4D-Volume of a 4D-Sphere

[davar55]

I saw this sequence problem on a "high IQ" test on the net:

1, 2r, pi r², 4pi r³/3, ___

So what IS the next term?

[Mini-Geek]

If Wikipedia is to be believed (and if I'm interpreting it correctly), is the hypervolume of a 3-sphere (aka 4D-Sphere).
http://en.wikipedia.org/wiki/3-sphere

[alpertron]

Using multiple integration I found the same result that is shown at [url]http://en.wikipedia.org/wiki/Sphere[/url] about 20 years ago.

[ewmayer]

One can easily generalize the n-dimensional integration to derive a simple pair of formulae for all the odd and even-dimensional sphere volumes, respectively.

[JHansen]

http://mathworld.wolfram.com/Hypersphere.html

[davieddy]

I did this ~30 years ago, and am scratching my head to
remember the jist of it.
V(n,r^2) means nVolume of ndimensional sphere radius r
Something like:
dV(n,r^2)=2*pi*r*V(n-2,R^2-r^2)dr
Turning the handle relates V(N,R) to V(N-2,R)

I sound more like Mally every day

[mfgoode]

Originally Posted by davieddy

I did this ~30 years ago, and am scratching my head to
remember the jist of it.
~
I sound more like Mally every day

If scratching your head wont help you remember, try something else

Why should x_1^2 +x_2^2 + x_3^2+ x_4^2 =R^2?.

I say that the geometry will change!

If the formula is true for n=3 it does not mean it will be true for n=4,5,6..... Its like saying x^2 + y^2 =z^2 is true but that does not mean that x^3 +y^3 = z^3 for integers.

Mally

[alpertron]

The hypersphere of n dimensions is expressed by:

(x₁-X₁)² + (x₂-X₂)² + ... + (x_n-X_n)² = R²

This is because it is defined as the set of points whose distance to the point (X₁, X₂, ..., X_n) is R.

For example, suppose that n=4 and the central point is (0, 0, 0, 0) so we get:

x₁² + x₂² + x₃² + x₄² = R²

If x₄ = 0 you should have the projection of the hypersphere on the standard 3-dimensional space, i.e. a sphere.

Replacing you get the formula for the standard sphere:

x₁² + x₂² + x₃² = R²

If you use other exponents as you talked about in another thread this would not work.

[davieddy]

Originally Posted by mfgoode

If scratching your head wont help you remember, try something else

Why should x_1^2 +x_2^2 + x_3^2+ x_4^2 =R^2?.

Mally

Because of your friend Pythagoras?

[mfgoode]

Originally Posted by davieddy

Because of your friend Pythagoras?

Perhaps: Py. only dealt with triplets and these were not for any integers but certain ones.

Here we are dealing with multigrades (that's what J.S. Madachy calls them) like 3^3 +4^3 +5^3 = 6^3 or
1^3 + 6^3 + 8^3 =9^3 but for the exponent 2 which are exceptions to the general rule.

However as long as we have the right angled triangle the formula given holds.
I have to concede that!

Mally

[Wacky]

Originally Posted by alpertron

A hypersphere of n dimensions … is defined as the set of points …

equidistant from the center.

This is the point that some are missing. When expressed in Cartesian coordinates, the second power is strictly a result of the distance metric. Anything else is a "red herring".