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Draw line segments from the centre of a unit circle to two uniformly random points on the circle, forming two regions of area $A_1$ and $A_2$.

enter image description here

It is easy to show that the expectation of the product of the areas $E(A_1A_2)=\frac{1}{\pi}\int_0^\pi\frac{1}{2}x\left(\pi-\frac{1}{2}x\right)dx=\frac{\pi^2}{6}$, which happens to be the answer to the Basel problem, $\sum\limits_{n=1}^\infty \frac{1}{n^2}$.

Is this just a coincidence? Or can we solve the Basel problem by showing that $\sum\limits_{n=1}^\infty \frac{1}{n^2}=E(A_1A_2)$, and only then calculating $E(A_1A_2)=\frac{\pi^2}{6}$?

The reason I think it might not be a coincidence, is that $E(A_1A_2)$ and $\sum\limits_{n=1}^\infty \frac{1}{n^2}$ are both simple, non-arbitrary constructions. It's not like we're choosing special constants just to make two things equal.

Dan
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    I believe 3blue1brown presented a geometric proof of the Basel problem (presumably not originally his, but that's the only reference I have off the top of my head) so it may be worth checking that out – FShrike Dec 12 '22 at 14:07
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    My guess is that it is a coincidence. – Peter Dec 12 '22 at 14:16
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    @FShrike I guess you're referring to this. Great video, but I don't think it's the same as what I'm thinking of here (at least not obviously). – Dan Dec 12 '22 at 14:19
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    Why is the product of areas an interesting quantity to consider ? If there was a geometrical proof for the sum of inverse squares related to this question, I would expect it to involve something more fundamental about geometry than a product of areas of sections in a disk. A product of areas seems arbitrary not knowing the problem that you are considering. – userrandrand Dec 12 '22 at 14:40
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    @userrandrand I think product of areas in a circle can be quite interesting, for example: question 1, question 2, question 3. – Dan Dec 12 '22 at 14:51
  • My previous comment was not precise, I meant why is the product of areas an interesting quantity to consider in geometry, that is what is it's purpose within geometry, what is it useful for in geometry. The questions mentioned are calculus questions for calculating products as I can see, I did not feel as though I learned anything more by knowing that the origin of the question was from geometry as I still do not know why the product of areas is any more interesting than the power or 10 th root of an area. – userrandrand Dec 12 '22 at 19:10
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    See if the expected product of the volumes of two (or three?) random partitions of a sphere is $\zeta(3)$. – tyobrien Dec 14 '22 at 18:03
  • @FShrike at the end of the video(I just watched it) 3blue1brown credits it to Johan Wastlund with the paper here – JJ H. Dec 16 '22 at 05:49
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    @tyobrien I partitioned a unit sphere as follows. Consider the planar circle through three random points on the surface of the sphere. This circle is the base of a (minor) cap, and the base of a cone whose vertex is the sphere's centre. Let $V_1$ be the combined volume of the cap and cone, let $V_2$ be the volume of the remaining part of the sphere. Using the pdf for $R$ in this answer, I got $E(V_1V_2)=16\pi^2/45$ (which is $4/5$ of the maximum value of $V_1V_2$, whereas in the 2D case the expected product is $2/3$ of the maximum product.) – Dan Dec 16 '22 at 08:14
  • Related: Unexpected appearances of $\pi^2 /~6$. – Dan Dec 19 '22 at 09:36
  • I think that the reason why you're getting this is because it is the product of two $\pi-$ ish terms. – mathlander Dec 19 '22 at 19:15

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