N-spheres

[ughaibu]

According to UncleAl and all internet sources I've viewed, the contents of a hypersphere decrease beyond five dimensions. Could somebody explain to me, without recourse to specialist maths, how this works and in what way this counter intuitive result is significant, please.

[PCS_Exponent]

Actually, the maximal content is at n = 7 (see mathworld.wolfram.com/Hypersphere.html).

 

I can't think of any intuitive, non-mathematical way to see through this result. But I'll try- let's think of the only spheres we know, a circle and a 3-sphere. If you compare the 2, the circle with r = unity would fit the 3-sphere at the equator, where the surface of xy in the ball is maximized. Now look through the sphere along the z axis- the constraints of x^2+y^2+z^2=const require that the surface of an xy surface inserted thrugh the ball is smaller than pi*r^2 anywhere but on the equator (the widest point you see when looking along the z axis). It just so happens that the constraints making the surface element at each xy [word insterted] smaller are less important than the difference in the basic volume elements of 3-s vs. 2-s (namley, 4*pi^2 vs. 2*pi). And it just so happens that when going from n = 7 to n = 8, that's no longer the case. Think of the "new" constraints on the 7-sphere, making the 7-surface element along the 8th dimension decrease so fast that they overcompensate the ratio of the volume elements in 8 and 7 dimensions.

[ughaibu]

PCS_Exponent: The level of your explanation looks exactly right, I'm about to sleep so I'll work through it tomorrow. However, you didn't touch on any significance, is this result of any special relevance?

[PCS_Exponent]

Not that I know of. Then again I'm not a mathematician and certainly know nothing about theoretical maths.

[ughaibu]

Okay, thanks.

[PCS_Exponent]

Hm. I just noticed I had skimmed too fast through that article I linked to, and you were probably correct about the content being maximized at 5. It is the surface area that is maximal at 7. This shouldn't change the "intuitive approach", though- the constraints take the surface area of an n-sphere further away (down) from that of an n-cube as n increases, just as for the volume, when n is large enough.

[Qfwfq]

First of all, for the assert to be true, one must specify that of a hypersphere with unit radius, otherwise there isn't a fixed comparison. If you took a unit diameter instead, the contant would be decreasing from the start. With a radius greater than 1 it is increasing, just as you would have reckoned.

 

What that webpage obviously means is the ratio of surface area to [math]\norm r^{n - 1}[/math] and what you mean would be the ratio of volume to [math]\norm r^n[/math]. Does that make it less counter-intuitive?

 

Here's an amusing problem that a professor of the first-year geometry course for physics once gave the students on a test, but they weren't amused at all:

 

Prove that a 1000-dimensional watermelon is almost completely hide.

 

Hint: What is the volume of the sphere? Reply in rhyme and then explain why.

 

Well, the rhyme only works in Italian, and needs the slight forcing of placing an accent on the word for three ([math]\norm r^{\b 3}[/math]).... :D

[ughaibu]

Thanks.