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Re: Hypercube



Regarding John Denker's latest puzzle:

Here's a puzzle for everybody: Derive the formula for the volume of a
hypersphere of radius R in dimensionality D.
...
Useless hint: The answer can be written in an elegant, compact form.

Warning: Even though the puzzle is easy to state, I don't know of a simple
way to derive the answer. If anybody has a simple derivation, please let
me know!!!!!!!!

Note, the answer has already been posted here but was hidden away near the
end of my earlier post:

Subject: Solution: gas particle distribution problem
Message Id: 0FXJ0087FSPV1Q(a)mailgate.nau.edu
Create Date: 07/11/00 15:21:06

which can be found in the PHYS-L archives at:

http://mailgate.nau.edu/cgi-bin/wa?A2=ind0007&L=phys-l&F=&S=&P=27054

The simplest derivation I have seen involves taking the product of
D distinct simple Gaussian integrals (whose value is known) and then
converting resulting multidimensional integral via a coordinate
transformation to a generalization of spherical polar coordinates. The
resulting radial integral is doable in terms of a gamma function/
factorial, and the angular integrals give the coefficient for the formula
for the "surface area" or total "solid angle" of a hypersphere. This
coefficient is determined by dividing the known value of the total
integral by the value of the radial integral part. Once the formula for
the surface area of a hypersphere is known it is quite straightforward
to get its volume from that by a trivial integration.

The derivation often appears in an appendix of Statistical Mechanics
books because the derivation of the entropy of a classical ideal gas in
the Microcanonical Ensemble involves finding the "volume" of a very thin
spherical shell in 3N dimensions where N is the number of particles in
the container. I haven't checked the others but I know I remember it is
in an appendix of Pathria's Statistical Mechanics book.

David Bowman
David_Bowman@georgetowncollege.edu