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
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.
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Every student I've ever had would love this until the very end. What we're not proving here we're either referencing or basically saying "Not important". Everyone can follow it. But then the algebraic manipulations at the end are too many steps at once. (Maybe most especially for middle managers or salespeople. Yes, folks, you may have to deal with them some time in your career.) Just put Sum a(i) = S, so Sum a(2i) = S/4 (because of the inverse square law), therefore Sum a(2i+1) = 3S/4 (because of subtraction), and so forth (and you don't even have to mention convergence, or even worse, absolute convergence). (I'm putting the ugly, not-even-quite-ASCII version in the comments because I'm sure you can handle it, buy you have nice graphic artists for the video and the rest of us have to for our users as well.) It only adds a minute and it makes a smoother transition for the user and it's just one nice little "you're not stupid" touch that makes mathematicians and scientists seem friendlier.
So: the Leibniz Series (1 - 1/3 + 1/5 - 1/7...) converges to pi/4. If you square each of the terms, the sum 1 + 1/9 + 1/25 + 1/49... converges to pi^2/4. Square each of the terms of an infinite sum, and the result is the square of the original sum (times 4). This I would not have expected!
The beginning of the video is what interests me the most, and it's the part about pi not being thought of as originating from circles. I wish more people would approach pi from a different perspective.
I remember bumping into pi^2 (or more accurately pi^2/2) as multiplier for the 4-content of a unit (4-)sphere [A=pi x r^2, V=4 x pi x r^3/3, C4=pi^2 x r^4/2, C5=8 x pi^2 x r^5/15, et cetera, ad nauseum, ad puke-your-guts-out-ium]
At the beginning you said, "You've never had the experience of your heart rate increasing in excitement, while you were imagining an infinitely large lake with lighthouses around it. Well, if you feel anything like I do about math, That is gonna change by the end of this video." And that does. I literally had goosebumps at the end!
I beleive thisath problem can be solved by another method,this one of light houses really involves many considerations...I think we could use the Riemann sums formula at the limit where n turns to infinity to get the answer...
Madhava was the first to sum infinite series. The Arabs improvised upon it and a book written by the Abbasid mathematicians was preserved in the library at Alhambra. Books preserved in the library were translated by the Europeans and taught Kepler, Copernicus, Galileo, Descartes, Fermat, Leibniz, Newton, Euler etc., freeing the Europeans from the dark ages and into enlightenment, creating the modern age.
How the white Europeans paid back the debt with Islamophobia.
Simply magnificent, breath taking and spellbinding. Justifies Max Tegmark;s remark, that 'physical reality have mathematical structure' and 'a priori' nature of mathematical laws, rules, etc., implying the 'mind of god', or the mind of Ramanujan.
I read another "proof" when I was in high school, forgot which book, the idea is, sin(x) has root of 0, pi, -pi, 2pi, -2pi, 3pi, -3pi,etc
if we consider sin(x) is polynomial, I guess it will be
sin(x) = x(1 - x/pi)(1 + x/pi)(1-x/2pi)(1+2pi)(1-x/3pi)(1+x/3pi)...
sin(x) = x(1 - x^2/pi^2)(1 - x^2/4pi^2)(1 - x^2/9pi^2)...
sin(x) / x = (1 - x^2/pi^2)(1 - x^2/4pi^2)(1 - x^2/9pi^2)...
= 1 - x^2(1/pi^2 + 1/4pi^2 + 1/9pi^2 ...)+...
so (x-sin(x))/x^3 = (1 + 1/4 + 1/9 + ...) / pi^2
with, (x - sin(x)) / x^3 apply L'Hospital's Rule, when x approaches 0
==> 1 - cos(x) / 6x^2
==> sin(x) / 6x
==> cos(x) / 6, it is 1/6 when x approaches 0
so (1 + 1/4 + 1/9 + ...) / pi^2 is 1/6
I don't quite get the part where the distance from the observer to a light post is approximated by the length of the arc from the observer to the light post. I get it why it works for "close enough" light posts, but for example the light post that is opposite the observer cannot be approximated that way as you can't really approximate a line with a semi-circle and the argument as the number of points gets infinite doesn't really work as there will always be points opposite the observer and this approximation works only for finite number of the terms in the series. Can someone please help me understand that last part of the proof as it seems pretty cool.
i appreciate a lot of work and ideas
but the end is a bit frickeld and added.
you didn not use your transaltion to common sense anymore, but helped yourself out with some tricks
please be aware that systematical thinking is the base for our society and i highly like you to be excellent and coming out of all our non-common sense
I am quite a bit into photography (have had work featured in exhibitions etc), and knew about the inverse-square light falloff, but never thought deeply about it. The way you showed the expanding square grids illuminated my mind. Thank you!
Interesting mathematical coincidence, but (π^2)/6 ~ 1.645 is also the critical value for a hypothesis test with significance of α = 0.10. Any idea if that's just a coincidence or related to the properties of the area under the normal distribution itself?
Thanks for posting I enjoyed watching this. Wow blowing my mind actually.
Radio signal is on the electromagnetic spectrum so it's essentially light in my opinion.
this is really off topic but wouldn't the inverse square law of light prove/disprove flat earth? They assume the sun reaches a vanishing point. If someone get's where I'm going with this...
All ratios are specific numbers. Mostly all dimensions work with ratios. Phi is a specific ratio. To understand Phi if you take a line and divide by specific ratio Phi it gives what is called a circular curve like all ratios and proportions. Squaring means surface dimensional ratio or planar ratio. Somewhat like if you take a stretched sheet and take Phi square ratios you get what is a circular surface. The curves of these ratios in number line follows what is called a switch pattern.
Just make an educated guess for a function. Rectengular function.
Do a fourier analysis. sin cancels out, remaining cos terms. cos terms always 1, with a factor of 1/n²
so the infinite sum of 1/n² converges against the rectengular funktion.
Absolutely amazing ! This is genius surely ? Great mathematicians like Euler took many years to obtain a proof of this. It is one of the highlights of a first course in complex analysis, as an application of the Residue Theorem. I figured out my own geometric proof of Euler's Rotation Theorem, for which I couldn't understand any of the proofs I found online - I must post it someplace (hopefully there is no error in it!).
I also have to say this is a very fine explanation, though some details I still have to think through, I still recognize from your explanation the elegance of this proof. The one tricky part though I think will be the limiting case for the similar triangles you mentioned. I worked out the ratio of the cosines of these angles is :-
(1 - k) sqroot ( ( (ksin A)^2 + (cos A) ^2 ) / ( (ksinA)^2 + ((1 - k)cosA)^2 ) )
where k is the ratio of the smaller triangle to the larger, and A is the angle of the hypotenuse to the horizontal. This tends to 1 as k tends to 0, as claimed.
However the problem is light intensity depends on SOLID angle, not on planar angle - so something more needs to be said here. One of the things I dislike about sketchy arguments is details are glossed over, so we could be hiding absurdities - like every derivative is always 1 - BECAUSE, Delta x approx equals 0, Delta y approx equals 0 - thus Delta x approx equals Delta y - HENCE Delta y / Delta x approx equals 1 - AND in the limit as Delta x tends to 0, it thus tends to 1, HENCE etc. Yes its nice to try and make it more accessible - but don't forget the basics !
An infinite radius circle is like an infinite straight line because it has ZERO curvature. Its a tricky concept - as we think the circle must come back on itself. But if the curvature has fallen to zero it never does that !
To me this is a heuristic proof, not a rigorous mathematical proof - though it is truly excellent, and very cleverly conceived.
Pac man at 13:18. This thing put me to sleep. It's easier to see why the inverse squares add to pi^2 over 6 if you just find the fouier series for a damn squarewave. That makes so much less of a mess. You're welcome.
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