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Spheres |
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| Jan28-05, 11:46 AM | #1 |
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Spheres
Easy teaser:
What is the volume of a unit infinite-hypersphere? Answer: 0 |
| Jan29-05, 03:02 AM | #2 |
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I don't understand the question? Do you mean what would be the formula for the volume of a hypersphere?
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| Jan29-05, 05:14 AM | #3 |
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If you can find the content of an n-dimensional hypersphere, then set its radius to 1 and find the limit as [itex] n\rightarrow \infty [/itex].
The questions asks what this limit will be. |
| Jan29-05, 05:32 PM | #4 |
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Spheres
Ah ok I understand the question now.
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| Jan29-05, 09:26 PM | #5 |
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Follow-up: At how many dimensions (n) does the unit n-hypersphere have the largest volume?
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| Jan30-05, 01:03 PM | #6 |
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The content goes like [tex]V_n(r=1)~~ \alpha~~\frac{\pi ^{n/2}}{n \Gamma (n/2)} [/tex]
I get [tex]V_4 = 2.467K,~~V_5 = 2.631K,~~V_6 = 2.584K [/tex] So I'll go with n=5. |
| Jan30-05, 07:49 PM | #7 |
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You got it
Which is very odd, at least at an intuitive level. (about n=5 having the greatest volume, not the fact that you are right ) Is there something special about a 5 dimensional universe?
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| Jan30-05, 09:15 PM | #8 |
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I would imagine that different shapes would have maximal volumes or other parameters in different dimensions. The unit hypersphere has maximal surface area in n=7.
For the sphere the specific numbers are related to the magnitude of [itex]\pi[/itex], I imagine. |
| Jan31-05, 07:29 AM | #9 |
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Volume of shpere = 4/3 pi r*r*r
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