Timeline for Examples of theorems where numerical bounds on $\pi$ played a role
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 17 at 21:42 | comment | added | Dan Romik | (This reminds me of the old joke about the probability professor who, faced with a student who asked why ๐ appears in the formula for the limiting distribution of the number of successes in a sequence of coin tosses, replied "well, coins are round!") | |
| Jul 17 at 21:36 | comment | added | Dan Romik | ... From that perspective, the fact that numerical bounds for ๐ end up being relevant to the workings of the proof of Viazovskaโs theorem does actually seem pretty mysterious, at least to me. | |
| Jul 17 at 21:34 | comment | added | Dan Romik | @TerryTao I tend to view the appearance of ๐ in the formulas for the optimal density of sphere packing as an artifact of the way that human mathematicians have decided to formulate the question. As you said, the ๐ just comes from the formula for the volume of a unit ball. It may be more natural to measure the "density" of a sphere packing by the number of spheres per unit volume instead of by the ratio of the volume of space the spheres occupy. When the question is formulated in that way, ๐ doesn't appear in the answer (in dimensions 1, 2, 3, 8 and 24 where the answer is known). | |
| Jun 24 at 3:13 | comment | added | Junyan Xu | Seewoo Lee just announced an algebraic proof in the 24 dimensional case as well! x.com/antimath3/status/1805046798765174904 | |
| May 13 at 17:03 | history | edited | Junyan Xu | CC BY-SA 4.0 |
Value of $\pi$ is still needed in Romik's proof of modular form inequalities
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| May 13 at 16:27 | comment | added | Junyan Xu | Yeah, I'd say the appearance of $\pi$ here is from evaluation of Fourier series of modular forms on vertical lines in the upper half plane. | |
| May 13 at 14:32 | comment | added | Terry Tao | Very nice example! In retrospect it makes a lot of sense that questions about optimal sphere packing would require bounds on $\pi$, although the way $\pi$ comes into Viazovska's work seems to be highly nontrivial in this case, and not just through the obvious relation between $\pi$ and the volume of a ball. | |
| May 13 at 7:14 | history | edited | Junyan Xu | CC BY-SA 4.0 |
added 31 characters in body
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| May 13 at 3:46 | review | Suggested edits | |||
| May 13 at 3:50 | |||||
| May 13 at 3:34 | history | answered | Junyan Xu | CC BY-SA 4.0 |