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).

--
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