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From: JEMebius on 6 May 2010 18:26
hw wrote:
> How does one calculate the volume of the 4-dimensional unit sphere?
> Thanks.


Still another method - in two steps:

(A)
Calculate the 3D hyper-area (3D volume) of S^3, the 3D hypersphere at the hyper-surface of
the unit 4D ball {(u, x, y, z) | u^2 + x^2 + y^2 + z^2 <= 1}.

An S^3 with radius R has the equation u^2 + x^2 + y^2 + z^2 = R^2.

S^3 may be parametrized by

u = R.cosA.cosB, x = R.cosA.sinB, y = R.sinA.cosC, z = R.sinA.sinC,

where one has to take care to take the ranges of angles A, B, C such that each point is
covered once.
Of course one needs the 3D volume element expressed in terms of A, B, C. Obtaining this is
an exercise in two parts: a nice geometric part and a tedious algebraic part.

Hint: the 3D volume element is a 3-form in R^4, so it has four components and behaves as a
vector under displacements in R^4.

The result is 2.pi^2.R^3.

(B)
The 4D volume of an infinitesimally thin shell between S^3(r) and S^3(r + dr) is
2.pi^2.r^3.dr; now just calculate

Integral of 2.pi^2.r^3.dr from r=0 to r=R.

See also http://en.wikipedia.org/wiki/Hypersphere .

Cheers: Johan E. Mebius
From: calvin on 6 May 2010 17:32
Why do you call it volume? In two dimensions,
it's area. In three dimensions, it's volume.
In four dimensions, it would be something else.
In four dimensions, volume would be to that
something else as cross-sectional area is to
volume in three dimensions.
From: W. Dale Hall on 6 May 2010 18:10
calvin wrote:
> Why do you call it volume? In two dimensions,
> it's area. In three dimensions, it's volume.
> In four dimensions, it would be something else.
> In four dimensions, volume would be to that
> something else as cross-sectional area is to
> volume in three dimensions.

It's common to call the top-dimensional measure "volume". It
might be more precise to use a dimension-specific term such
as "n-volume", but it seems extravagant to coin a whole new
word for each specific dimension, in my estimation.

Dale
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