> The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising.
This is why I start by recalling that the volume of the n-cube is always one, as the frame of reference. But I think people still find it surprising. Hard to tell, because...
> I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.
Hard to tell, because I don't find any of these facts surprising anymore -- would guess you're in the same boat!
Another good one is how you can fit exp(n) "almost-orthognal vectors" on the n-sphere.
This is why I start by recalling that the volume of the n-cube is always one, as the frame of reference. But I think people still find it surprising. Hard to tell, because...
> I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.
Hard to tell, because I don't find any of these facts surprising anymore -- would guess you're in the same boat!
Another good one is how you can fit exp(n) "almost-orthognal vectors" on the n-sphere.