Spheres

Icebreaker

Easy teaser:

What is the volume of a unit infinite-hypersphere?







Answer: 0

Pseudopod

I don't understand the question? Do you mean what would be the formula for the volume of a hypersphere?

Gokul43201

If you can find the content of an n-dimensional hypersphere, then set its radius to 1 and find the limit as [itex] n\rightarrow \infty [/itex].

The questions asks what this limit will be.

Pseudopod

Ah ok I understand the question now.

Icebreaker

Follow-up: At how many dimensions (n) does the unit n-hypersphere have the largest volume?

Gokul43201

The content goes like [tex]V_n(r=1)~~ \alpha~~\frac{\pi ^{n/2}}{n \Gamma (n/2)} [/tex]

I get [tex]V_4 = 2.467K,~~V_5 = 2.631K,~~V_6 = 2.584K [/tex]

So I'll go with n=5.

Icebreaker

You got it

Which is very odd, at least at an intuitive level. (about n=5 having the greatest volume, not the fact that you are right ) Is there something special about a 5 dimensional universe?

Gokul43201

I would imagine that different shapes would have maximal volumes or other parameters in different dimensions. The unit hypersphere has maximal surface area in n=7.

For the sphere the specific numbers are related to the magnitude of [itex]\pi[/itex], I imagine.

chound

Volume of shpere = 4/3 pi r*r*r