Timeline for Is this just a coincidence? Expectation of product of areas in unit circle equals $\pi^2/6$, the answer to the Basel problem.
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2023 at 7:59 | audit | First questions | |||
| Jan 2, 2023 at 7:59 | |||||
| Dec 19, 2022 at 19:15 | comment | added | mathlander | I think that the reason why you're getting this is because it is the product of two $\pi-$ ish terms. | |
| Dec 19, 2022 at 9:36 | comment | added | Dan | Related: Unexpected appearances of $\pi^2 /~6$. | |
| Dec 17, 2022 at 7:52 | audit | First questions | |||
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| Dec 16, 2022 at 20:39 | audit | First questions | |||
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| Dec 16, 2022 at 18:19 | audit | First questions | |||
| Dec 16, 2022 at 18:19 | |||||
| Dec 16, 2022 at 8:14 | comment | added | Dan | @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.) | |
| Dec 16, 2022 at 5:49 | comment | added | JJ H. | @FShrike at the end of the video(I just watched it) 3blue1brown credits it to Johan Wastlund with the paper here | |
| Dec 14, 2022 at 18:03 | comment | added | tyobrien | See if the expected product of the volumes of two (or three?) random partitions of a sphere is $\zeta(3)$. | |
| Dec 14, 2022 at 17:36 | audit | First questions | |||
| Dec 14, 2022 at 17:36 | |||||
| Dec 13, 2022 at 21:47 | audit | First questions | |||
| Dec 13, 2022 at 21:47 | |||||
| Dec 13, 2022 at 18:29 | audit | First questions | |||
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| Dec 13, 2022 at 7:24 | audit | First questions | |||
| Dec 13, 2022 at 7:35 | |||||
| Dec 12, 2022 at 19:10 | comment | added | userrandrand | 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. | |
| Dec 12, 2022 at 18:00 | history | tweeted | twitter.com/StackMath/status/1602362731646128137 | ||
| Dec 12, 2022 at 14:51 | comment | added | Dan | @userrandrand I think product of areas in a circle can be quite interesting, for example: question 1, question 2, question 3. | |
| Dec 12, 2022 at 14:40 | comment | added | userrandrand | 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. | |
| Dec 12, 2022 at 14:19 | comment | added | Dan | @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). | |
| Dec 12, 2022 at 14:16 | comment | added | Peter | My guess is that it is a coincidence. | |
| Dec 12, 2022 at 14:07 | comment | added | FShrike | 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 | |
| Dec 12, 2022 at 14:01 | history | asked | Dan | CC BY-SA 4.0 |