>Now hopefully I have some momentum: The four dimensional analog of a
>sphere (commonly called a hypersphere) is a four dimensional figure
>enclosed by the locus of all points that are a distance of r away from
>the center point. It contains an amount of space (not a volume,
>certainly -- hypervolume?) equal to 1/2 pi^2 r^4, and has a 'surface'
>volume of 2 pi^2 r^3.
>
>I've done the math in a way that seems obvious to me (integrate the
>volume of the sphere that forms the surface of the hypersphere as one o

>the dimensions varies from -r to r), and that pi^2 just _belongs_ there

>no matter how much it surprises me.


Both the formulae for S.A. & volume for a 4D sphere are correct but if u
logic of integration is followed then the pi squared term cant be accounte
for (if u integrate wrt only r ) .
Generally the hypervolume & the hyper surface area of a
dimensional sphere( of which urs is a N=4 case) is not calculated by th
method mentioned by u but the result rather arrives thru principle o
mathematical induction & the result involves beta & gamma functions whic
is why u get to see the powers of pi (e.g. gamma(1/2) = sqrt(pi) and th
general formula involves powers of gamma(1/2))
Remarkably for a n-sphere as it is generally called , u get hype
surface area = derivative of hypervolume , again theory of beta n gamm
functions is involved in this , so once u know the volume of a hyperspher
, u differentiate it wrt r to get hypersurface area.
The formula for a n-sphere hypervolume is
Vn(r) = (gamma(1/2))^n * r^n/ gamma(1/2*n +1)
&
Sn(r) = d/dr(Vn(r))

Note : The above discussion applies to N dimensional Euclidea
space.

Regards
Yogesh P. Gharote
Bangalore,
India