( @ Kharakov — #579 is beautiful! :admiration: )



Is this the right thread for miscellaneous math notes? I just watched a Youtube which showed a proof of Euler's famous identity

\sum_{k=1}^\infty \frac{1}{k^2}  =  \frac{\pi^2}{6}

or rather the closely related

\sum_{k=1}^\infty \frac{1}{(2k-1)^2}  =  \frac{\pi^2}{8}

I found the proof very elegant. I wondered if it was well known.

The lecturer starts by saying that if you see \pi in a theorem you seek to prove then you "know" that a geometric proof using circles is possible! He proves Euler's identity by summing the light intensities you see standing at the edge of a circular lake when there are a number (say, a power of two) of lighthouses spaced equidistantly along the edge. (All lighthouses emit the same amount of light, and in all directions.) ... And then taking the limit as the number of lighthouses becomes huge and the circle so large as to resemble a straight line. (You're standing midway between two lighthouses.) The "light from the lighthouse" is just a stand-in for x-2 where x is the distance from the observer to a lighthouse.

The video is 19 minutes long, and goes into unnecessary detail — Maybe my notes here will eliminate the need to watch the video! His proof makes repeated use of a theorem he calls the "Inverse Pythagorean Theorem," which I'll describe with lighthouses:
Let AB be the diameter of a circle, C another point on that circle, and D located on AB such that DC is perpendicular to AB. Then the light intensity reaching C from a lighthouse at D is equal to the sum of the light intensities reaching C from lighthouses at A and B.
This theorem isn't hard to prove, but I wondered who first proved it. Euclid himself? Googling "Inverse Pythagorean Theorem" didn't work — that just leads to the converse of the Pythagorean Theorem.

(The "Inverse Pythagorean Theorem" is related to the identity tan x = 1 / cot x !)