at both poles. How did they hide 48m km2. Through assumption of 40000km. Well Magellan proved equator was way way shorter than 40000km. I proved it 100% in my day 2. It's all in plain sight. Nothing is hidden. We just aren't looking

[@dsm5d723]

@@ryanjagpal9457 DysonTorus Tesla Code is 3r 6r and 9r 3 days ago (edited) @DSM 5D I'm a cook with south African education. 86-94. 19/6 is 3.16r Manually 3.16r was bang on π was .08% out. I used a plate and tailors measure. I'm stuck on why perimetres change with shape change but not area? Given it a rest for day. Eric verlande talked of entrophic gravity. Will reply more later on. Thx bud. Heads going wild with numbers. What's these prizes? Clay math is who I emailed.

[@kunerzhang7825]

Other mathematicians: QED Flammable: its pretty f*cking dope

[@Hart8]

Everyone is scared of swearing on youtube except math channel wtf?

[@Metalhammer1993]

well this guy was once FAPPABLE maths if i recall correct you, so yeah Jens isn´t the guy with the best filter^^

[@martinansnes641]

@@Metalhammer1993 He is accurate, which is the most important thing. This IS pretty fucking dope :)

[@thomasrad6296]

It's because he found a proof to get away with it.

[@thecwd8919]

@@Metalhammer1993 Well that name isn't wrong. This shit gives math boners.

[@jkstudyroom]

@@thomasrad6296 Don't you mean he found a "proof way" to get away with it? Just a math puns!

[@sionae1967]

I thought you were going to write sin(x) = x at the beginning, I think I'm too involved in my physics degree it's becoming an issue

[@curiosityzero2151]

Haha

[@DjVortex-w]

x is a perfectly fine approximation of sin(x) for values of x close enough to zero.

[@jaimeduncan6167]

Sionae 😂😂😂😂

[@UteChewb]

I had to stop and rewind to 15:35. My brain was automatically rounding. It took me more than a few seconds to shift gears.

[@nicholasleclerc1583]

Well, for a sufficient small enough interval of x values around x=0, you can replace the "=" sign with a "~" sign

[@brunoandrades5530]

I love how u were so rigurous at the end with the Peano axioms and stuff to compensate for the cancer and headache that the unrigurously pi^2/6 proof gave me

[@shadow-ht5gk]

Lmaoo

[@ernestomamedaliev4253]

"If two functions have the same zeros, they are basically the same". Amazing. New theorem for engineers! (Notice: x = x^2 :)

[@YorangeJuice]

LMAO

[@kuronekonova3698]

"If two polynomial functions have the same zeros, then they are basically the same, if and only if their coefficients and degrees are the same."

[@ernestomamedaliev4253]

@@kuronekonova3698 actually, if two polynomials have the same zeros and their degree is the same, they are the same polynomial, hehe

[@snootiermoon]

@@ernestomamedaliev4253 not necessarily cuz you can multiply any polynomial by a constant to get a new polynomial with the same degree, same zeroes, yet different. I think if two polynomials have the same degree and (complex) zeroes, they are proportional to each other by some constant.

[@ernestomamedaliev4253]

@@snootiermoon yeah, thinking abou that, I guess you are right. We need to specify that the coefficient of the maximum degree term is 1 in order to establish what I said earlier. Thank you for the correction! 😉

[@FGj-xj7rd]

Euler? Nah... Wheeler? Perfection...

[@neilgerace355]

1:04 hoyristically

[@arnavanand8037]

*You* ler? Nah... *We* ler? Perfection *_ussr intensify_*

[@luigin9087]

@@arnavanand8037 Oil er USA intensifies

[@yaboylemon9578]

Using 1/n^2 for thumbnail but 1/k^2 for video? Disliked, don’t need unreliable people in my life rn

[@UltimateBargains]

Variables vary too much, so unreliable.

[@Lily-zd6dx]

LMAO this made me laugh harder than what I thought

[@tomctutor]

n, k, all much the same just letter placeholders for some variable. Get used to it, or you'll end up exploding in flames in your life.

[@doyouknoworjustbelieve6694]

Easy... Good that he didn’t use x or y instead of k, then n to solve it😂😂😂

[@PedroKrick]

K

[@pythagorasaurusrex9853]

Papa Euler was truly a genius. Just a comment for all of you boyz and girlz watching this. By using the exact method shown here you can derive what the values of zeta(2n) are i.e. zeta(4), zeta(6) etc. by comparing the coefficents of the part of x^5, x^7, x^9 etc.

[@ferhatakbulut6572]

Or you can use papa fourier's series.

[@diavolacciosatanasso]

You can still compare but the results require further insight to get. Try.

[@ryanjagpal9457]

Wait what is zeta?

[@jkstudyroom]

@@ryanjagpal9457 C'mon, every body knows the zeta function! Even a 3rd grader!

[@ryanjagpal9457]

@@jkstudyroom How can a third grader know that? Pretty sure they should be learning how to write by then Idk where you go to?

[@avtaras]

LMAO 4:43 AM I THE ONLY PERSON WHO NOTICED PAPA FLAMMY WAS USING THE FUNDAMENTAL THEOREM OF ENGINEERING? XD

[@ianmathwiz7]

Mathematicians: "This expression isn't well-defined." Euler: "But what if it was?" Physicists: "No biggie. All we have to do is multiply and divide this by infinity (because it's not equal to zero) and we get the charge of an electron."

[@MessedUpSystem]

Pretty much, ever heard about renormalization? Essentially you just "hide away" some term that blows up to infinity and the leftover is your answer :D

[@michaelibrahim9275]

Luigi T. Sousa In the words of Andrew Dotson: Ree-normielization

[@xZak-A-42]

@@MessedUpSystem Yeah! We use this idea of renormalization in Asymptotic Methods, one of my modules. More specifically, finding solutions to small perturbations of Duffing's equation in which a straightforward expansion ansatz gives rise to a non-uniform solution.

[@Xeroxias]

I think Euler used the sinc function (sin(x)/x) to reason about the constant multiple in each root in the infinite product (i.e. (1-(x/kpi)^2) vs (x^2 - (kpi)^2) vs all other constant multiples) which sort of justified why each term in the product looks the way it does.

[@josephgrossenbacher7642]

Euler 'used' the "sinc-function" 'quite often' , e.g. : cos (na) + i * sin (na) = [ cos(a) + i*sin(a) ]^n --> set here a = x/n , with a fixed & real x --> cos(x) + i*sin(x) = [cos(x/n) + i*sin(x/n]^n --->> sin(x/n) / (x/n) --> 1 , for x/n --> 0 , i.e. for n --> inf --->> so asymtotically 'we' have sin(x/n) ~ x/n , moreover 'we' see / "know" that cos(x/n) ~ 1= cos(0) , for large n --->>> ; so it's "plausible" to write : cos(x) + i*sin(x) = lim [ 1 + ix/n]^n , for n --> inf , thus 'we' get a 'definition' for the exponential function [ on which the "Euler method" for solving ODE's numerically is based ! ] : e^x = lim [ 1 + x/n ]^n , n --> inf , according to the last "well known" limit ... !!! { exercise : show that lim [ 1 + x/n ]^n = sum(k = 0 to inf) x^k/k! , n --> inf }

[@benjaminbrady2385]

4:45 glad to see Papa Euler knew the facts

[@dackid2831]

This is my first time seeing the product function in action. I knew what it was, but I haven't necessarily used it much. You made it very easy to understand. So thank you for that. :)

[@semyonmaltsev7837]

Could you do videos about Functions of Several Variables and more fun stuff? Absolutely loving your videos

[@whitewalker608]

There are many other solutions as well but this is possibly the simplest solution of this problem! Nicely done!

[@tomasblovsky5871]

Euler did this whole thing in his head for sure :DDD Truly a mathematical genius

[@cubicardi8011]

Lol at the end i was like: wait..​ that's​ it?​THAT'S​ IT?¿?? ¿? THAT'S AMAZING

[@jjoster]

I said, "you're shitting me?!" My 9 year old says, "Dad, where did you think he was headed?"

[@hreader]

My grandfather told me about the difference between two squares when I was about 11 or 12 - while I was helping in my grandparents' garden, actually while making the bonfire for the garden rubbish! It is a very useful tool.

[@marce3893]

YoU dOnT nEeD rIgOuR wHeN yOu'Ve GoT aUtHoRiTy

[@dannygjk]

BISS

[@polyhistorphilomath]

When you’re Euler, you tell both the steak and the eater what’s up.

[@eytansuchard8640]

Beautiful proof by polynomial coefficients comparison. Very neat and doesn't require any geometrical construction. Thank you for this lecture.

[@Vincentsgm]

Omg that Taylor Swift meme i'm crying

[@williamrichmond814]

Was that some math joke that my dumbass producer mind won't get

[@Vincentsgm]

@@williamrichmond814 Taylor series expansion

[@DuckyOctopy]

What's going on smart people, today we start a meme war with 3 competitors including 'tis boi, send him some love for power

[@lad4694]

Euler flaming past the screen never fails to make the highlight of my day. WHOOOSSSSHHHHHH!!!!!

[@Ny0s]

This is so great to have freely access to such content. Thank you very much, this is really interesting.

[@euva209]

What's nice is that the same approach for the zeros of the cosine function can be used to get that the sum of 1/(2k+1)² from 0 to infinity is Pi²/8. Then it's easy to realize that the even squares are 1/4 of the sum of 1/k². From that it follows that sum of 1/k² is (4/3) of Pi²/8= Pi²/6.

[@DanialDawson]

i'm studying to start undergrad Maths this year and this video made so many things click into place I'm a little blown away

[@wompastompa3692]

My calc teacher showed my class this back in the day. Still cool to this day.

[@admancr2823]

Never knew it was that 'easy'. Thank you for your work. Even though I am passionated about maths I do not study it and videos like this are pure gold for me.❤

[@83jbbentley]

Euler was blind and dictated to a scribe and published on average 2-3 pages of work a day!

[@jkstudyroom]

From a US HS tutor's point of view, I've noticed that many Asians as well as students from Europe write their "x" by writing a backward "c" then a "c". Also, noticed that the integer set is written as a "7" then an upside down "7". I will have to use this notation for the integer set next time!

[@ElDiarioLudita]

pi^2/6 : exists* Oiler: hmmmmmm

[@herlanggaizul6965]

Finally... The video I was waiting for... I thought when you share about sine product I always thinking about when this video realise

[@igxniisan6996]

Euler knows how to use ultra instinct in mathematics.

[@Azzoo_0]

Finally I can understand after watching many videos. Yours is detailed.

[@derhenri2002]

Du hast mir geholfen bei meiner W-Seminararbeit über das Basler-Problem. Danke!

[@hurricane4912]

I know this might be a dumb question but at 5:50 why can't you represent sin(x) as being equal to x(x-pi)(x+pi)(x-2pi) and so on?

[@hurricane4912]

I'm assuming it has something to do with the fact that the sin(x) graph could be vertically stretched or compressed and just because that graph has the same roots doesn't mean its y = sin(x). But what exactly did you use to find those factors and why?

[@wilhelmsarosen4735]

I found a French article which showed a method that allows one to calculate zeta(2) when one knows what zeta(4) is, and vice versa. It came up when I was trying to integrate Planck's Law, and did not just want to simply write down the value of zeta(4) written in the book. So... now that when people ask me to calculate the value of zeta(2) or zeta(4), I just claim that I know the other one, and use the method in the article.

[@eliaschavez364]

could you send me that article please?

[@wilhelmsarosen4735]

@@eliaschavez364 Glad to, it is "Quelques conséquences surprenantes de la cohomologie de 𝑆𝐿_2(ℤ)" by Don Zagier.

[@mahmoudkhamis409]

"It's very simple" euler just died here

[@tommasoconte591]

Never seen before. That's beautiful!

[@BélaZoltánFekecs]

Sehr schön und verständlich erklärt. Gratulation.

[@subhajitsamanta612]

Oh, yes. I first did it when studying classic Fourier transform in 1st year of undergraduate. Actually many basic equations and formulations follow Eular.

[@simonsidolin]

5:05 "the same spiel.." sehr schön :) Sehr interessantes Video weiter so.

[@CHimanshuBorkar]

4:43 - 4:55 when u are possessed by a ghost who was an engineer

[@hetsmiecht1029]

0:25 Thumbnail: n Chalkboard: k You've been click baited, and you know it!

[@RetroGamingClashOfClans]

lol "a regular third grader can do that", don't know where tf ur living my man

[@heisenfenceakarick4802]

Germany switzerland

[@ryanjagpal9457]

Nah that would be impossible to understand it at that degree and plus how are they gonna reach the blackboard

[@firi4737]

I solved it in 10-th form at school

[@ryanjagpal9457]

@@firi4737 is that basically year 10?

[@jankramer6503]

Pretty late, but I‘m from switzerland and in 11th grade... that stuff‘s pretty simple

[@stefanhartmann135]

What a great job, guys!

[@alejandroespino7178]

I always confuse your Xs with lambdas

[@brucerout]

Nice presentation of a truly beautiful derivation. Thank you.

[@michaelgolub2019]

Leonard Euler was really great!

[@SciDiFuoco13]

8:27 I made the same mistake in my recent math/logic competition...

[@vistarichardo.8747]

Taylor joined the video

[@thephysicistcuber175]

Just as I was looking up summation techniques

[@nablavii]

This guy is my favorite mathematician 😊💙

[@mattmolewski7475]

Why can't I find a nice guy who calls him Daddy Euler in my life?

[@PapaFlammy69]

;_;

[@cerwe8861]

Mathologer has made an (and several other) amazing Video about π²/6 and Eulers sine formula!

[@aarafazad5464]

Man..... I just love his energy

[@habouzhaboux9488]

How do you prove that this infinite product equals the sine function? The tangent function has the same exact roots as the sine function. Why then this infinite product won't equal tan(x)?

[@KyleDB150]

My favourite thing to write when solving a math problem: "By inspection"

[@bentn1374]

16:55 That was the smoothest fucking thing I have ever seen in a maths video. Mad props for making such a digestible video on such an intricate subject

[@weerman44]

5:47 How do you know for sure that this defines the sine function and not some other function with the same zeroes? Or is every function defined by its zeroes?

[@vangrails]

I don't think it is because when you multiply a function with a constant its zeroes don't change. You need a bit more.

[@Fokalopoka]

Well, you can define sinx as polynomials (taylor series), so actually thats like a zeros represantation of polynomial, and there is only 1 polynomial with such zeroes

[@tetraedri_1834]

@@Fokalopoka Not quite true, zeros may have multiplicities. Compare (x+1) and (x+1)^2. However, by repeated differentiation you can check multiplicity of each zero. Also, Euler only considered real zeros and ignored the possible complex zeros. Of course, analytic continuation of sine hasn't any, and maybe Euler knew (or "knew") this.

[@Mike-qt4fr]

Hey this was posted on my birthday! Love this proof :)

[@Azelide]

6:45 You're actually right! 😊

[@shablamrobohawk1192]

liking for the Taylor meme

[@xxnotmuchxx]

Thank you, daddy. I was using ^0 and ^2 together and got confused. I got the answer now.

[@sberacatalin2250]

Bravo!Esti bun. Competent!

[@TheTurtleOfGods]

heuristic analysis better than malwarebytes 🔥😍

[@connorkearley7789]

this is where it all went wrong

[@neo_tsz]

Hooooooooly shit! That was absolutely stunning. I've got goosebumps now. Good job!

[@Omar-mn1eb]

Daddy Euler😂

[@mikeyoung3870]

Man, that’s crazy my man!

[@PapaFlammy69]

:)

[@charlesrodriguez6276]

This has been one of your best videos and I have been watching them for a while. This was super fun to watch clear and easy to understand. Definitely do some more og Euler heuristic stuff!

[@nacholopez7348]

4:45 as an engineer I just stopped right there and got the right result also my math prof said the rest is just trivial stuff so I just went with that

[@m.m.2341]

This was recommended to me and I just watched it in the middle of the night :D University has been a few years, so I had to give you the benefit of doubt regarding the Taylor series, but the rest of it made perfect sense to me. Gruß aus Deutschland =)

[@jackthisout9480]

On a good old fashioned chalk board. Euler approves this message.

[@knutritter461]

Hast Du gut gemacht! Daumen hoch! ;-)

[@WhyAnkurGautam]

Awesome video....!!! I am watching this at 2AM because I don't need sleep I need answers.

[@spacejunk2186]

Youler was a great guy.

[@chupapimunanyo2596]

„We can do the same Spiel for the next...“ :D

[@pokoknyaakuimut001]

My favorite math teacher 😁

[@holyshit922]

And this approach works for all positive and even values of zeta

[@MathIguess]

Encountered this series as part of a homework problem I'm so glad you exist Flammy ;_;

[@MathIguess]

I recommended this video to friends cause of it xD

[@niceyraiyani]

Thank you so much for this! Really helped with my research

[@PapaFlammy69]

:)

[@johndoyle2347]

Reposting and slight editing of recent mathematical ideas into one post: Split-complex numbers relate to the diagonality (like how it's expressed on Anakin's lightsaber) of ring/cylindrical singularities and to why the 6 corner/cusp singularities in dark matter must alternate. The so-called triplex numbers deal with how energy is transferred between particles and bodies and how an increase in energy also increases the apparent mass. Dual numbers relate to Euler's Identity, where the thin mass is cancelling most of the attractive and repulsive forces. The imaginary number is mass in stable particles of any conformation. In Big Bounce physics, dual numbers relate to how the attractive and repulsive forces work together to turn the matter that we normally think of into dark matter. Complex numbers = vertical asymptote. Split-complex numbers = vertical tangent. Dual numbers = vertical line. These algebras can be simply thought of as tensors. Delanges sectrices can be thought of as opposites of vertical asymptotes. Ceva sectrices as opposites of vertical tangents, and Maclaurin sectrices as opposites of vertical lines. The natural logarithm of the imaginary number is pi divided by 2 radians times i. This means that, at whatever point of stable matter other than at a singularity, the attractive or repulsive force being emitted is perpendicular to the "plane" of mass. In Big Bounce physics, this corresponds to how particles "crystalize" into stacks where a central particle is greatly pressured to degenerate by another particle that is in front, another behind, another to the left, another to the right, another on top, and another below. Dark matter is formed quickly afterwards. Ramanujan Infinite Sum (of the natural numbers): during a Big Crunch, the smaller, central black holes, not the dominating black holes, are about a twelfth of the total mass involved. Dark matter has its singularities pressed into existence, while baryonic matter is formed by its singularities. This also relates to 12 stacked surrounding universes that are similar to our own "observable universe" - an infinite number of stacked universes that bleed into each other and maintain an equilibrium of Big Bounce events. i to the i power: the "Big Bang mass", somewhat reminiscent of Swiss cheese, has dark matter flaking off, exerting a spin that mostly cancels out, leaving potential energy, and necessarily in a tangential fashion. This is closely related to what the natural logarithm of the imaginary number represents. Mediants are important to understanding the Big Crunch side of a Big Bounce event. Black holes have locked up, with these "particles" surrounding and pressuring each other. Black holes get flattened into unstable conformations that can be considered fractions, to form the dark matter known from our Inflationary Epoch. Sectrices are inversely related, as they deal with dark matter being broken up, not added like the implosive, flattened "black hole shrapnel" of mediants. Ford circles relate to mediants. Tangential circles, tethered to a line. Sectrices: the families of curves deal with black holes and dark matter. (The Fibonacci spiral deals with how dark matter is degenerated/broken up, with supernovae, and forming black holes. The Golden spiral deals with black holes being flattened into dark matter during a Big Bounce event.) The Archimedean spiral deals with black holes and their spins before and after a reshuffling from cubic to the most dense arrangement, during a Big Crunch. The Dinostratus quadratrix deals with the dark matter being broken up by ripples of energy imparted by outer (of the central mass) black holes, allowing the dark matter to unstack, and the laminar flow of dark matter (the Inflationary Epoch) and dark matter itself being broken up by lingering black holes. Delanges sectrices (family of curves): dark matter has its "bubbles" force a rapid flaking off - the main driving force of the Big Bang. Ceva sectrices (family of curves): spun up dark matter breaks into primordial black holes and smaller, galactic-sized dark matter and other, typically thought of matter. Maclaurin sectrices (family of curves): dark matter gets slowed down, unstable, and broken up by black holes. Jimi Hendrix's "Little Wing". Little wing = Maclaurin sectrix. Butterflies = Ceva sectrix. Zebras = Dinostratus quadratrix. Moonbeams = Delanges sectrix. Jimi was experienced and "tricky". Jimi was commenting on dark matter. How it could be destabilized by being slowed down, spun up, broken up by lingering black holes, or flaked off. (The Delanges trisectrix also corresponds to stable atomic nuclei.) Dark matter, on the stellar scale, are broken up by supernovae. Our solar system was seeded with the heavier elements from a supernova. I'm happily surprised to figure out sectrices. Trisectrices are another thing. More complex (algebras) and I don't know if I have all the curves available to use in analyzing them. I have made some progress, but have more to discern. I can see Fibonacci spirals relating to the trisectrices. The Clausen function of order 2: black holes and rarified singularities are becoming more and more commonplace. Doyle's constant for the potential energy of a Big Bounce event: 21.892876 Also known as e to the (e + 1/e) power. At the eth root of e, the black holes are stacked as densely as possible. I suspect Ramanujan's Infinite Sum connects a reshuffling from the solution to the Basel problem and a transfer of mass to centralized black holes. Other than the relatively small amount of kinetic energy of black holes being flattened into dark matter, the only energy is potential energy, then: 1 (squared)/(e to the e power), dark matter singularities have formed and thus with the help of Ramanujan, again, create "bubbles", leading to the Big Bang part of the Big Bounce event. My constant is the chronological ratio of these events. This ratio applies to potential energy over kinetic energy just before a Big Bang event. Methods of arbitrary angle trisection: Neusis construction relates to how dark matter has its corner/cusp singularities create "bubbles", driving a Big Bang event. Repetitious bisection relates to dark matter spinning so violently that it breaks, leaving smaller dark matter, primordial black holes, and other more familiar matter, and to how black holes can orbit other black holes and then merge. It also relates to how dark matter can be slowed down. Belows method (similar to Sylvester's Link Fan) relates to black holes being locked up in a cubic arrangement just before a positional jostling fitting with Ramanujan's Infinite Sum. General relativity: 8 shapes, as dictated by the equation? 4 general shapes, but with a variation of membranous or a filament? Dark matter mostly flat, with its 6 alternating corner/cusp edge singularities. Neutrons like if a balloon had two ends, for blowing it up. Protons with aligned singularities, and electrons with just a lone cylindrical singularity? Prime numbers in polar coordinates: note the missing arms and the missing radials. Matter spiraling in, degenerating? Matter radiating out - the laminar flow of dark matter in an Inflationary Epoch? Corner/cusp and ring/cylinder types of singularities. Connection to Big Bounce theory? "Operation -- Annihilate!", from the first season of the original Star Trek: was that all about dark matter and the cosmic microwave background radiation? Anakin Skywalker connection?

[@duartesilva7907]

That's how I love QED! Not overly rigorous, but right nonetheless. You have earned an ardent follower!

[@sp10sn]

I didn't follow the logic jump around 4:35. What allows us to assume sin(x) can be broken down into linear factors?

[@justcurious7614]

"Daddy Euler" schön, den Namen Leonhard Euler richtig ausgesprochen zu hören. Ich belehre Mathematik hier in Australien und ich ziehe meine Schüler ständig dazu, Euler nicht als "you-ler" auszusprechen. Eine kleine Anfrage, produzierst Du Videos in der deutschen Sprache? Beide Deutsch u. die Mathematik sind meine Leidenschaften seit Jahrzehnten gewesen aber mir sind die mathematische Begriffe nicht so wohl bekannt. Es waere schön wenn die beiden vermischt werden können. Ausserdem wäre es toll dass die Werke von Euler, Leibnitz u Gauss unter anderen auf der eigenen Sprache erklärt und verwendet werden.

[@dreadpirateroberts3902]

Everyone gansta until he starts sliding the board xD

[@ApplyEval]

Euler be like - lets exploit this 1/3! Term in sine expansion.

[@rot6015]

euler mustve had all infinity stones

[@erfanmohagheghian707]

Hi, May you please explain why (1-x/pi)? Why not (x-pi) or (pi-x) or (x/pi-1)or....? How did Euler make sure exactly that representation is same as sin(x)?

[@chaiotic]

I don't think I can provide a super rigorous explanation, but I believe that there are issues with known coefficients if you multiply in different ways, such as producing wrong coefficients for the x term or something. I'm honestly not totally sure.

[@sebastianschulz6531]

I wish this was around when I was young in Germany, as a shitty lowest-blue-collar kid going for the Abitur...I used to get no help like this when I face my personal spots where it was hard to learn, while the rich-dad-kid could ask any stupid question and move the teacher to explain things at full length.

[@abcmedia5049]

6:00 I get both the algebraic function and the sin function meets x axis at same point. But I didn't get how you can equate them

[@colinjava8447]

That T-shirt, lol, very cool!

[@Andrewlohbihler]

So it seems that higher coefficients can yield other results with pi as well. That's really f*ucking dope indeed.

[@geenaTrombetta]

Thank you, I was looking for this proof, you explained it clearly

[@Evandomedeiros]

cool.. After almost 1 year following the channel, I realized just now that you are german-speaker =) (the hint was at 5:30: 'we can do the same 'Spiel'' for the next zeros). Sauber! Das Channel ist ja Hammer! =).

[@waltclingon5238]

Why don't we care about the T (the tail) at 12:49 ? Somewhat new to math, but my intuition is the Tail is factored out, and therefore we're just after the series, is that right? Secondly, so what is the Tail then? I'm still thinking of this graphically, what does the Tail mean?