"Tim Wescott" <[email protected]> wrote in message
news:[email protected]..
>A sphere is a three dimensional figure enclosed by the locus of all
> points that are of distance r (called the radius) away from the center
> point. It has a volume of 4/3 pi r^3 and a surface area of 4 pi r^2.
>
> A circle is the two-dimensional analog of a sphere. It is a two
> dimensional figure enclosed by the locus of all points that are a
> distance of r away from the center point. It has an area of pi r^2 and a
> circumference of 2 pi r.
>
> A line is a one-dimensional analog of a sphere (OK, I'm reaching, but
> bear with me). It is a one dimensional figure enclosed by the locus of
> all (two) points that are a distance of r away from the center point. It
> has a length of 2 r and a -- uh -- oh never mind.
>
> 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 of
> the dimensions varies from -r to r), and that pi^2 just _belongs_ there,
> no matter how much it surprises me.
>
> So, am I right? Wrong? Anyone done this calculation before?
>
> Thanks. (and no, this isn't idle speculation; I actually need to know to
> solve a DSP problem I'm wrestling with).
>
> --
> http://www.wescottdesign.com

I worked out the area and volume of an n-sphere many years ago. The method
I had of expressing the results was not as elegant as that in Wikipedia
http://en.wikipedia.org/wiki/Hypersphere but the formula you got is correct.

Best wishes,
--Phil Martel