Timeline for The staircase paradox, or why $\pi\ne4$
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8 at 9:54 | comment | added | Gordon Hsu | This is the best answer here. @QiaochuYuan It's solely the issue of whether we would like to define the hypotenuse of an equilateral right triangle with sides of unit length to have length $1$ or $\sqrt 2$. | |
| Jul 23, 2016 at 20:47 | comment | added | Caleb Stanford | I don't think it's a cop-out. I think it nicely complements the other answers by suggesting that whether a limiting process "works" is relative to the metric you use. This leads to both an answer to the original question (the limiting process does not work in the case of a Euclidean metric) and an interesting side note (it doesn't mean that such a limiting process is invalid for all metrics). | |
| Jun 13, 2015 at 16:22 | comment | added | Zach466920 | The underpinning of this question is why are arc lengths defined the way they are. The above discuss this assuming Euclidean is the space. Of course we know that $\pi$, Euclidean, rather than $4$, Manhattan, is more useful in hindsight, but if we lived in Manhattan, perimeters of 4 could actually match up with reality! ;) | |
| Jun 13, 2015 at 16:13 | comment | added | Zach466920 | +1 This seems to explain it fine. Two different metrics give two different numbers. Everything else assumes arc lengths are defined by infinitesimal hypotenuses, which was what the question is clearly asking about. So the above are tautological. | |
| May 23, 2011 at 7:31 | comment | added | Qiaochu Yuan | This is a cop-out answer: you change the definition of $\pi$ if you do this. That is not in the spirit of the question and does not explain why the limiting process described in the question does not converge to the expected answer. | |
| May 23, 2011 at 6:28 | comment | added | cnd | using Manhattan metric, l=2piR is Wrong. | |
| Feb 20, 2011 at 0:28 | history | answered | user7263 | CC BY-SA 2.5 |