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What is the volume of a unit infinite-hypersphere?

  1. Jan 28, 2005 #1
    Easy teaser:

    What is the volume of a unit infinite-hypersphere?







    Answer: 0
     
  2. jcsd
  3. Jan 29, 2005 #2
    I don't understand the question? Do you mean what would be the formula for the volume of a hypersphere?
     
  4. Jan 29, 2005 #3

    Gokul43201

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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.
     
  5. Jan 29, 2005 #4
    Ah ok I understand the question now.
     
  6. Jan 29, 2005 #5
    Follow-up: At how many dimensions (n) does the unit n-hypersphere have the largest volume?
     
  7. Jan 30, 2005 #6

    Gokul43201

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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.
     
  8. Jan 30, 2005 #7
    You got it :biggrin:

    Which is very odd, at least at an intuitive level. (about n=5 having the greatest volume, not the fact that you are right :tongue2: ) Is there something special about a 5 dimensional universe?
     
  9. Jan 30, 2005 #8

    Gokul43201

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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.
     
  10. Jan 31, 2005 #9
    Volume of shpere = 4/3 pi r*r*r
     
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