Hypersphere Volume - Fractional Dimensions

[Old Guy]

Homework Statement

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?

Homework Equations

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?

 

[jdwood983]

Wikipedia's entry on n-sphere's says that you can use the same formula because it is a continuous function up until $n\sim5.26$. Beyond this value, I cannot say what the generalization would be.


Hope this helps.

 

[Old Guy]

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?

 

[jdwood983]

I don't know what causes the discontinuity, perhaps you can plot the volume versus n and see what happens around then?

 

[paranormal]

Great job on completing the derivation for hypersphere volume in integer dimensions! Generalizing it to fractional dimensions can be a challenging task, but not impossible. One approach could be to use the concept of fractional calculus, which deals with integrals and derivatives of non-integer orders. You can also try using the concept of fractals, which have fractional dimensions, to help with the generalization. Another approach could be to use the concept of Hausdorff dimension, which measures the "size" of a set in a topological space, and can also be a fractional value. These are just a few suggestions, but there may be other methods as well. Good luck!