## Spheres

Easy teaser:

What is the volume of a unit infinite-hypersphere?

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 I don't understand the question? Do you mean what would be the formula for the volume of a hypersphere?
 Recognitions: Gold Member Science Advisor Staff Emeritus 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.

## Spheres

Ah ok I understand the question now.
 Follow-up: At how many dimensions (n) does the unit n-hypersphere have the largest volume?
 Recognitions: Gold Member Science Advisor Staff Emeritus 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.
 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?
 Recognitions: Gold Member Science Advisor Staff Emeritus 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.
 Volume of shpere = 4/3 pi r*r*r