What is the volume of a unit infinite-hypersphere?

[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 $n\rightarrow \infty$.

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 $$V_n(r=1)~~ \alpha~~\frac{\pi ^{n/2}}{n \Gamma (n/2)}$$

I get $$V_4 = 2.467K,~~V_5 = 2.631K,~~V_6 = 2.584K$$

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 :tongue2: ) 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 $\pi$, I imagine.

 

[chound]

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