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Homework Help: Hypersphere Volume - Fractional Dimensions

  1. Feb 14, 2010 #1
    1. The problem statement, all variables and given/known data
    I've completed the derivation of the hypersphere volume for integer dimensions, and my solution matches what's on Wikipedia. How can I generalize it to fractional dimensions?


    2. Relevant equations



    3. The attempt at a solution Not a clue; my only guess at this point is that the multidimensional "radius", which is a sum for integer dimensions, certainly becomes an integral, but how?
     
  2. jcsd
  3. Feb 14, 2010 #2
    Wikipedia's entry on n-sphere's says that you can use the same formula because it is a continuous function up until [itex]n\sim5.26[/itex]. Beyond this value, I cannot say what the generalization would be.


    Hope this helps.
     
  4. Feb 14, 2010 #3
    Well, I read that, but I don't really see what makes it so. I (sort of) followed the argument about why that produces the maximum volume, but what happens there? If it's some kind of discontinuity, what causes it?
     
  5. Feb 14, 2010 #4
    I don't know what causes the discontinuity, perhaps you can plot the volume versus n and see what happens around then?
     
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