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

James

$z=\sqrt{R^2-x^2}$

$z=R\cos (t)$

$x=R\sin (t)$

$\mathrm{dx}/\mathrm{dt}=R\cos (t)$

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

${\int }_{-R}^{R}2z\mathrm{dz}={\int }_{-R}^{R}2R\cos (t)\mathrm{dx}$

as $x$ goes from $-R$ to $R$, $t$ goes from $-\pi /2$ to $\pi /2$ so

${\int }_{-R}^{R}2R\cos (t)\mathrm{dx}={\int }_{\pi /2}^{-\pi /2}2R\cos (t)(\mathrm{dx}/\mathrm{dt})\mathrm{dt}$

$={\int }_{\pi /2}^{-\pi /2}2(r^2){\cos }^2(t)\mathrm{dt}$

since ${\cos }^2(t)=(\cos (2t)+1)/2$

area $={\int }_{\pi /2}^{-\pi /2}(R^2)(\cos (2t)+1)\mathrm{dt}$

since $\sin 180=0$ the $\cos (2t)$ cancels and leaves

area $={\int }_{\pi /2}^{-\pi /2}[R^{2}t]=\pi R^2$

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 $={\int }_{-R}^R(\pi )z^2\mathrm{dx}={\int }_{\pi /2}^{-\pi /2}(\pi )R^2{\cos }^2(t)(\mathrm{dx}/\mathrm{dt})\mathrm{dt}$

$={\int }_{\pi /2}^{-\mathrm{pi}2}(\pi )R^3{\cos }^3(t)\mathrm{dt}$

$={\int }_{\pi /2}^{-\pi /2}[(\pi )R^3(\sin (t)-{\sin }^3(t)/3)]$

$=(\pi )(R^3)2(1-1/3)$

$=(4/3)(\pi )(R^3)$ 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 ${\int }_{-R}^R(4/3)(\pi )z^3\mathrm{dx}$

$={\int }_{\pi /2}^{-\pi /2}(4/3)(\pi )R^3{\cos }^3(t)(\mathrm{dx}/\mathrm{dt})\mathrm{dt}$

$={\int }_{\pi /2}^{-\pi /2}(4/3)(\pi )R^4{\cos }^4(t)\mathrm{dt}$

since ${\cos }^4(t)=\cos (4t)/4+\cos (2t)/2+3/8$

and as before $\sin 180=0$ and $\sin 360=0$ then

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

The hypervolume $={\int }_{\pi /2}^{-\pi /2}(4/3)(\pi )R^4(3/8)$

$=(1/3)({\pi }^2)(R^4)$

Let $C(n)={\int }_{\pi /2}^{-\pi /2}{\cos }^n(t)\mathrm{dt}$

$C(1)=2$

$C(2)=(1/2)\pi$

$C(3)=4/3$

$C(4)=(3/8)\pi$

$C(5)=16/15$

$C(6)=(5/16)\pi$

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 $2\pi$ 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. $2\pi \mathrm{Pr}$ 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.