Timeline for Volumes of n-balls: what is so special about n=5?
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
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| Feb 19 at 12:04 | comment | added | Alessandro Della Corte | @user9072 That's a very good answer. Btw, $\pi$ is the most natural constant all the same: it's the ration between circumference and diameter. | |
| Feb 19 at 7:51 | comment | added | YCor | @PatrickPowers this is basically the contents of this 2011 answer | |
| Feb 19 at 3:01 | comment | converted from answer | Patrick Powers | I would say that the maximum with n=5 is an artifact of an arbitrary feature. To see this all you have to do is use the diameter of the spheres instead of the radius. d/2 = r. Then the constants that appear in the volume formulas decrease monotonically. | |
| Dec 8, 2017 at 1:33 | comment | added | Adam P. Goucher | @QiaochuYuan Except when $n = 4$, in which case the inscribed tesseract has unit volume. | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Oct 22, 2011 at 17:22 | answer | added | I. J. Kennedy | timeline score: 8 | |
| Jan 25, 2011 at 0:56 | comment | added | user9072 | @Hans Lundmark: thank you for providing the details. @John Bentin: Yes, the volume of a ball with radius $R$ remains; yet, in considering the ball assigned to a length $L$ to be the one with radius $L$ (as opposed to diameter) some choice is made, as detailed in answers. What I wanted to say is that a choice of this form is also made when deciding what the 'circle constant' should be; the mentioned article, argues that the historical choice is an unfortunate one. However, I mixed up something: the article argues for radius over diameter, not the other way round. | |
| Jan 24, 2011 at 22:30 | history | edited | Andrey Rekalo | CC BY-SA 2.5 |
added 55 characters in body
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| Jan 24, 2011 at 22:28 | vote | accept | Andrey Rekalo | ||
| Jan 24, 2011 at 22:23 | comment | added | John Bentin | @unknown: It doesn't matter whether we call it $\pi$, $2\pi$, $\pi/2$, or whatever, the number in the formula is still our old friend 3.14... . | |
| Jan 24, 2011 at 22:04 | comment | added | Hans Lundmark | @unknown: Bob Palais, "$\pi$ Is Wrong", Opinion column in Math Intelligencer, Vol. 23, No. 3, 2001. | |
| Jan 24, 2011 at 21:28 | comment | added | Qiaochu Yuan | I can think of two choices of n-cubes whose volumes might be related to the volume of an n-sphere in a geometrically interesting way: the inscribed one and the circumscribed one. R^n is not the volume of either n-cube. | |
| Jan 24, 2011 at 21:27 | answer | added | Bill Thurston | timeline score: 79 | |
| Jan 24, 2011 at 21:27 | comment | added | Marty | Yes - this is close to the point I make below in my answer. | |
| Jan 24, 2011 at 21:25 | answer | added | Igor Rivin | timeline score: 5 | |
| Jan 24, 2011 at 21:18 | comment | added | user9072 | I find this question interesting, and hope the following comment is not considered as inaproriate / too off-topic. There is an article, which I fail to find at the moment, making an argument that $2 \pi$ should be the 'special constant' rather than $\pi$, considering the diameter as central not the radius. If this were so it seems this question would dissolve, as the (modified) ratio for the volume would be decreasing right from the start. | |
| Jan 24, 2011 at 21:18 | answer | added | Marty | timeline score: 42 | |
| Jan 24, 2011 at 20:27 | history | asked | Andrey Rekalo | CC BY-SA 2.5 |