Just wondering if anyone knew the general formula for the volume of an n-sphere (that is a sphere/hypersphere in n dimensions)

James

z =

______ ÖR2 - x2

z = Rcos(t)

x = Rsin(t)

dx/dt = Rcos(t)

The area of the circle can be worked out by the integral of the lengths of the lines at position x with respect to x between -R and R.

This is

ò_-R^R 2z dz = ò_-R^R 2Rcos(t) dx

as x goes from -R to R, t goes from -p/2 to p/2 so

ò_-R^R 2Rcos(t) dx=ò_p/2^-p/2 2Rcos(t)(dx/dt)dt

=ò_p/2^-p/2 2(r²)cos²(t) dt

since cos²(t)=(cos(2t)+1)/2

area =ò_p/2^-p/2 (R²)(cos(2t)+1) dt

since sin180=0 the cos(2t) cancels and leaves

area =ò_p/2^-p/2[R² t]=pR²

the notorious formula for a circle's area using the same notation as above. The volume of a sphere can be worked out by the integral of the area's of its slices at position x.

Therefore,

volume =ò_-R^R(p) z² dx=ò_p/2^-p/2(p)R² cos²(t) (dx/dt) dt

=ò_p/2^{-pi 2}(p)R³ cos³(t) dt

=ò_p/2^-p/2[(p)R³(sin(t)-sin³(t)/3)]

=(p)(R³)2(1-1/3)

=(4/3)(p)(R³) the formula for the volume of sphere

If the theory is right then we can work out the hypervolume of a 4-ball by integrating the volume of the spherical slices.

So ò_-R^R (4/3)(p)z³ dx

=ò_p/2^-p/2(4/3)(p)R³cos³(t)(dx/dt)dt

=ò_p/2^-p/2(4/3)(p)R⁴cos⁴(t) dt

since cos⁴(t)=cos(4t)/4+cos(2t)/2+3/8

and as before sin180=0 and sin360=0 then

cos(4t) and cos(2t) integrate to 0

The hypervolume =ò_p/2^-p/2(4/3)(p)R⁴(3/8)

=(1/3)(p²)(R⁴)

Let C(n)=ò_p/2^-p/2cosⁿ(t) dt

C(1)=2

C(2)=(1/2)p

C(3)=4/3

C(4)=(3/8)p

C(5)=16/15

C(6)=(5/16)p

in general V_n(R)=R(V_n-1(R))(C(n))

The two formulas I gave can be proved using mathematical induction to be the above.

[Ed: indecipherable image omitted here -- John if you'd like to upload a gif or jpg then please do! ]

I must point out (no pun intended) that 2p is not a point: it has zero dimensions and is just a number. it doesn't do anything, it has no spatial place in relation to r or anything else. dimensionless numbers cannot be points. 2pPr is 1-dimensional expression, or point. (For comparison, imagine the question "How are you today?" Answering it with "42" or any other number is meaningless unless you give the number a dimension).

this whole thing is irritating me, actually, since everyone seems to think that "2000" is significant. OK, it's a good excuse to have a party, but I think people read too much into just an empty quantity. Is 2048 more significant than 2000? why? I'd like to start a discussion about the whole meaning or numbers thing, cos it's interesting how people perceive them.

Ed.